Phase-First Gaussian Modulation for Resilient Continuous-Variable Quantum Communication Under Adversarial Disturbances

Abstract

Continuous-variable quantum communication (CVQC) operates under finite-resolution inference (finite data windows, calibration uncertainty, and estimator tolerances) and hardware control/readout limits that can be exploited by structured and adversarial disturbances. We study a feedback-inspired phase-space modulation strategy for implementation-layer resilience under DoS-like receiver-observable stress (e.g., fluctuation inflation, phase reference destabilization, or interface non-idealities), rather than proposing a protocol-level security proof. We propose a phase-first framework in which the defender selects a phase-space rotation angle $\theta$ (and, in principle, a squeezing parameter r) to minimize a receiver-observable centered second-moment degradation proxy, emphasizing containment rather than disturbance inversion. Because platforms expose different native observables, we evaluate phase-first modulation using two complementary tracks: (i) in theory/simulation, we monitor basis-dependent quadrature variance and covariance-derived summaries formed from mean-subtracted second moments so that $\Delta E_{\mathrm{cov}}$ reflects covariance inflation rather than coherent displacement; (ii) in the X8_01 hardware workflow, the readout is Fock sampling; thus, we use the shot-to-shot standard deviation ${\sigma }_N\left(\theta \right):=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)}$, where $N\left(\theta \right)$ denotes the shot-level detected count random variable at fixed $\theta$. In the reported hardware workflow, this shot-level count is formed by aggregating the returned Fock counts prior to postprocessing. We emphasize that ${\sigma }_N\left(\theta \right)$ is not claimed to estimate $Tr\left(V\right)$; it is an implementation-layer variability proxy aligned with the available readout. Our experimental validation is restricted to phase-only control instantiated as offline phase selection via one-dimensional grid search over $\theta$. Across numerical simulations and hardware phase-angle scans on Xanadu’s X8_01 photonic quantum processor, we find that static operating points can be brittle under strong DoS-like stress, whereas optimized phase selection can materially reduce a receiver-observed degradation proxy even without real-time feedback. Since $Tr\left(V\right)$ is invariant under pure rotations for phase-independent additive noise and ideal photon-number probabilities are invariant under a terminal Fock-basis phase gate, any observed $\theta$-dependence is interpreted operationally as evidence of a phase-dependent effective disturbance/measurement channel at the receiver interface. Simulation-only analyses indicate additional upside when squeezing is available, motivating future extensions incorporating higher-rate re-optimization, feedback-assisted architectures, and extended Gaussian control when available.

1. Introduction

Robustness in deployed quantum links is governed as much by finite-resolution inference (finite samples, calibration uncertainty, and estimator tolerances) and hardware control/readout limits as by idealized protocol models [1,2,3]. In continuous-variable quantum communication (CVQC), where information is encoded in optical phase-space quadratures [4,5], these operational constraints shape what a receiver can infer from finite ensembles of outcomes, shaping both performance and the physical-layer attack surface. Following the receiver-centric terminology in [6], we use disturbance to denote a physical-layer perturbation (benign or adversarial) and reserve noise for its effective stochastic representation in receiver-estimated statistics. This distinction matters in practice, as deterministic physical actions can appear stochastic once filtered through measurement-basis choice, LO/phase-reference dynamics, finite-window estimation, drift, and receiver-interface non-idealities.

A central vulnerability in finite-resolution settings is operational indistinguishability: distinct underlying mechanisms can yield statistically indistinguishable receiver observations over finite datasets, creating tolerance-induced regions where sub-threshold structure can be exploited or learned (e.g., [7]). Building on this viewpoint, [6] introduced a receiver-centric threat model and an operational regime taxonomy (reconnaissance/exploratory/ denial-of-service) expressed directly in terms of receiver-accessible phase-space and covariance statistics. The present paper is positioned as a complementary implementation-layer resilience study; rather than proposing a new protocol-level security proof or attempting mechanism identification, we evaluate a lightweight and hardware-compatible mitigation knob within the same receiver-observable framing.

CVQC is a key primitive for transporting and processing bosonic quantum states across optical hardware and networks, including distribution of nonclassical states and entanglement, interconnects between quantum processors, and repeater/networking architectures that rely on faithful state transmission and interfacing between photonic modes [8,9,10]. In these settings, practical implementations are sensitive to excess variance, phase instability, calibration and local-oscillator (LO) effects, and receiver limitations, which can degrade coherence, distort effective second-moment structure, and reduce operational reliability [8,11,12,13,14]. Beyond uncontrolled environmental fluctuations, these sensitivities define a physical-layer attack surface in which disturbances may be structured or timed to probe stability and/or disrupt operation. Denial-of-service (DoS)-like regimes are especially relevant operationally because they can overwhelm the receiver’s measurement/inference pipeline (e.g., via effective fluctuation inflation, phase-reference loss, or receiver-interface nonlinearities), forcing abort, loss of usable states, or forced re-initialization.

In this work, we investigate phase-first Gaussian modulation as a practical physical-layer strategy for improving resilience in CVQC under high-intensity (DoS-like) stressed conditions. The core control action is a defender-selected phase-space rotation, parameterized by an angle $\theta$, which re-orients how a phase-dependent or anisotropic effective disturbance/measurement channel projects onto receiver-observable statistics. Although joint control over rotation and squeezing $(\theta ,r)$ is a natural extension, we treat squeezing as prospective and restrict experimental validation to phase-only modulation. Importantly, under an ideal phase-invariant additive noise model, a pure rotation would not change rotation-invariant summaries such as $Tr\left(V\right)$; therefore, any observed $\theta$-dependence of a receiver-side degradation proxy is interpreted operationally as evidence of a phase-dependent effective channel at the receiver interface (e.g., anisotropic disturbance statistics, quadrature/mode mixing, LO/phase-reference coupling, calibration drift, or other non-idealities).

1.1. Receiver-Observable Evaluation (Simulation vs. Hardware)

A practical constraint in near-term platforms is that the native receiver readout may not consist of quadrature samples. Accordingly, we evaluate phase-first modulation using two complementary centered second-moment proxies (Section 2.3): (i) in theory/simulation, we monitor the basis-dependent quadrature variance $Var\left({\widehat{X}}_{\theta }\right)$ (Section 2.3.1) using mean-subtracted second moments so that the metric tracks variance/covariance inflation rather than coherent displacement; (ii) in the X8_01 access mode used here, the native readout is Fock sampling (MeasureFock) and we quantify phase sensitivity using the shot-to-shot standard deviation ${\sigma }_N\left(\theta \right):=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)}$, where $N\left(\theta \right)$ denotes the shot-level detected count random variable at fixed $\theta$. In the reported X8_01 workflow, the shot-level count is formed by aggregating the returned Fock counts before postprocessing; however, the reported hardware observable is the scalar quantity ${\sigma }_N\left(\theta \right)$ itself. This statistic is centered by construction and is not claimed to estimate $Tr\left(V\right)$; it is an implementation-layer receiver-observable stability/degradation proxy aligned with the available measurement primitive.

1.2. Offline Phase Selection (Proof-of-Concept)

The “feedback-inspired” aspect of this study is instantiated as offline parameter selection rather than real-time closed-loop control, reflecting current hardware/interface constraints. Operationally, the implemented phase search is a one-dimensional grid search over $\theta$, with cost dominated by data acquisition (shots per tested angle). Offline selection is meaningful when the effective disturbance/measurement behavior is approximately stationary over the acquisition window; fast nonstationarity motivates periodic re-optimization or closed-loop updates, which we treat as future work. We validate this phase-first mechanism on Xanadu’s X8_01 photonic quantum processor because it provides programmable phase operations (Rgate) and repeated sampling access suitable for systematic phase-angle scans [15,16]. While the location of ${\theta }^{*}$ and the achievable improvement are platform- and condition-dependent, the operational conclusion, i.e., that phase choice can materially affect a receiver-observed degradation proxy under phase-dependent effective conditions, is expected to transfer to other photonic CV platforms with programmable phase control and stable measurement primitives.

1.3. Threat-Model Boundary for Re-Optimization Cadence

The present proof-of-concept assumes that the effective disturbance/measurement behavior is approximately quasi-static over the phase-grid evaluation window used to evaluate candidate settings. Accordingly, the demonstrated offline selection rule is intended for stressed regimes in which the adversary’s effective action is fixed, slowly varying, or at least sufficiently persistent that a beneficial minimizer ${\theta }^{*}$ remains identifiable over the integration window. If the disturbance changes on a timescale shorter than the offline acquisition interval, then the selected ${\theta }^{*}$ can become stale before deployment and re-optimization must occur on a shorter cadence than the characteristic drift/variation time. In that regime, the appropriate problem is no longer a static offline minimization but a dynamic estimation–control task requiring coarse-to-fine search, interleaved sampling across angles, or closed-loop tracking. Therefore, we frame the present hardware result as an offline proof-of-concept under quasi-static stressed conditions rather than as a claim of real-time protection against rapidly varying adversarial dynamics. Equivalently, the present manuscript studies a fixed-condition problem of the form ${min}_{\theta \in {\Theta }_K}J(\theta ;a)$ for an effectively constant stressed condition a over the sweep, whereas a rapidly adaptive adversary would lead naturally to a repeated-game or minimax formulation ${min}_{\theta \in {\Theta }_K}{max}_{a\in \mathcal{A}}J(\theta ;a_t)$, which lies outside the scope of the current hardware demonstration.

1.4. Positioning Relative to Adjacent Literature

This manuscript sits at the intersection of two nearby literature corpora. On one side are resilience and implementation security studies in CV-QKD and coherent CV receivers, which typically evaluate degradation through quadrature-level excess-noise, estimator reliability, abort probability, and receiver-interface vulnerabilities [1,2,6,12,13]. On the other side are adaptive-control, phase-stabilization, and hardware-tuning studies in photonic platforms, which emphasize actuator choice, calibration-sensitive operation, and reconfiguration under practical device constraints [11,15,17]. The present contribution is positioned as an implementation-layer bridge between these viewpoints; we retain a receiver-centric security framing while studying a hardware-compatible control action, namely, offline phase selection based on a receiver-observable degradation proxy.

1.5. What Is Novel Here, and Why That Combination Matters

The novelty is not claimed to lie in only one ingredient taken in isolation; rather, it lies in how three ingredients are connected into a single operational pipeline and why that connection is necessary under current hardware constraints. First, we adopt a receiver-observable threat model framing for stressed operation, meaning that the control objective is defined in terms of what the receiver can actually estimate under finite data and finite resolution [1,2,6]. Second, we translate that framing into a phase-first control rule that is compatible with near-term photonic hardware, namely, offline phase-grid search rather than assumed real-time closed-loop Gaussian control [11,15]. Third, we keep the interpretation observable-consistent by explicitly separating the quadrature-moment track used in theory/simulation from the Fock sampling track used on X8_01; in this way, the hardware results are not overstated as direct covariance reconstruction. This combination matters because it answers three distinct practical questions at once: why phase choice can matter (a phase-dependent effective channel at the receiver interface), how the defender can act with currently exposed hardware controls (offline phase-only selection), and how the resulting evidence should be interpreted without claiming more than the available readout supports.

1.6. Resilience, Robustness, and Security Guarantees in This Paper

In the terminology of this paper, resilience means the existence of a controllable reduction in a receiver-observable degradation proxy under stressed operating conditions. Robustness means the persistence of that benefit under finite-shot uncertainty, phase-setting error, grid quantization, and drift over the acquisition window. By contrast, a security guarantee would require a protocol-specific proof that links the controlled observables to accepted security parameters and threat-model assumptions. Accordingly, the present manuscript demonstrates implementation-layer resilience and discusses robustness through local sensitivity, finite-resolution, and drift-aware interpretation, but does not claim a protocol-level security guarantee.

1.7. Contributions

The contributions of this work are threefold:

1.

We formulate a receiver-centric phase-first modulation framework in which resilience is defined operationally via minimization of a receiver-observable centered second-moment degradation proxy under disturbance.

2.

We provide simulation evidence (including phase-space/Wigner context and a basis-dependent variance sweep) that clarifies the mechanism of phase-first improvement as a projection/orientation effect under structured directional disturbance models.

3.

We experimentally demonstrate on X8_01 that an offline one-dimensional phase-grid search can identify non-intuitive phase settings that reduce a directly observed hardware proxy ${\sigma }_N\left(\theta \right)$ relative to static reference angles, establishing feasibility of phase-first containment under current control and readout constraints. The present hardware result is reported at the level of a shot-level scalar observable, with mode-resolved extensions left to future work.

Together, these results establish phase-first modulation as a lightweight and hardware-compatible countermeasure that enhances implementation-layer resilience against adversarial disturbances in CVQC systems, motivating future extensions incorporating higher-rate re-optimization, feedback-assisted control, and extended Gaussian actuation when available.

Table 1. Notation used throughout the manuscript.

Symbol	Meaning
$\widehat{\mathbf{r}}={(\widehat{x},\widehat{p})}^{\mathsf{T}}$	Single-mode quadrature vector.
$\mathbf{d}=\langle \widehat{\mathbf{r}}\rangle$	Displacement (first-moment) vector.
G	Effective linear action in the affine receiver-observable model.
$\xi$	Effective zero-mean random perturbation in estimator space.
N	Covariance of $\xi$; effective fluctuation matrix in estimator space.
V	Centered quadrature covariance matrix.
$V^{\prime }$	Disturbed/effective covariance after channel or interface action.
$V_0$	Reference covariance under nominal operation.
$V_{\theta }$	Covariance after phase-space rotation by $\theta$.
$R\left(\theta \right)$	Phase-space rotation matrix.
${\widehat{X}}_{\theta }$	Rotated/monitored quadrature at angle $\theta$.
${\mathbf{u}}_{\theta }$	Unit vector ${(cos\theta ,sin\theta )}^{\mathsf{T}}$ used in $Var\left({\widehat{X}}_{\theta }\right)$.
$Tr\left(V\right)$	Trace of the covariance matrix; total variance summary.
$\Delta E_{\mathrm{cov}}$	Trace-based covariance-inflation proxy.
$\mathcal{J}\left(\theta \right)$	Generic receiver-observable objective used for phase selection.
${\mathcal{J}}_{\mathrm{cov}}\left(\theta \right)$	Quadrature/covariance-track objective.
${\mathcal{J}}_{\mathrm{pc}}\left(\theta \right)$	Photon-count-track objective.
${\mathcal{C}}_{\theta }$	Allowed phase-control set/phase constraint set.
$\theta$	Defender-selected phase rotation angle.
${\theta }_k$	k-th tested phase angle in the discrete grid.
${\theta }^{*}$	Selected minimizing phase setting.
$\tilde{\theta }$	Implemented phase after phase-setting error.
${\epsilon }_{\theta }$	Phase-setting calibration error.
$\Delta \theta$	Phase-grid spacing.
${\Theta }_K={\left\{{\theta }_k\right\}}_{k=1}^K$	Discrete grid of candidate phase settings.
K	Number of tested phase angles in the offline grid search.
r	Squeezing parameter (prospective/simulation-side control).
$N^{\left(s\right)}\left(\theta \right)$	Shot-level detected count at phase $\theta$ in the hardware workflow.
$N\left(\theta \right)$	Random variable corresponding to $N^{\left(s\right)}\left(\theta \right)$ over repeated shots at fixed $\theta$.
${\sigma }_N\left(\theta \right)$	Shot-to-shot standard deviation of the hardware count observable at fixed $\theta$.
$n_{\mathrm{shots}}$	Number of repeated shots collected per tested phase angle.
$N_{\mathrm{shots},\mathrm{tot}}$	Total shot budget across the full offline grid search.
${\rho }_0$	Reference vacuum state in simulation.
${\rho }_{\mathrm{atk}}$	Directional disturbance state (displacement-mixture model).
${\alpha }_{\mathrm{atk}}$	Displacement amplitude in the structured simulation disturbance model.
d	Displacement magnitude in quadrature units in the simulation model.
${\varphi }_{\mathrm{atk}}$	Disturbance direction in phase space in the simulation model.
${\mu }_{\theta }$	Projected component mean shift in the monitored quadrature.
${\theta }_{\mathrm{atk}}$	Fixed stress phase offset used in the hardware stressed configuration.
$T_{\mathrm{search}}$	Approximate end-to-end time required for the offline search.
${\tau }_{\mathrm{eval}}$	Effective wall-clock latency per tested angle.
${\tau }_{\mathrm{drift}}$	Characteristic drift timescale of the effective device/interface mapping.
$T_{\mathrm{run}}$	Total duration of the reported hardware run/acquisition window.
$\widehat{\mathcal{J}}\left(\theta \right)$	Estimated hardware objective from finite-shot data.
${\sigma }_{\mathrm{shot}}\left(\theta \right)$	Finite-sample estimator uncertainty at fixed $\theta$.
$S_{\theta }\left(\theta \right)$	Local first-order phase sensitivity of the measured objective.
${\kappa }_{\theta }\left(\theta \right)$	Local second-order phase curvature/sensitivity of the measured objective.
$\delta {\mathcal{J}}_{\mathrm{sens}}\left(\theta \right)$	First-order change in the objective due to phase-setting error.
$\delta {\mathcal{J}}_{\mathrm{drift}}(\theta ;t)$	Time-dependent drift contribution at fixed phase.
${\sigma }_{\mathrm{drift}}(\theta ;T_{\mathrm{run}})$	Effective drift scale over the acquisition window.
${\sigma }_{\mathrm{obs}}(\theta ;T_{\mathrm{run}})$	Total observed spread in the measured hardware proxy.
${\mathcal{C}}_{\mathrm{off}}\left(V\right)$	Off-diagonal covariance-energy summary used in the multi-mode scaling discussion.

summarizes the notation used throughout the manuscript.

2. Receiver-Observable Model and Evaluation Metrics

2.1. Receiver-Centric Viewpoint and Terminology

Continuous-variable quantum communication (CVQC) systems are governed not only by idealized protocol models but also by what the receiver can actually observe and reliably estimate from finite data under practical hardware constraints. In deployed optical CV links, receiver decisions and health checks are mediated by a measurement-and-inference pipeline that includes coherent detection (homodyne/heterodyne) relative to a local oscillator (LO) and phase reference, digitization and finite resolution, calibration and reference-setting, finite-window estimation, and hardware non-idealities such as saturation or clipping [8,12,13,14].

Therefore, this paper adopts a receiver-observable stance in which we model adversarial behavior in terms of its impact on statistics available to the receiver (and on the receiver’s estimator reliability), rather than attempting unique identification of an underlying physical mechanism. This perspective is particularly important in adversarial settings because distinct physical actions can induce statistically similar signatures once filtered through finite-resolution measurement, finite-window estimation, and calibration dynamics, especially when the receiver relies on tolerance-based acceptance logic (as in parameter-estimation-driven operation and monitoring) [1,2,6].

2.1.1. Terminology: Disturbance vs. Noise

Throughout this manuscript, we separate the physical-layer cause from its statistical appearance at the receiver:

1.

Disturbance (physical-layer perturbation). A disturbance denotes a perturbation applied to the CVQC link or receiver interface (benign or adversarial), which may be structured, adaptive, nonstationary, or deterministic at the physical layer.

2.

Noise (effective stochastic representation). Noise denotes the effective stochastic representation of a disturbance as seen through receiver-estimated statistics, e.g., an inferred excess-variance contribution, a time-varying fluctuation level over monitoring windows, or an effective additive term in a phenomenological model.

3.

Perturbation (effect in estimator space). Perturbation refers to the induced change in receiver observables or estimators (e.g., changes in estimated variances/covariances, confidence intervals, or other receiver-defined monitoring statistics) caused by a disturbance.

This distinction matters operationally. Even when an adversary’s action is deterministic (e.g., a repeatable physical perturbation aligned with a device sensitivity), the receiver may observe an effectively stochastic signature due to uncontrolled internal degrees of freedom, LO/phase-reference dynamics, finite sampling, drift, windowing, and hardware non-idealities. Accordingly, this work characterizes adversarial behavior at the level of receiver-observable statistics under finite data and finite resolution, consistent with a receiver-centric threat-modeling viewpoint [6].

2.1.2. DoS-like Conditions as a Receiver-Observable Regime

In this paper, denial-of-service (DoS)-like conditions are defined operationally as a receiver-observable regime in which the measurement/inference pipeline becomes unreliable or fails. We intentionally do not equate DoS with “adding coherent intensity” (i.e., increasing a coherent displacement amplitude), because coherent displacement primarily changes first moments (mean field), and does not by itself constitute excess quadrature variance or estimator destabilization. Under ideal models, a phase rotation does not reduce the intrinsic quadrature variance of a coherent state, and rotation-invariant summaries such as $Tr\left(V\right)$ are unchanged by a pure rotation for fixed V.

Instead, DoS-like behavior is characterized by one or more of the following receiver-facing consequences:

1.

Effective fluctuation inflation (excess-variance growth). Disturbance-driven increases in second-moment statistics that degrade estimator precision and push receiver monitoring beyond tolerance/acceptance margins (e.g., excess-variance growth in receiver parameter estimation and link-health monitoring) [1,2].

2.

Phase-reference/LO destabilization. Disturbances that impair coherent detection by disrupting the phase reference, inducing phase-dependent estimator failure or rapid basis-dependent degradation [8,11,12].

3.

Receiver-interface non-idealities and nonlinear failure. High-stress conditions that trigger saturation/clipping/blinding-like behavior, or other hardware-interface effects that invalidate nominal inference assumptions and can lead to abrupt loss of operability [18,19,20,21].

Connection to Coherent-Receiver Stress Mechanisms

Operationally, DoS-like behavior in coherent receivers often arises through receiver/interface limits (e.g., saturation or clipping in the detection chain, or LO/reference-path coupling) that invalidate nominal inference assumptions and can induce phase-dependent effective statistics. In this work, we remain agnostic to the underlying physical mechanism and evaluate stress through receiver-observable degradation proxies.

We use the qualifier “DoS-like” to emphasize that our experiments and analysis are anchored in observable operational stress (estimator/interface breakdown and degradation proxies), not in a claim of a single physical attack mechanism. In particular, multiple physical families can produce similar receiver-observable symptoms under finite resolution and finite monitoring windows. Therefore, the aim of the present study is to evaluate whether a lightweight and hardware-compatible phase-space re-orientation can contain receiver-observed degradation under such high-stress conditions, while making explicit the invariance caveats that apply under idealized phase-independent noise models.

2.2. Effective Observable-Level Model (Baseline)

2.2.1. Affine Receiver-Observable Model in Estimator Space

To summarize the receiver-observable impact of physical-layer disturbances in a compact and implementation-relevant way, we adopt a baseline observable-level model that acts directly on the quadrature vector $\widehat{\mathbf{r}}={(\widehat{x},\widehat{p})}^{\mathsf{T}}$. In this representation, the net effect of channel/interface behavior on receiver-estimated first moments is written as an affine transformation

$${\widehat{\mathbf{r}}}^{\prime }=G\widehat{\mathbf{r}}+\xi ,$$

(1)

where $G\in {\mathbb{R}}^{2\times 2}$ is a real matrix capturing the effective linear action inferred at the receiver (e.g., rotation/mixing, gain/attenuation, or other linear distortions in the receiver-observable description) and $\xi$ is an effective zero-mean random vector summarizing disturbance-driven fluctuations in estimator space. We denote the (receiver-inferred) second-order structure of $\xi$ by a positive semidefinite matrix $N⪰0$,

$$\mathbb{E}\left[\xi \right]=\mathbf{0},\mathbb{E}\left[\xi {\xi }^{\mathsf{T}}\right]=N.$$

(2)

Equation (1) is understood as a moment-level effective relation (a map on receiver-estimated first/second moments over a finite window), not as a claim of a unique underlying physical mechanism.

When quadrature moments are available and the receiver forms a mean-subtracted covariance matrix $V:=\mathbb{E}\left[(\widehat{\mathbf{r}}-\mathbf{d}){(\widehat{\mathbf{r}}-\mathbf{d})}^{\mathsf{T}}\right]$ with displacement $\mathbf{d}:=\mathbb{E}\left[\widehat{\mathbf{r}}\right]$, Equation (1) implies the standard covariance update rule

$$V^{\prime }=GV{G}^{\mathsf{T}}+N,$$

(3)

where $V^{\prime }$ is the receiver-observable covariance after the effective channel/interface action. Equation (3) is used in this paper as a baseline bookkeeping model for how disturbances can manifest in receiver-estimated second moments and how phase-space control choices can re-orient the monitored basis before those statistics are formed.

We emphasize that Equation (1) is not restricted to adversarial behavior; it also provides a convenient summary of benign non-idealities (e.g., drift, calibration offsets, or environmental perturbations) as they appear to the receiver over a monitoring window. In adversarial settings, the same form can represent the receiver-observable footprint of structured interference while remaining agnostic to the underlying physical mechanism.

2.2.2. Non-Identifiability and Scope of the Model

Non-Identifiability (Equivalence-Class Viewpoint)

The pair $(G,N)$ in Equations (1)–(3) should be interpreted as an equivalence-class description in estimator space: multiple distinct physical mechanisms can induce statistically indistinguishable receiver-observable effects on first and second moments over finite data windows. This non-identifiability is strengthened in practice by finite measurement resolution, finite sample size, calibration uncertainty, and LO/phase-reference dynamics, which can blur mechanistic distinctions and create tolerance-induced indistinguishability regions in estimator space [1,2,6].

Consequently, this work does not attempt to infer attacker capability, identify a unique physical attack mechanism, or claim that any observed phase dependence of a proxy corresponds to a specific exploit; instead, our objective is operational. We evaluate whether a defender-controlled phase-space re-orientation can reduce a receiver-observed degradation proxy under fixed stressed conditions, and we interpret any observed $\theta$-dependence as evidence of a phase-dependent effective disturbance/measurement channel at the receiver interface (e.g., anisotropic statistics, quadrature/mode mixing, LO/reference coupling, calibration drift, or other non-idealities).

Scope Limitations (What Is and Is Not Captured)

The baseline model above is intentionally restricted to what is accessible at the level of first and second moments and, where relevant, to windowed estimation:

1.

Covariance-level view. The model captures how disturbances manifest in mean-subtracted covariances and trace-like summaries, but does not uniquely characterize higher-order moments, non-Gaussian tails, or multimodal distributions. Disturbances with signatures that reside primarily in higher-order structure may not be well described by $(G,N)$ alone [8].

2.

Nonstationarity and temporal correlation. The model can be interpreted window-by-window, allowing $(G,N)$ to vary with time (or with the phase setting) to reflect drift and nonstationarity. However, the explicit modeling of memory effects across windows (e.g., correlated disturbances that reduce effective sample size) is not developed here; such effects motivate finite-rate adaptation and closed-loop extensions discussed later.

3.

Receiver-interface nonlinearities. Under sufficiently strong stress (DoS-like conditions), hardware effects such as saturation, clipping, or blinding-like behavior can invalidate linear-Gaussian assumptions. In such cases, Equation (1) should be read as a net observable summary rather than a faithful physical model of the receiver response [12,19,20].

Interpretation of Phase Dependence

A recurring caveat is that under an ideal phase-independent additive-noise model, a pure rotation does not change rotation-invariant summaries: for a fixed covariance V and rotation $R\left(\theta \right)$, $Tr\left(RV{R}^{\mathsf{T}}\right)=Tr\left(V\right)$; therefore, any empirical $\theta$-dependence of a trace-based quadrature proxy should be interpreted operationally as evidence that the effective receiver-observable channel depends on $\theta$ through $G\left(\theta \right)$ and/or $N\left(\theta \right)$ (e.g., anisotropic or basis-dependent disturbance statistics, quadrature mixing, LO/reference coupling, calibration dynamics, or other interface non-idealities). This interpretation is consistent with the equivalence-class viewpoint emphasized above and with the receiver-centric threat modeling framework in [6].

2.3. Metrics Used in This Paper: Two Observable Tracks

A central practical constraint in this study is that the simulation/theory analysis and the X8_01 hardware workflow expose different native observables. Accordingly, we evaluate phase-first modulation using two complementary centered second-moment tracks. The common design principle is to quantify disturbance-driven fluctuation inflation (variance/covariance growth about an estimated mean), not changes in coherent displacement. This avoids a common “amplitude-trick” pitfall: attenuation or phase-dependent mean-field changes must not be misread as noise reduction.

When first-moment changes are operationally relevant, they should be tracked separately from the centered degradation proxy. In quadrature language, this means monitoring the displacement vector $\mathbf{d}=\langle \widehat{\mathbf{r}}\rangle$ alongside the centered covariance V; in the Fock-sampling workflow used here, it means pairing ${\sigma }_N\left(\theta \right)$ with the empirical mean $\mathbb{E}\left[N\right(\theta \left)\right]$. Therefore, a candidate phase setting should not be preferred solely because it lowers the centered proxy if it simultaneously induces an unacceptable coherent displacement, count-bias, or saturation-adjacent mean shift.

2.3.1. Quadrature-Moment Track (Theory/Simulation)

When a Gaussian-moment description is available (analytical model or simulation with access to quadrature statistics), we work directly with mean-subtracted second moments of $\widehat{\mathbf{r}}={(\widehat{x},\widehat{p})}^{\mathsf{T}}$. Let $\mathbf{d}:=\langle \widehat{\mathbf{r}}\rangle$ denote the displacement vector and define the (mean-subtracted) covariance matrix

$$V:=〈(\widehat{\mathbf{r}}-\mathbf{d}){(\widehat{\mathbf{r}}-\mathbf{d})}^{\mathsf{T}}〉.$$

(4)

All quadrature-level degradation metrics reported in this track are computed from V (or from windowed estimates $\widehat{V}$ formed after subtracting empirical first moments), so that the metric captures excess variance/covariance inflation rather than coherent amplitude. In the quantum Gaussian formalism, V is understood as the symmetrized covariance, $V_{ij}:=\frac{1}{2}\langle \Delta {\widehat{r}}_i\Delta {\widehat{r}}_j+\Delta {\widehat{r}}_j\Delta {\widehat{r}}_i\rangle$; for the single-quadrature variances used here, this coincides with the usual centered second moment.

Basis-Dependent Variance Under Phase Re-Orientation

For a phase-space rotation $R\left(\theta \right)$, the rotated quadrature

$${\widehat{X}}_{\theta }:=\widehat{x}cos\theta +\widehat{p}sin\theta$$

(5)

has variance

$$Var\left({\widehat{X}}_{\theta }\right)={\mathbf{u}}_{\theta }^{\mathsf{T}}V{\mathbf{u}}_{\theta },{\mathbf{u}}_{\theta }:={(cos\theta ,sin\theta )}^{\mathsf{T}},$$

(6)

which provides a rotation-sensitive diagnostic of anisotropy and basis-dependent inflation. In this paper, $Var\left({\widehat{X}}_{\theta }\right)$ (or its windowed estimate) serves as a primary simulation-side objective for illustrating projection/orientation effects under structured disturbances. Therefore, the phase-resolved simulation quantity $N^{\left(s\right)}\left(\theta \right)$ is referred to in prose as this rotated-quadrature variance diagnostic, rather than introducing it as a stand-alone displayed equation.

Trace-Based Covariance-Inflation Proxy (Rotation-Invariant Severity Coordinate)

When a scalar summary is useful, we report a centered covariance-inflation proxy

$$\Delta E_{\mathrm{cov}}:={\displaystyle \frac{Tr\left(V^{\prime }\right)-Tr\left(V_0\right)}{4}},$$

(7)

where $V_0$ denotes the mean-subtracted reference covariance (nominal operation) and $V^{\prime }$ the mean-subtracted covariance under disturbance. By construction, $\Delta E_{\mathrm{cov}}$ is insensitive to coherent displacement and captures total covariance inflation (excess variance) in shot-noise units under the adopted SNU convention (vacuum covariance $I_2$). For an m-mode system with $V\in {\mathbb{R}}^{2m\times 2m}$, the same definition applies with $Tr\left(V\right)$ aggregating variance across modes; however, in multi-mode settings $Tr\left(V\right)$ should be complemented by cross-covariance monitoring when mode coupling is important (Section 2.6).

Interpretation Caveat (Rotation Invariance)

For a fixed covariance V and a pure rotation $R\left(\theta \right)$, $Tr\left(RV{R}^{\mathsf{T}}\right)=Tr\left(V\right)$. Therefore, under an ideal phase-independent additive-noise model, $\Delta E_{\mathrm{cov}}$ would be invariant to $\theta$. Any observed $\theta$-dependence of quadrature-level proxies is interpreted operationally as evidence of a phase-dependent effective disturbance/measurement channel (e.g., anisotropy, quadrature mixing, LO/reference coupling, calibration drift, or other interface non-idealities), consistent with the equivalence-class viewpoint of Section 2.2.2.

2.3.2. Photon-Count Track (X8_01 Workflow)

In the X8_01 access mode used for our experiments, the native measurement primitive is Fock-basis sampling (MeasureFock), not quadrature readout. Therefore, we evaluate phase dependence using a tomography-light centered scalar count observable computed directly from the measured photon-number outcomes.

Let $N^{\left(s\right)}\left(\theta \right)$ denote the shot-level detected count in shot s at phase setting $\theta$ and let $N\left(\theta \right)$ denote the corresponding random variable over repeated shots at fixed $\theta$. In the reported X8_01 run, $N^{\left(s\right)}\left(\theta \right)$ is formed by aggregating the returned Fock counts across the program outputs before postprocessing.

Our hardware observable is the centered dispersion statistic

$${\sigma }_N\left(\theta \right):=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)},$$

(8)

where $\widehat{Var}(·)$ denotes the population variance estimator over shots at fixed $\theta$ (normalization by $n_{\mathrm{shots}}$), consistent with the default numpy var used in our analysis code [22]. This quantity is centered by construction (variance about the sample mean); therefore, it is used here to track shot-to-shot fluctuation inflation in the measured outcomes rather than changes in the coherent mean field itself.

What ${\sigma }_N$ Is (and Is Not)

We emphasize that ${\sigma }_N\left(\theta \right)$ is not claimed to estimate $Tr\left(V\right)$ or any protocol-specific excess-noise parameter. It is an implementation-layer variability proxy tied to the platform’s available readout, used here to quantify phase sensitivity under fixed stressed conditions. Its role is to support the proof-of-concept question of whether a phase setting exists that measurably reduces a directly observed receiver-side fluctuation proxy under the available hardware interface.

Interpretation Caveat (Terminal-Phase Equivalence)

Under ideal photon-number measurement, a terminal phase gate $U\left(\theta \right)=e^{i\theta \widehat{n}}$ is diagonal in the Fock basis and commutes with the measurement projectors; therefore, the photon-number outcome probabilities are invariant to $\theta$ (Section 2.4.2). Consequently, any observed $\theta$-dependence of ${\sigma }_N\left(\theta \right)$ in our experiments indicates that the commanded phase change is not operationally equivalent to a terminal phase gate with respect to the measured effective modes under the compiled program and device interface. Equivalently, the overall input–output map realized by the compiled program plus device interface depends on $\theta$. We interpret this as a receiver-interface/implementation effect, not as mechanism identification.

Connection to Protocol-Level Reliability (e.g., CV-QKD)

In protocols that include parameter estimation and tolerance-based acceptance logic (e.g., CV-QKD), increases in receiver-estimated fluctuation levels reduce estimator reliability, and can also increase abort probability and degrade achievable secret key rates. While this work does not compute protocol-specific key rates, the centered second-moment proxies used here are chosen to track the same receiver-facing phenomenon, namely, fluctuation inflation, that drives those operational failures under finite data and finite resolution [1,2,12,13].

The justification for the centered fluctuation proxy used here is monotonic rather than identity-based. Larger centered second moments imply less stable finite-window estimation, wider uncertainty on monitored receiver statistics, and reduced operational margin under tolerance-based acceptance logic. Thus, although ${\sigma }_N\left(\theta \right)$ is not a direct estimator of $Tr\left(V\right)$, a reduction in this proxy still indicates reduced receiver-observed variability in the native readout available to the hardware workflow. What we do not claim here is a formal protocol-level bound mapping ${\sigma }_N$ to secret-key rate, abort probability, or other end-to-end figures of merit; establishing such bounds would require a protocol-specific model and access to the corresponding quadrature-level parameters.

2.4. Invariance and Interpretation of Phase Dependence

Phase-first modulation is meaningful only when interpreted through the correct invariances. In an ideal CV model with phase-independent additive noise, a pure rotation cannot reduce intrinsic quadrature noise; likewise, under ideal Fock readout, a terminal phase gate cannot change photon-number probabilities. Therefore, any empirically observed dependence of a receiver-side degradation proxy on the chosen phase setting $\theta$ must be interpreted operationally as evidence that the effective disturbance/measurement channel at the receiver interface is phase dependent (anisotropic, mixing, calibration-dependent, or otherwise non-ideal) rather than as a claim of fundamental noise reduction by rotation alone. This section formalizes these invariances and states how we interpret $\theta$-dependent outcomes throughout the paper.

2.4.1. Rotation Invariance Under Ideal Phase-Independent Additive Noise

Consider a single-mode phase-space rotation $R\left(\theta \right)\in {\mathbb{R}}^{2\times 2}$ acting on the quadrature vector $\widehat{\mathbf{r}}={(\widehat{x},\widehat{p})}^{\mathsf{T}}$. At the covariance level, the rotated covariance is

$$V_{\theta }=R\left(\theta \right)V{R}^{\mathsf{T}}\left(\theta \right)$$

(9)

and the trace is invariant:

$$Tr\left(V_{\theta }\right)=Tr\left(R\left(\theta \right)V{R}^{\mathsf{T}}\left(\theta \right)\right)=Tr\left(V\right).$$

(10)

Now, consider an ideal phase-independent additive-noise model in which the effective disturbance contributes an additive covariance term $N⪰0$ that does not depend on $\theta$. Then,

$$V_{\theta }^{\prime }=R\left(\theta \right)V{R}^{\mathsf{T}}\left(\theta \right)+N,$$

(11)

therefore,

$$Tr\left(V_{\theta }^{\prime }\right)=Tr\left(V\right)+Tr\left(N\right),$$

(12)

which is also independent of $\theta$. Consequently, the centered covariance-inflation proxy

$$\Delta E_{\mathrm{cov}}\left(\theta \right):={\displaystyle \frac{Tr\left(V_{\theta }^{\prime }\right)-Tr\left(V_0\right)}{4}}$$

(13)

is invariant under a pure rotation in this idealized setting. This makes explicit the key point raised in peer review: phase rotation does not reduce intrinsic quadrature variance under a phase-independent isotropic additive-noise model.

Rotation can, however, change rotation-sensitive diagnostics when the covariance is anisotropic or when the monitored quantity depends on a particular quadrature basis. For example, the basis-dependent variance $Var\left({\widehat{X}}_{\theta }\right)$ varies with $\theta$ whenever V is not proportional to the identity, even though $Tr\left(V\right)$ remains constant (Section 2.3.1).

2.4.2. Fock-Readout Invariance Under a Terminal Phase Gate

In the X8_01 workflow used here, the native readout is photon-number (Fock) sampling. Let the single-mode phase gate be $U\left(\theta \right)=e^{i\theta \widehat{n}}$, where $\widehat{n}$ is the photon-number operator. This unitary is diagonal in the Fock basis:

$$U\left(\theta \right)|n\rangle =e^{in\theta }|n\rangle .$$

(14)

For any state $\rho$, applying a terminal phase gate immediately before an ideal photon-number measurement does not change the outcome probabilities:

$$\begin{array}{cc}\hfill p_{\theta }\left(n\right)& =Tr\left(|n\rangle \langle n|U\left(\theta \right)\rho U^{†}\left(\theta \right)\right)=Tr\left(U^{†}\left(\theta \right)|n\rangle \langle n|U\left(\theta \right)\rho \right)=Tr\left(|n\rangle \langle n|\rho \right)=p\left(n\right),\hfill \end{array}$$

(15)

since $U^{†}\left(\theta \right)|n\rangle \langle n|U\left(\theta \right)=|n\rangle \langle n|$. The same argument extends directly to the multi-mode case with $U\left(\theta \right)={⨂}_{i=1}^M{e}^{i{\theta }_i{\widehat{n}}_i}$ and projectors $|\mathbf{n}\rangle \langle \mathbf{n}|$.

Therefore, under an ideal model in which the applied phase operation is effectively a terminal phase gate with respect to the measured modes, photon-number probabilities and all photon-number statistics are invariant to $\theta$. In particular, the photon-count dispersion proxy

$${\sigma }_N\left(\theta \right)=\sqrt{Var\left(N\left(\theta \right)\right)}$$

(16)

would be invariant under such an idealized terminal-phase scenario.

2.4.3. Operational Interpretation of Observed $\theta$-Dependence

The invariances in Section 2.4.1 and Section 2.4.2 imply a strict interpretive rule used throughout this paper:

Interpretive Rule (No “Intrinsic Noise Reduction” Claim from Rotation Alone)

If a receiver-observable proxy (quadrature-based $\Delta E_{\mathrm{cov}}$, basis-dependent $Var\left({\widehat{X}}_{\theta }\right)$, or hardware ${\sigma }_N$) exhibits $\theta$-dependence, we interpret this as evidence that the effective disturbance/measurement channel relevant to the receiver is phase dependent, rather than as evidence that a pure rotation reduces intrinsic quadrature noise under an ideal phase-independent model.

Operationally, $\theta$-dependence can arise from several non-ideal but deployment-relevant effects, including:

1.

Anisotropic or quadrature-dependent effective disturbance. The disturbance may preferentially inflate one quadrature (or a low-dimensional subspace in multi-mode phase space), so that changing $\theta$ changes which component is emphasized in the monitored statistic.

2.

Quadrature/mode mixing and phase-reference coupling. LO phase-reference dynamics, imperfect phase tracking, and coupling between signal and reference paths can induce phase-dependent mixing, which appears effectively as $G\left(\theta \right)$ and/or $N\left(\theta \right)$ in the observable model of Section 2.2.1.

3.

Calibration drift and finite-window estimation. Slowly varying calibration parameters (gain, offsets, normalization) can make the effective mapping from the applied control $\theta$ to estimated statistics nonstationary over the acquisition window.

4.

Receiver-interface nonlinearities. Under stressed (DoS-like) conditions, saturation/clipping or other nonlinear response can create strongly phase-dependent measurement distortion even if the underlying state rotation is ideal.

5.

Non-equivalence to a terminal phase gate on the measured effective modes. In a realistic photonic processor, the commanded Rgate need not be operationally equivalent to a terminal phase gate with respect to the measured effective modes under the compiled program and device interface (e.g., due to interferometric compilation, loss/mode mismatch, or drift), so photon-number statistics can acquire $\theta$-dependence even though terminal-phase invariance holds in the idealized setting.

Why This Interpretation Supports (Rather than Weakens) the Resilience Claim

The goal of this paper is implementation-layer resilience: identifying whether a controllable phase setting can reduce a receiver-observed centered fluctuation proxy under realistic phase-dependent effective conditions. The existence of $\theta$-dependent degradation is itself a statement about the practical receiver interface and its vulnerability surface. Therefore, phase-first modulation is framed as a containment knob that exploits phase dependence in the effective channel, not as a universal method that reduces variance under ideal phase-independent additive-noise assumptions.

Connection to the Offline Proof-of-Concept Scope

Because the effective channel can drift over time, the optimal phase ${\theta }^{*}$ identified by a finite sweep is only guaranteed to be meaningful when the effective behavior is approximately stationary over the acquisition window. Accordingly, throughout this manuscript we treat phase-first selection as an offline proof-of-concept, and explicitly separate (i) existence of a beneficial ${\theta }^{*}$ under fixed conditions (what we demonstrate) from (ii) real-time tracking under fast nonstationarity (future work).

Transition From Observables to Control Rule

Section 2.3, Section 2.4 and Section 2.5 are intended to separate three layers of the argument. Section 2.3 defines the receiver-observable degradation proxies, while Section 2.4 states the invariance rules needed to correctly interpret any observed phase dependence. Now, Section 2.5 uses those proxies and caveats to define the actual offline phase-selection rule implemented in theory and hardware. This ordering is meant to make explicit that the control rule is built from receiver-observable quantities only after the corresponding interpretive caveats have been stated.

2.5. Phase-First Selection and Practical Implementation (Offline)

This section specifies how the phase-first idea is instantiated in the present study under current hardware/interface constraints. Although the framework is adaptive in principle (estimation → parameter selection), our experimental realization is deliberately offline and phase-only: we evaluate a discrete set of phase angles on a phase grid $\left\{{\theta }_k\right\}$, compute a receiver-observable degradation proxy at each setting, and select the minimizing phase. This implementation avoids assuming low-latency real-time parameter identification during an ongoing DoS-like event. Instead, we demonstrate the existence and feasibility of phase selection under approximately stationary stressed conditions, leaving closed-loop tracking and squeezing-enabled extensions to future work.

2.5.1. Receiver-Observable Objective and Phase-Only Instantiation

Because different platforms expose different native observables, we define phase-first selection using a generic receiver-observable cost $\mathcal{J}\left(\theta \right)$ computed from the statistics available under the chosen readout. The operational phase-first rule is

$${\theta }^{*}\in arg\underset{\theta \in {\mathcal{C}}_{\theta }}{min}\mathcal{J}\left(\theta \right),$$

(17)

where ${\mathcal{C}}_{\theta }$ encodes the allowable phase range and any hardware constraints.

Quadrature-Moment Instantiation (Theory/Simulation)

When quadrature-level modeling (or quadrature samples) are available, we evaluate centered second moments and define a covariance-inflation proxy such as

$${\mathcal{J}}_{\mathrm{cov}}\left(\theta \right):=\Delta E_{\mathrm{cov}}\left(\theta \right)={\displaystyle \frac{Tr\left({\widehat{V}}^{\prime }\left(\theta \right)\right)-Tr\left(V_0\right)}{4}},$$

(18)

where ${\widehat{V}}^{\prime }\left(\theta \right)$ is computed from mean-subtracted statistics so that $Tr\left(\widehat{V}\right)$ tracks excess variance/covariance inflation rather than coherent displacement. In the simulation results section, we also report basis-dependent quadrature variance $Var\left({\widehat{X}}_{\theta }\right)$ to make orientation/projection effects explicit (Section 2.3.1).

Photon-Count Instantiation (X8_01 Workflow)

In the experimental workflow used here, readout is Fock sampling, so the receiver-observable objective is defined directly from the shot-level detected count:

$${\mathcal{J}}_{\mathrm{pc}}\left(\theta \right):={\sigma }_N\left(\theta \right)=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)},$$

(19)

where $N^{\left(s\right)}\left(\theta \right)$ denotes the shot-level detected count used in the hardware workflow; in the reported X8_01 run, it is formed by aggregating the returned Fock counts before postprocessing. As emphasized in Section 2.4, this is an implementation-layer variability proxy aligned with the available measurement primitive, and is not claimed to estimate $Tr\left(V\right)$.

Phase-Only Scope

In the present hardware experiments, squeezing is not actuated in the demonstrated access mode. Accordingly, we set $r=0$ (or treat r as fixed) and perform phase-only selection using Equation (17). Simulation-only sections may explore prospective $(\theta ,r)$ behavior as an upper-bound reference, but the experimental claims are restricted to phase-only modulation.

2.5.2. Offline Grid Search, Cost Scaling, and Stationarity Assumption

Discrete Phase-Grid Evaluation

We instantiate Equation (17) by evaluating $\mathcal{J}\left(\theta \right)$ on a discrete grid of K candidate angles,

$${\Theta }_K:={\left\{{\theta }_k\right\}}_{k=1}^K,{\theta }^{*}\in arg\underset{{\theta }_k\in {\Theta }_K}{min}\mathcal{J}\left({\theta }_k\right).$$

(20)

This is a one-dimensional line search suitable for near-term interfaces in which phase parameters can be specified per job but integrated low-latency measurement-to-actuation feedback is not available.

Cost Scaling: Computation vs. Sampling

Postprocessing cost is lightweight: we compute a sample variance (and optionally additional summaries) for each ${\theta }_k$, which is $\mathcal{O}\left(n_{\mathrm{shots}}\right)$ per angle and $\mathcal{O}\left(K{n}_{\mathrm{shots}}\right)$ overall. The dominant practical cost is data acquisition: if $n_{\mathrm{shots}}$ shots are collected per angle, the total sampling cost is

$$N_{\mathrm{shots},\mathrm{tot}}=K{n}_{\mathrm{shots}}.$$

(21)

This separation addresses the reviewer request for clearer accounting of computational effort: phase-first selection is not computationally heavy, and is experimentally heavy only through the shot budget.

For the representative hardware phase-angle scan used in Figure 1, $K=36$ and $n_{\mathrm{shots}}=100$, giving a total acquisition budget of $N_{\mathrm{shots},\mathrm{tot}}=K{n}_{\mathrm{shots}}=3600$ Fock samples. If ${\tau }_{\mathrm{eval}}$ denotes the effective wall-clock latency required to acquire and return one estimate of $J\left({\theta }_k\right)$, then the corresponding sweep time is approximately

$$T_{\mathrm{search}}\approx K{\tau }_{\mathrm{eval}}.$$

Therefore, the offline phase-selection assumption is most credible in the regime $T_{\mathrm{search}}\ll {\tau }_{\mathrm{drift}}$, where ${\tau }_{\mathrm{drift}}$ is the characteristic time over which the effective device/interface mapping changes appreciably. We do not estimate ${\tau }_{\mathrm{drift}}$ directly in the present experiment, so the proposal is interpreted as an offline proof-of-concept rather than a demonstrated real-time control architecture. As such, a priority next step is to benchmark the end-to-end wall-clock latency of the search workflow against experimentally estimated drift timescales, which would allow the offline regime studied here to be separated more clearly from any future real-time operating regime.

Stationarity Assumption (What Offline Selection Does and Does Not Claim)

Offline phase selection is meaningful when the effective disturbance/measurement behavior is approximately stationary over the acquisition window. Concretely, we assume that the induced map from the commanded setting $\theta$ to the measured objective $\mathcal{J}\left(\theta \right)$ does not drift substantially during acquisition of the K angle points (or, more weakly, that drift is slow enough that a minimum remains identifiable). If the effective channel varies faster than the offline acquisition window, then a fixed offline ${\theta }^{*}$ may become stale. This limitation motivates future work on faster re-optimization, coarse-to-fine search, or closed-loop tracking; however, establishing the existence of a beneficial phase under controlled stressed conditions is the proof-of-concept objective of this study.

Nonstationary Extensions (Brief)

When disturbance nonstationarity is significant, two practical adaptations are natural: (i) reduce K by using a coarse grid followed by local refinement near the best angle, and/or (ii) interleave angles in time (round-robin sampling) to reduce bias from slow drift. These strategies retain $\mathcal{O}\left(K\right)$ structure while improving robustness to time variation.

2.5.3. Robustness to Phase-Setting Error and Finite Resolution

Phase-first selection is implemented with finite phase resolution (grid spacing) and may be affected by phase-setting error (hardware calibration and compilation variability). Therefore, we treat ${\theta }^{*}$ as an operational minimizer subject to quantization and uncertainty, and we report robustness considerations that enable replication.

Grid Quantization and Local Refinement

Let $\Delta \theta$ denote the grid spacing. Then, the selected ${\theta }^{*}$ in Equation (20) is quantized to within $\pm \Delta \theta /2$ of the true minimizer of the underlying (unknown) smooth objective. In practice, one can improve accuracy without changing the overall workflow by:

1.

Coarse-to-fine grid search: Choose a coarse grid to locate an approximate minimizer, then re-evaluate locally on a finer grid with a finer $\Delta \theta$.

2.

Local model fit: Fit a quadratic (or sinusoidal) model to $\mathcal{J}\left(\theta \right)$ in a neighborhood of the minimum and estimate a sub-grid minimizer.

Both methods preserve the offline nature of the experiment and reduce sensitivity to grid choice.

Phase-Setting Error and Sensitivity

Let the implemented phase be $\tilde{\theta }=\theta +{\epsilon }_{\theta }$, where ${\epsilon }_{\theta }$ captures phase-setting error (control calibration, compilation, drift). A first-order robustness characterization is obtained from the local slope/curvature of the measured objective: if $\mathcal{J}\left(\theta \right)$ is flat near ${\theta }^{*}$, then modest ${\epsilon }_{\theta }$ produces only small performance loss. Operationally, we recommend reporting either (i) a confidence band on $\mathcal{J}\left(\theta \right)$ across repeated runs or (ii) an uncertainty interval for ${\theta }^{*}$ obtained by bootstrapping the shot data used to compute $\mathcal{J}\left(\theta \right)$. This directly addresses the reviewer request to discuss how deviations in $\theta$ are handled.

Finite-Sample Uncertainty (Shot-Noise of the Estimator)

Because $\mathcal{J}\left(\theta \right)$ is itself estimated from finite data, its uncertainty decreases with the number of shots per angle. For the photon-count proxy, one may estimate uncertainty by standard variance-estimator error bars or by nonparametric bootstrap over shots: resample ${\left\{N^{\left(s\right)}\left(\theta \right)\right\}}_{s=1}^{n_{\mathrm{shots}}}$ with replacement, recompute ${\sigma }_N\left(\theta \right)$, and report percentile intervals. Analogous resampling applies to quadrature-based statistics in simulation.

A short local sensitivity analysis can be built from the empirical landscape near the selected minimum. In particular, finite-difference slope and curvature estimates around ${\theta }^{*}$ indicate whether the minimum is broad (hence, relatively robust to quantization, drift, and phase-setting error) or sharp (hence, more sensitive to implementation uncertainty). In a fuller experimental study, this local geometry should be reported together with bootstrap confidence intervals for ${\sigma }_N\left(\theta \right)$ at the best grid point and nearby reference angles.

Practical Selection Rule with Robustness Margin

When multiple nearby angles have statistically indistinguishable objectives, a conservative rule is to choose the best angle within a tolerance band and optionally prefer angles with lower empirical variance across repeats. This reflects a resilience-oriented design that prioritizes stable improvement over a potentially fragile sharp optimum.

These robustness steps ensure that the reported phase-first improvement is not an artifact of a single grid point or a brittle calibration, and provide a clear replication path under finite phase resolution and finite shot budgets.

2.6. Implementation Note on Aggregation in the Hardware Workflow

The hardware result reported in this paper is not a mode-resolved analysis. In the X8_01 workflow used here, the returned Fock counts are aggregated into a shot-level scalar $N^{\left(s\right)}\left(\theta \right)$ before postprocessing, and the reported hardware observable is the shot-to-shot standard deviation ${\sigma }_N\left(\theta \right)$. Mode-resolved extensions are possible, but are outside the scope of the present proof-of-concept.

3. Methods

This section describes the simulation and experimental methodologies used to evaluate phase-first Gaussian modulation as an implementation-layer countermeasure against structured and adversarial disturbances in continuous-variable quantum communication (CVQC) systems. Our methodological objective is to express all performance claims in terms of receiver-observable statistics while making a clear separation between (i) the broader control framework (joint phase-space rotation and squeezing, $(\theta ,r)$) and (ii) what is exercised in the present hardware workflow (phase-only rotation).

Although the phase-first framework is adaptive in principle (estimation → parameter selection), the hardware validation reported here is deliberately instantiated as an offline proof-of-concept. We evaluate a receiver-observable cost on a discrete grid of phase settings and select the minimizing phase, without real-time closed-loop updates. This choice reflects current interface constraints in the demonstrated X8_01 access mode, where programmable phase operations are available but low-latency measurement-to-actuation feedback and dynamic squeezing actuation are not.

Because the simulation environment and the X8_01 workflow expose different native observables, we report two complementary centered second-moment evaluation tracks. In theory/simulation, we quantify degradation using mean-subtracted quadrature second moments (basis-dependent variance and covariance-derived summaries), so that reported metrics capture covariance inflation (excess variance) rather than coherent displacement. In the hardware experiments, the native readout is Fock sampling; thus, we use a tomography-light count-standard-deviation proxy computed directly from photon-number outcomes. Accordingly, the experimental hardware observable is treated as an implementation-layer variability proxy rather than as a direct estimate of the quadrature covariance trace.

Both simulation and hardware studies emphasize DoS-like receiver-observable stress as a conservative benchmark. This stress-focused methodology is aligned with our paper’s containment goal of evaluating whether a lightweight and hardware-compatible phase re-orientation can reduce a receiver-observed degradation proxy under fixed high-stress conditions while clarifying the limits of phase-only control when stronger Gaussian actuation (e.g., squeezing) and real-time adaptation are unavailable.

3.1. Theoretical Framework

3.1.1. Adversarial Disturbances and Qumode Response

Continuous-variable quantum states encoded in qumodes are sensitive to environmental and adversarial disturbances. In this work, we characterize disturbances operationally through their impact on receiver-observable statistics under finite data and practical receiver constraints (Section 2), rather than attempting unique mechanism identification. In particular, reconnaissance, exploratory, and DoS-like regimes correspond to progressively stronger receiver-facing deviations, including fluctuation inflation in centered second moments, phase-reference dependence, and receiver-interface non-idealities.

Two Receiver-Observable Evaluation Tracks

A key methodological constraint is that the theory/simulation analysis and the X8_01 hardware workflow expose different native observables. Accordingly, we evaluate degradation using two complementary centered second-moment proxies (Section 2.3):

1.

Quadrature-moment track (theory/simulation). When quadrature statistics are available, we work with mean-subtracted second moments of $\widehat{\mathbf{r}}={(\widehat{x},\widehat{p})}^{\mathsf{T}}$ and the centered covariance $V=\langle (\widehat{\mathbf{r}}-\mathbf{d}){(\widehat{\mathbf{r}}-\mathbf{d})}^{\mathsf{T}}\rangle$ with $\mathbf{d}=\langle \widehat{\mathbf{r}}\rangle$ (Equation (4)). This ensures that reported variance/covariance changes reflect fluctuation inflation rather than coherent displacement. We report (i) the basis-dependent monitored-quadrature variance $Var\left({\widehat{X}}_{\theta }\right)$ (Equation (6)); when a scalar summary is useful, we also report (ii) the covariance-inflation proxy $\Delta E_{\mathrm{cov}}$ (Equation (7)) under the adopted shot-noise-unit (SNU) convention.

2.

Photon-count track (hardware). In the X8_01 workflow, the native readout is Fock sampling (MeasureFock). Therefore, we quantify phase sensitivity using the shot-to-shot standard deviation ${\sigma }_N\left(\theta \right)=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)}$ (Equation (8)), where $N\left(\theta \right)$ denotes the shot-level detected count random variable used in the hardware workflow. This statistic is centered by construction (variance about the sample mean) and is treated as an implementation-layer variability proxy, not as an estimator of $Tr\left(V\right)$ or any protocol-specific excess-noise parameter (Section 2.3.2).

Interpretation of any observed $\theta$-dependence follows the invariances and operational rule in Section 2.4.

3.1.2. Phase-First Modulation Strategy

We investigate a phase-first control strategy in which the defender selects a phase-space rotation angle $\theta$ to minimize a receiver-observable degradation proxy. The framework is adaptive in principle, but the present implementation is instantiated offline via a discrete phase sweep. While the broader framework admits joint Gaussian control $(\theta ,r)$, we deliberately restrict the experimental validation to phase-only selection in order to test the minimum viable hardware-compatible mitigation knob under realistic interface constraints. In the demonstrated X8_01 access mode, phase rotations are programmable and repeatable across modes, whereas dynamic squeezing actuation and low-latency measurement-to-actuation feedback are not exposed to user programs during a sweep. Therefore, focusing on phase-only control isolates the effect of basis re-orientation on receiver-observed degradation proxies and provides a conservative proof-of-concept that can transfer to other near-term photonic CV platforms where phase control is accessible but higher-bandwidth Gaussian actuation is limited. This choice emphasizes resilience under constrained control resources: the defense should remain effective even when only low-dimensional tuning is available and when rapid online estimation is impractical during DoS-like stress.

Observable-Dependent Objective

We instantiate the phase-first selection rule in Equation (17) using an observable-dependent cost $\mathcal{J}\left(\theta \right)$, with the specific instantiation determined by the available measurement track:

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\mathrm{sim}}\left(\theta \right)& :=Var\left({\widehat{X}}_{\theta }\right)\left(\mathrm{primary}\right)\mathrm{and}\mathrm{we}\mathrm{report}\Delta E_{\mathrm{cov}}\left(\theta \right)\mathrm{as}\mathrm{a}\mathrm{severity}\mathrm{summary},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\mathcal{J}}_{\mathrm{pc}}\left(\theta \right)& :={\sigma }_N\left(\theta \right)=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)},\hfill \end{array}$$

(23)

where $Var\left({\widehat{X}}_{\theta }\right)$ is defined in Equation (6), $\Delta E_{\mathrm{cov}}$ in Equation (7), and ${\sigma }_N$ in Equation (8).

Offline Grid Search (Hardware Instantiation)

Operationally, we evaluate ${\mathcal{J}}_{\mathrm{pc}}\left(\theta \right)$ on a discrete grid ${\Theta }_K={\left\{{\theta }_k\right\}}_{k=1}^K$ and select the minimizing grid point (see Equation (27)). The postprocessing cost is linear in the shot count per angle, while the dominant cost is data acquisition (shots per tested angle). This offline instantiation demonstrates existence and feasibility of phase selection under approximately stationary stressed conditions; real-time tracking under faster nonstationarity is treated as a future architectural extension.

3.2. Simulation Methodology

3.2.1. Simulation Setup

Numerical simulations were performed using the Strawberry Fields (SF) library [16] to (i) prepare single-mode Gaussian states, (ii) visualize phase-space structure via Wigner-function panels, and (iii) sanity-check the shot-noise unit (SNU) normalization on the Gaussian backend. For the phase-sweep objective, we use an analytic/quasi-analytic model for $Var\left({\widehat{X}}_{\theta }\right)$ and Monte Carlo sampling to emulate finite-shot estimation, which keeps the sweep objective transparent and reproducible: the expected curve is available in closed form, and finite-shot effects are emulated via seeded Monte Carlo sampling. This design keeps the simulation objective aligned with the receiver-observable metric emphasized throughout, with the phase-first sweep minimizing a centered second-moment proxy, here the monitored-basis quadrature variance $Var\left({\widehat{X}}_{\theta }\right)$ (Section 2.3.1).

In simulation, the phase sweep is performed using the basis-dependent objective $Var\left({\widehat{X}}_{\theta }\right)$; $\Delta E_{\mathrm{cov}}$ is reported only as a rotation-invariant severity summary and is not used as the sweep objective.

Structured Disturbance Model (Directional Displacement Mixture)

To represent an intentionally structured directional disturbance rather than independent and identically distributed isotropic Gaussian noise, we model a DoS-like stress pattern as a two-component displacement mixture aligned along an adversary-selected phase-space direction ${\varphi }_{\mathrm{atk}}$. Starting from the vacuum state ${\rho }_0$ (SNU), we define

$${\rho }_{\mathrm{atk}}=\frac{1}{2}D(+{\alpha }_{\mathrm{atk}}){\rho }_0{D}^{†}(+{\alpha }_{\mathrm{atk}})+\frac{1}{2}D(-{\alpha }_{\mathrm{atk}}){\rho }_0{D}^{†}(-{\alpha }_{\mathrm{atk}}),$$

(24)

where ${\alpha }_{\mathrm{atk}}$ is chosen such that the displacement magnitude is d (in quadrature units) and its direction in phase space is ${\varphi }_{\mathrm{atk}}$. This mixture is mean-centered by construction and yields a bimodal Wigner-function signature, providing a simple and interpretable phenomenological model of a repeatable directional stress pattern without claiming a unique mechanism-level realization on any specific device.

Phase-First Sweep Observable

For each defender-chosen phase setting $\theta$, we consider the rotated quadrature

$${\widehat{X}}_{\theta }=\widehat{x}cos\theta +\widehat{p}sin\theta$$

(25)

and evaluate the receiver-observed variance $Var\left({\widehat{X}}_{\theta }\right)$ under ${\rho }_{\mathrm{atk}}$ as the simulation-side phase-first objective.

Simulation Procedure

The simulation workflow consists of the following:

1.

Reference normalization (vacuum). Initialize a single-mode vacuum state in SF and verify the expected SNU normalization on the Gaussian backend (vacuum covariance $I_2$), implying $Var\left({\widehat{X}}_{\theta }\right)=1$ for all $\theta$ in the ideal model.

2.

Construct disturbance components and Wigner visualization. Generate the displaced components $D(\pm {\alpha }_{\mathrm{atk}}){\rho }_0{D}^{†}(\pm {\alpha }_{\mathrm{atk}})$ (via Dgate) and form Wigner-function panels for each component on a common grid. Because Equation (24) is an incoherent mixture, the mixture Wigner function is computed as the equal-weight sum of the two component Wigner functions.

3.

Phase sweep and finite-shot realism. For the displacement-mixture model, the expected monitored-quadrature variance admits the closed form

$$\mathbb{E}\left[Var\left({\widehat{X}}_{\theta }\right)\right]=1+{\mu }_{\theta }^2,{\mu }_{\theta }=dcos(\theta -{\varphi }_{\mathrm{atk}}),$$

(26)

where the constant 1 is the vacuum variance (SNU) and ${\mu }_{\theta }$ is the component mean shift in the monitored quadrature. To emulate finite-shot estimation, we sample $n_{\mathrm{shots}}$ outcomes from the corresponding one-dimensional mixture distribution $\frac{1}{2}\mathcal{N}(+{\mu }_{\theta },1)+\frac{1}{2}\mathcal{N}(-{\mu }_{\theta },1)$ and compute the sample variance as a function of $\theta$.

Parameter Sweeps and Reproducibility

We sweep $\theta \in [-\pi ,\pi ]$ on a uniform grid with fixed angular resolution (e.g., ≈10°), holding $(d,{\varphi }_{\mathrm{atk}})$ fixed within each sweep. All random draws used for finite-shot sampling are seeded to enable exact reproduction of plotted markers and summary tables.

Shot Notation

In both simulation and experiment, $n_{\mathrm{shots}}$ denotes the number of repeated samples collected per tested phase setting.

3.2.2. Performance Metrics

We report two complementary simulation-side metrics:

1.

Receiver-observed monitored-quadrature variance (primary). The primary objective is $Var\left({\widehat{X}}_{\theta }\right)$, reported as both the expected curve in Equation (26) and the finite-shot estimate from mixture sampling. This is a centered second-moment proxy that captures disturbance-driven fluctuation inflation in the monitored basis.

2.

Wigner-function structure (supporting). Wigner panels provide a qualitative phase-space depiction of the vacuum baseline and the structured disturbance mixture, supporting interpretation of phase-first improvement as a projection/orientation effect.

Simulation results are benchmarked against reference angles (e.g., $\pi /2$, $\pi /4$) and summarized by reporting the grid-selected minimizer ${\theta }^{*}=arg{min}_{\theta \in {\Theta }_K}Var\left({\widehat{X}}_{\theta }\right)$, with ${\Theta }_K$ the discrete phase grid used in the offline run, the corresponding minimum $Var\left({\widehat{X}}_{{\theta }^{*}}\right)$, and relative improvements versus the reference angles. Because the disturbance model is directional by construction, the optimum is interpretable; it occurs near the measurement axis orthogonal to the disturbance direction (up to grid resolution), consistent with the projection-based mitigation mechanism emphasized in the Results section.

3.3. Experimental Methodology

3.3.1. Experimental Design

Platform and Scope

Experiments were performed on Xanadu’s X8_01 photonic quantum processor to test whether offline phase-first selection can reduce a receiver-observed degradation proxy under a fixed high-stress (DoS-like) configuration. We use X8_01 because it provides (i) programmable phase rotations (Rgate) across multiple optical modes and (ii) native Fock-basis sampling (MeasureFock) suitable for repeated phase-angle scans during the offline run [15,16]. In the access mode used here, the workflow does not expose shot-based quadrature readout (homodyne/heterodyne samples) required to estimate a full covariance matrix $V\left(\theta \right)$ from quadrature second moments. Accordingly, the hardware study is framed as an implementation-layer proof-of-concept that quantifies phase dependence using a centered count-standard-deviation statistic computed directly from Fock samples, rather than as a quadrature-covariance evaluation.

Although Rgate$\left(\theta \right)$ is specified as a single-mode phase rotation at the circuit level, the compiled implementation and effective measured modes need not render it operationally equivalent to an ideal terminal phase gate immediately before measurement; accordingly, phase-dependent statistics can arise from device/interface effects.

Operational Sources of Observed Phase Sensitivity

We do not claim unique mechanism identification from the present hardware data; however, the observed phase dependence is consistent with several concrete classes of non-ideal receiver/interface behavior: (i) interferometric compilation that causes the commanded Rgate$\left(\theta \right)$ to act internally rather than as an isolated terminal Fock-basis phase shift on the effective measured modes, (ii) phase-dependent loss or mode mixing in the realized optical network, (iii) calibration and phase-reference drift during the sweep, and (iv) nonlinear interface/readout effects under stressed conditions. This supports a more predictive interpretation of the hardware landscape: if the effective receiver interface couples disturbance energy anisotropically into the measured observable, then re-orienting the accessible phase basis can systematically reshape the degradation curve $\theta \mapsto {\sigma }_N\left(\theta \right)$ rather than merely producing an unexplained empirical correlation.

Compiled Program Versus Device Interface

By “compiled program” we mean the concrete gate sequence and interferometric realization submitted after compilation, including internal decompositions, mode transformations, and placement of phase operations within the realized optical network. By “device interface” we mean the effective input–output behavior seen at measurement, including calibration state, phase-reference behavior, loss, mode mismatch, control electronics, and readout response. Terminal-phase invariance applies only when the applied phase operation is equivalent to a final diagonal Fock-basis phase gate on the measured effective modes. An internal compiled rotation followed by device-interface effects need not satisfy that equivalence, which is why a commanded Rgate$\left(\theta \right)$ can produce $\theta$-dependent observed statistics even though an ideal terminal phase gate would not.

Offline (Proof-of-Concept) Adaptivity

Because the user-facing workflow does not provide an integrated low-latency measurement-to-actuation loop and because dynamic squeezing actuation is not available during the sweep, the experimental instantiation is restricted to phase-only control with offline parameter selection. Operationally, we evaluate a degradation proxy on a discrete grid of phase settings and select the minimizing phase without closed-loop updates during execution. This isolates phase choice as a standalone hardware-compatible mitigation knob.

Centered-Statistic Interpretation (Hardware Count-Standard-Deviation Proxy)

Because the observable available in this workflow is photon-number samples rather than quadrature outcomes, the experimental objective is a centered second-moment proxy of shot-to-shot count fluctuations. We denote the shot-level detected count by $N^{\left(s\right)}\left(\theta \right)$ and evaluate the hardware observable ${\sigma }_N\left(\theta \right)$ as defined in Equation (8), where $Var(·)$ is taken over shots at fixed $\theta$. Therefore, throughout the hardware sections “improvement” denotes reduced receiver-observed count variability in this fixed workflow, not universal noise cancellation and not a direct measurement of quadrature covariance inflation.

Unless stated otherwise, $Var(·)$ is computed as the population variance over shots at fixed $\theta$ (i.e., normalization by $n_{\mathrm{shots}}$, consistent with the default numpy implementation used in our analysis), and ${\sigma }_N\left(\theta \right)$ is reported as the corresponding standard deviation.

3.3.2. Experimental Procedure

We evaluate $K=36$ candidate phase angles ${\theta }_k$ on a uniform grid over $\theta \in [-\pi ,\pi ]$ (approximately $10.{29}^{\circ }$ spacing); for each tested angle, we collect $n_{\mathrm{shots}}=100$ repeated Fock samples, giving a total acquisition budget of $N_{\mathrm{shots},\mathrm{tot}}=3600$ shots across the full offline run. For each tested angle ${\theta }_k$, we execute the following steps:

1.

Program compilation and phase-grid evaluation. Compile a fixed-depth eight-output program that applies Rgate$\left(\theta \right)$ identically across the returned outputs, implementing a one-dimensional grid search over $\theta$.

2.

Fixed stressed configuration. To emulate a reproducible DoS-like stressed operating configuration within the available gate set, apply an additional fixed stress phase offset Rgate$\left({\theta }_{\mathrm{atk}}\right)$ with ${\theta }_{\mathrm{atk}}=0.5\mathrm{rad}$ (held constant across the offline run) identically within the program prior to measurement. This should be interpreted as a reproducible, phase-sensitive stress configuration within the compiled program/device interface chosen to produce a high-variability operating point for evaluation, not as a claim that an adversary physically applies Rgate to a deployed channel. In an ideal single-mode model, Rgate$\left(\theta \right)$ followed by Rgate$\left({\theta }_{\mathrm{atk}}\right)$ is equivalent to a single phase rotation by $\theta +{\theta }_{\mathrm{atk}}$; thus, any observed $\theta$-dependence under Fock readout is interpreted as an implementation/interface effect rather than an intrinsic property of an ideal terminal phase gate.

3.

Fock measurement and data acquisition. Measure the program in the Fock basis (MeasureFock) and collect $n_{\mathrm{shots}}$ repetitions per $\theta$. For each shot s, record the returned Fock sample and form the shot-level detected count $N^{\left(s\right)}\left(\theta \right)$ used in the hardware workflow.

4.

Compute the phase-dependent proxy. Compute ${\sigma }_N\left(\theta \right)$ as defined in Equation (8) (using $Var(·)$ over shots at fixed $\theta$) and store the result for the full phase-angle profile.

Normalization, Recalibration, and Evidentiary Scope

The present hardware dataset should be interpreted as a representative offline sweep obtained under one fixed workflow configuration. We did not implement interleaved pilot/reference states, a per-angle baseline normalization measurement, or a dedicated between-angle recalibration protocol in which the cadence is itself analyzed as an experimental variable. Likewise, this manuscript does not claim that Figure 1 averages multiple independently recalibrated sessions or that the reported minimum has been separated from all possible temporal drift contributions by a dedicated control experiment. The evidentiary claim is narrower: within the fixed acquisition window used here, the receiver-observed proxy ${\sigma }_N\left(\theta \right)$ exhibited a phase-dependent landscape, motivating future studies with explicit drift tracking, reference-state normalization, repeated-session error bars, and controlled recalibration schedules. Future hardware studies will incorporate interleaved reference-state measurements, explicit recalibration schedules, and repeated-session sweeps so that phase-selection effects can be disentangled more rigorously from temporal hardware fluctuations.

Phase Selection and Resolution

Because $\theta$ is evaluated on a discrete grid, the selected minimizer ${\theta }^{*}$ is quantized to the grid spacing. A straightforward refinement is to perform a coarse grid search to localize the minimum followed by a finer local grid search or local interpolation near the best grid point; for transparency and reproducibility, we report the baseline uniform-grid evaluation.

3.3.3. Data Collection and Analysis

The phase-first selection rule is instantiated as

$${\theta }^{*}\in arg\underset{\theta \in {\Theta }_K}{min}{\sigma }_N\left(\theta \right),$$

(27)

with ${\Theta }_K$ being the discrete phase grid. Because the measured observable is photon counts rather than quadrature samples, we do not claim reconstruction of $V\left(\theta \right)$ or estimation of $Tr\left(V\right)$ in this workflow.

Phase dependence is interpreted operationally: under an ideal terminal phase shift immediately before photon-number measurement, photon-number statistics are invariant to $\theta$ for any input state. Therefore, any observed $\theta$-dependence of ${\sigma }_N\left(\theta \right)$ indicates phase-sensitive effective behavior in the compiled program/device interface (e.g., phase-dependent coupling/mode mixing, calibration drift, phase-reference effects, or other non-idealities). Within this receiver-centric framing, offline phase selection provides a practical mitigation knob for reducing a directly observed variability proxy under the fixed stressed configuration.

Limitation Under Fast Drift

Because the hardware study does not implement real-time feedback, an offline ${\theta }^{*}$ can become stale if the effective phase-sensitive behavior drifts on timescales shorter than the sweep. This motivates future work on repeated sweeps for uncertainty quantification, shorter-window re-optimization, and closed-loop phase tracking when available.

4. Results

This section reports results from receiver-observable theoretical analysis, numerical simulation, and hardware measurements that collectively evaluate phase-first modulation as a practical implementation-layer countermeasure against adversarial disturbances in CVQC systems. Consistent with the scope clarified throughout the manuscript, the adaptive/feedback-inspired framework is instantiated experimentally as offline phase selection (no real-time closed-loop updates). On hardware, the only validated control action is phase-space rotation. Squeezing modulation is treated as a simulation-only extension that provides an upper-bound reference for what additional Gaussian control could achieve under idealized actuation, and is not claimed as an experimentally demonstrated capability here.

4.1. Two Complementary Centered Second-Moment Proxies

Because the simulation environment and the X8_01 access mode expose different native observables, we report two closely related centered second-moment degradation proxies: (i) in simulation, we track the receiver-observed monitored-quadrature variance as a function of measurement-basis angle, $Var\left({\widehat{X}}_{\theta }\right)$ (and equivalent covariance-derived summaries), which captures disturbance-driven variance/covariance inflation in the monitored basis; (ii) in hardware, we use the count-standard-deviation proxy from Section 2.3.2, ${\sigma }_N\left(\theta \right)$ (Equation (8)), computed from the shot-level detected count used in the hardware workflow. In the reported X8_01 run, this shot-level count is formed by aggregating the returned Fock counts before postprocessing. Here, $\widehat{Var}(·)$ denotes the population variance across shots (normalization by $n_{\mathrm{shots}}$), consistent with the default numpy implementation used in our analysis. Both metrics are mean-centered by construction (variance about an estimated mean). Accordingly, throughout this section “improvement” means reduced receiver-observed fluctuation inflation in a centered second-moment proxy, not noise cancellation, and does not represent a protocol-level security claim.

4.2. Interpretation: Phase Dependence Implies an Effective Phase-Dependent Channel

A key interpretive point is that under an ideal phase-invariant additive-noise model, a pure rotation would not change rotation-invariant summaries; likewise, under ideal Fock readout, a terminal phase gate would not change photon-number statistics. Therefore, any observed dependence of $Var\left({\widehat{X}}_{\theta }\right)$ or ${\sigma }_N\left(\theta \right)$ on $\theta$ is interpreted operationally as evidence that the effective disturbance/measurement channel seen by the receiver is phase-dependent, e.g., via anisotropic disturbance statistics, quadrature/mode mixing, LO/phase-reference coupling, calibration drift, or other receiver-interface non-idealities. Within that receiver-centric framing, phase selection provides a lightweight mitigation knob by re-orienting the effective projection so that the disturbance couples less strongly to the monitored observable, resulting in extended operational margin under DoS-like receiver-observable stress.

4.3. Roadmap of Results

Section 4.4 presents an analytic/simulation phase-angle scan in which we compute $Var\left({\widehat{X}}_{\theta }\right)$ across $\theta \in [-\pi ,\pi ]$, compare finite-shot estimates to the expected curve, visualize representative Wigner panels at baseline as well as at the selected ${\theta }^{*}$, and summarize key parameters (including improvement versus conventional reference angles such as $\pi /2$ and $\pi /4$). Section 4.5 then reports the corresponding phase-angle scan on Xanadu’s X8_01 platform using offline phase selection and Fock readout, quantifying the observed phase dependence via ${\sigma }_N\left(\theta \right)$ and emphasizing that the measured optimum is device- and condition-dependent.

4.4. Simulation Results

Figure 2 and Figure 3 summarize the simulation-side evidence for phase-first mitigation under the structured directional displacement-mixture disturbance model defined in Section 3.2.1 [8]. Rather than modeling the disturbance as independent and identically distributed additive Gaussian noise, we use a mean-centered mixture of displaced components aligned along a phase-space direction ${\varphi }_{\mathrm{atk}}$. Operationally, this captures the idea of a repeatable, directional stress pattern in phase space, while the receiver-facing signature is basis-dependent fluctuation inflation.

4.4.1. Qualitative Phase-Space Picture (Wigner Panels)

The vacuum baseline (left panel) is isotropic. Under the structured disturbance (middle panel), the Wigner function becomes bimodal rather than simply elliptically broadened, consistent with an incoherent displacement-mixture model. The right panel shows the same disturbed state visualized in the defender-selected basis ${\theta }^{*}$, emphasizing the mechanism of phase-first control in which phase selection does not remove the disturbance but rather changes how its structure projects onto the monitored quadrature.

4.4.2. Phase-Angle Scan Objective (Receiver-Observed $Var\left({\widehat{X}}_{\theta }\right)$)

To quantify this projection effect directly, we sweep the measurement-basis angle $\theta \in [-\pi ,\pi ]$ and evaluate the receiver-observed monitored-quadrature variance $Var\left({\widehat{X}}_{\theta }\right)$. For a directional displacement mixture, $Var\left({\widehat{X}}_{\theta }\right)$ is strongly phase dependent; it is maximized when the monitored quadrature aligns with the disturbance direction and minimized when it is orthogonal (up to the finite grid resolution). This produces the characteristic sinusoidal dependence in Figure 3 and yields a clear offline optimum ${\theta }^{*}$.

In the representative phase-angle scan shown here (with $d=2$ and ${\varphi }_{\mathrm{atk}}=0.90\mathrm{rad}$), the grid-selected optimum occurs at ${\theta }^{*}=-{40}^{\circ }$, yielding a near-vacuum minimum $Var\left({\widehat{X}}_{{\theta }^{*}}\right)\approx 1.003$ (vacuum-referenced units). In contrast, the same disturbance configuration yields substantially larger variance at common reference angles, e.g., $Var\left({\widehat{X}}_{\pi /2}\right)\approx 3.45$ and $Var\left({\widehat{X}}_{\pi /4}\right)\approx 4.95$ (from the analytic curve). This corresponds to a reduction of about $71\%$ versus $\pi /2$ and about $80\%$ versus $\pi /4$ under the fixed disturbance configuration used in the offline run, illustrating that static operating points can be highly suboptimal under structured directional stress.

4.4.3. Finite-Shot Realism and Implementation Relevance

The markers in Figure 3 show that finite-shot estimates closely track the expected curve. In this study, finite-shot realism is emulated by sampling $n_{\mathrm{shots}}$ outcomes from the corresponding one-dimensional mixture distribution $\frac{1}{2}\mathcal{N}(+{\mu }_{\theta },1)+\frac{1}{2}\mathcal{N}(-{\mu }_{\theta },1)$ implied by Equation (26), and computing the sample variance. Strawberry Fields is used to generate and visualize the underlying phase-space states (Wigner panels) and to sanity-check the adopted vacuum/SNU normalization [16].

4.4.4. Key Takeaways from Simulation

1.

Phase-first mitigation is a projection effect. The benefit arises because $Var\left({\widehat{X}}_{\theta }\right)$ depends on basis choice under a structured directional disturbance; choosing $\theta$ changes how strongly the disturbance appears in the monitored quadrature.

2.

The optimum is interpretable. The minimizing ${\theta }^{*}$ occurs near the quadrature orthogonal to the disturbance direction (up to grid resolution), providing a physically transparent explanation for the observed minimum.

3.

No claim of universal noise cancellation. The disturbance is not removed; the receiver-observed degradation proxy is reduced in the chosen basis under a fixed disturbance configuration.

Taken together, these simulation results motivate the hardware phase-angle scan study presented next: if the effective disturbance/measurement behavior at the receiver interface is phase dependent (e.g., via anisotropy, mixing, LO/phase-reference coupling, calibration drift, or other non-idealities), then an offline phase sweep can reveal a substantially better operating point ${\theta }^{*}$ even without real-time feedback.

4.5. Experimental Results

Experimental measurements on Xanadu’s X8_01 photonic quantum processor provide a proof-of-concept demonstration that offline phase selection can materially change a receiver-observed degradation proxy under a fixed stressed configuration [15]. Consistent with Section 3.3, the hardware study implements phase-only control (programmable phase rotations) with no integrated low-latency feedback and no squeezing actuation. Unlike the simulation section, which evaluates a quadrature-variance objective, the X8_01 workflow uses Fock-basis sampling; accordingly, we quantify phase sensitivity using a tomography-light count-standard-deviation statistic computed directly from measured samples.

4.5.1. Run Structure and Acquisition Details

Figure 1 corresponds to a single offline sweep over $K=36$ uniformly spaced phase settings on $\theta \in [-\pi ,\pi ]$, with $n_{\mathrm{shots}}=100$ Fock samples collected per angle, for a total acquisition budget of 3600 shots. For each tested angle, the reported quantity is ${\sigma }_N\left(\theta \right)=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)}$, where $N^{\left(s\right)}\left(\theta \right)$ is the shot-level detected count used in the hardware workflow. In the reported X8_01 run, this scalar is formed by aggregating the returned Fock counts before post-processing. Under ideal terminal-phase invariance this statistic would be $\theta$-independent; thus, the observed variation is interpreted operationally as phase-sensitive effective behavior in the compiled program/device interface rather than as intrinsic variance reduction from a terminal phase gate.

4.5.2. Measured Statistic (Hardware Proxy)

For each phase setting $\theta$, we collect $n_{\mathrm{shots}}=100$ repetitions. From the raw Fock samples returned by the program, we form the shot-level detected count $N^{\left(s\right)}\left(\theta \right)$ and report the hardware observable ${\sigma }_N\left(\theta \right)$ defined in Equation (8). In the analysis code, $\widehat{Var}$ is implemented as the population variance estimator (numpy.var, ddof = 0) [22], and ${\sigma }_N\left(\theta \right)$ is the corresponding standard deviation.

4.5.3. What the Offline Phase-Grid Evaluation Implements (and Why ${\theta }_{\mathrm{atk}}$ Is a Stress Knob, Not a Mechanism Claim)

In the implementation used here, the “stress” parameter is a deterministic phase offset applied via Rgate$\left({\theta }_{\mathrm{atk}}\right)$, not a stochastic noise injection. In this experiment, the program applies Rgate$\left(\theta \right)$ followed by a fixed offset Rgate$\left({\theta }_{\mathrm{atk}}\right)$ with ${\theta }_{\mathrm{atk}}=0.5\mathrm{rad}$ identically on each measured mode. In an ideal single-mode model, these operations compose to Rgate$(\theta +{\theta }_{\mathrm{atk}})$, and under ideal Fock readout all photon-number statistics would remain invariant to a terminal phase shift. Therefore, any observed $\theta$-dependence in Figure 1 is interpreted operationally as evidence that the compiled program/device interface induces phase-sensitive effective behavior for the measured modes under this stressed configuration (Section 2.4), rather than as an intrinsic property of an ideal terminal phase gate.

4.5.4. Drift, Finite-Shot Uncertainty, and Evidentiary Scope

Figure 1 should be interpreted as a single representative offline sweep. The observed phase-angle profile reflects finite-shot estimator uncertainty, and may also reflect slow drift over the acquisition window. Therefore, we treat the reported curve as evidence of a phase-dependent empirical landscape in the realized workflow, not as a fully drift-separated estimate of a stationary response curve. Repeated-session sweeps, interleaved-angle acquisition, and bootstrap uncertainty intervals are left to future work.

4.5.5. Key Observations

1.

Existence of a phase-dependent optimum under fixed stress. The statistic exhibits a clear minimizer ${\theta }^{*}$ consistent with the offline selection rule ${\theta }^{*}\in arg{min}_{\theta \in {\Theta }_K}{\sigma }_N\left(\theta \right)$.

2.

Static phase choices can be brittle. Reference angles such as $\pi /2$ and $\pi /4$ do not generally coincide with the minimizing setting, underscoring that fixed operating points can be suboptimal when the effective interface behavior is phase sensitive.

3.

The hardware result is a phase-dependent scalar landscape. The reported curve should be read as a phase-dependent empirical landscape of the shot-to-shot count standard deviation in the realized workflow, not as a mode-resolved analysis.

4.

Containment via phase choice, not universal noise cancellation. The observed reduction is consistent with basis-/configuration-dependent containment in a phase-sensitive effective hardware channel, not with intrinsic noise reduction from rotation alone.

Overall, the X8_01 measurements establish a hardware-feasible proof-of-concept: even without real-time feedback and without squeezing control, an offline phase-grid search can identify non-intuitive settings that reduce a directly observed receiver-side fluctuation proxy under a fixed stressed configuration. This motivates follow-on studies that (i) repeat offline runs across sessions to quantify robustness/uncertainty of ${\theta }^{*}$, and (ii) inspect mode-resolved count statistics and cross-correlations when diagnosing mode-selective or correlated effects.

5. Discussion

5.1. Interpretation of Key Results

The results in this paper support phase-first modulation as a practical implementation-layer containment mechanism under DoS-like receiver-observable stress, provided that the effective disturbance/measurement behavior seen by the receiver is phase dependent (e.g., anisotropic coupling to quadratures, phase-reference/LO effects, mode mixing, calibration drift, or other interface non-idealities). Consistent with the invariance arguments in Section 2.4, our central claim is not that a phase rotation reduces intrinsic rotation-invariant noise under an ideal phase-independent model; rather, appropriate phase-space orientation can reduce a receiver-observed centered second-moment degradation proxy in realistic, hardware-mediated conditions.

A key empirical observation is the pronounced dependence of the measured proxies on the applied phase setting $\theta$. By sweeping $\theta$ over $[-\pi ,\pi ]$ and selecting the minimizer ${\theta }^{*}$ (Section 2.5), we identify non-intuitive operating points that do not generally coincide with conventional reference angles such as $\pi /2$ or $\pi /4$. Under ideal models, a pure rotation would not change rotation-invariant quadrature summaries (Section 2.4.1), and an ideal terminal phase gate would not change photon-number statistics under Fock readout (Section 2.4.2). Therefore, the observed $\theta$-dependence is interpreted operationally as evidence that the compiled program plus device/receiver interface realizes a phase-dependent effective channel in the receiver-observable sense, consistent with the receiver-centric threat-modeling viewpoint of [6].

5.1.1. Simulation vs. Hardware Observables (Two Complementary Tracks)

To avoid conflating distinct measurement primitives, we emphasize that the simulation and hardware studies evaluate different but aligned receiver-observable proxies (Section 2.3). In simulation, we directly monitor basis-dependent quadrature second moments, in particular $Var\left({\widehat{X}}_{\theta }\right)$ (Equation (6)) under the structured directional displacement-mixture disturbance model (Section 3.2.1). This makes the phase-first mechanism explicit as a projection/orientation effect: changing $\theta$ changes how strongly structured disturbance energy appears in the monitored quadrature. In the X8_01 workflow, the native measurement primitive is Fock sampling (MeasureFock), so we quantify phase sensitivity using the centered count-standard-deviation proxy ${\sigma }_N\left(\theta \right)$ (Equation (8)), which we compute from the shot-level detected count used in the hardware workflow. In the reported hardware workflow, this shot-level count is formed by aggregating the returned Fock counts before post-processing. Accordingly, the reported hardware improvement is interpreted strictly as a reduction in receiver-observed count variability in this workflow, not as a direct measurement of quadrature covariance inflation or a reduction in $Tr\left(V\right)$.

5.1.2. Offline Phase Selection and Experimental Meaning

The hardware validation instantiates the feedback-inspired framework as offline phase selection (Section 2.5.2), i.e., a one-dimensional grid search with no real-time closed-loop updates and no squeezing actuation. Within that scope, the X8_01 phase-angle profile exhibits a clear minimizer ${\theta }^{*}$ and a substantial reduction of the measured proxy relative to a static reference angle in the representative run (Figure 1). This should be read as a proof-of-concept demonstration that a phase setting exists that improves a directly observed receiver-side fluctuation proxy under a fixed stressed configuration. Robustness of ${\theta }^{*}$ under drift and finite-shot variability motivates repeated sweeps and uncertainty quantification as follow-on work.

5.1.3. Generality and Security-Oriented Relevance

We chose X8_01 because it exposes programmable phase control suitable for systematic sweeps under repeated sampling; however, the qualitative conclusion is not intended to be device-specific: phase-first containment applies to any CVQC receiver for which the effective disturbance/measurement behavior is phase dependent. The location of ${\theta }^{*}$ and the achievable improvement are platform- and condition-dependent, and must be re-estimated per operating regime. Finally, although this work is not a protocol-level security proof, the operational relevance to security-oriented CVQC settings (including CV-QKD) is that phase agility can act as a physical-layer preconditioning knob that improves margin under phase-sensitive effective conditions; mapping these improvements to protocol-level parameters (e.g., quadrature excess-noise estimates and finite-size bounds) requires a workflow that measures (or reliably infers) the relevant quadrature statistics, and as such is treated as a priority direction for future integration studies.

5.2. Relation to Prior Work and Broader Implications

This paper contributes to the robustness literature in CVQC by targeting implementation-layer resilience under structured and potentially adversarial disturbances expressed in terms of receiver-observable statistics under finite data and hardware constraints [6,8,12,13]. To avoid ambiguity, we emphasize that the framework is adaptive in principle, whereas the validated instantiation in this manuscript is an offline phase-grid search/selection proof-of-concept (no real-time feedback, no dynamic squeezing).

5.2.1. Relation to CV-QKD Robustness and Finite-Size Security Analyses

In CV-QKD, operational reliability and security margins depend on receiver-estimated quadrature statistics and finite-size parameter estimation [1,2,23,24,25]. Accordingly, the most direct way to quantify the protocol-level impact of phase-first selection is through workflows that measure (or reliably infer) the quadrature-level quantities entering excess-noise bounds. Our simulation track speaks directly to this quadrature-level intuition (basis-dependent variance/covariance inflation). By contrast, the X8_01 hardware workflow demonstrated here uses Fock sampling, so the experimental objective is a tomography-light count-standard-deviation proxy; therefore, the contribution is best viewed as establishing that phase-angle scans can reveal non-intuitive and better-performing operating points under a receiver-observable metric in a phase-sensitive effective environment, rather than as directly suppressing a protocol-specific excess-noise parameter.

5.2.2. Relation to Phase Stabilization and Receiver-Interface Attacks

A large body of work addresses phase noise, LO/phase-reference stability, and calibration drift as key practical limitations in coherent CV receivers [11,12,14]. In parallel, security-oriented studies highlight that receiver-interface non-idealities (e.g., saturation/clipping and blinding-like behaviors) can create attack surfaces and operational failure modes under strong stress [3,18,19,20,21]. Phase-first modulation is complementary to these approaches; rather than assuming a single stabilized operating point remains optimal, it treats phase agility as a lightweight containment knob that can be invoked when the receiver-observed channel becomes effectively phase dependent (through anisotropy, mixing, LO/reference coupling, or interface nonlinearities).

5.2.3. Relation to Squeezing- and State-Engineering-Based Robustness

Static squeezing and optimized Gaussian preparation can improve signal-to-noise ratios in well-characterized channels and are central resources in many CV protocols [8,17]. However, these approaches typically assume that the disturbance is sufficiently stationary (and the hardware sufficiently stable) for a fixed operating point to remain near-optimal. Phase-first selection targets a different regime, using a control primitive that is widely available on near-term photonic platforms (phase rotation) to re-orient how a phase-sensitive effective channel projects onto receiver-observable degradation proxies, and also remains meaningful even when squeezing or high-bandwidth feedback is unavailable.

5.2.4. Broader Implications for Quantum Communication and Networks

As quantum networking architectures scale, the operational burden shifts toward maintaining stable receiver inference under calibration drift, nonstationarity, and adversarial stress [9,10]. Phase agility provides a protocol-agnostic physical-layer mechanism that can be composed with stabilization and monitoring. When receiver-observable degradation increases, a (periodic or triggered) phase re-optimization can recover margin without modifying protocol logic. In multi-mode settings, the same principle applies with expanded observables (e.g., covariance block structure under quadrature readout, or mode-resolved count moments/correlations under Fock readout), consistent with the scaling note in Section 2.6.

5.2.5. Broader Implications for Quantum Computing and Bosonic Architectures

Continuous-variable and bosonic encodings store logical information in oscillator modes, making them sensitive to hardware-mediated disturbances that can increase decoding overhead and reduce fault-tolerance margin [26,27,28]. Phase-first modulation is not a substitute for QEC; rather, it can act as a pre-conditioning step that reshapes how disturbances appear in receiver-accessible observables before verification, decoding, or acceptance tests, potentially reducing the effective disturbance burden presented to higher layers.

5.2.6. Cybersecurity Workflow Analogy: Detection Plus Containment

Finally, the results align with a standard cybersecurity lesson: detection alone is insufficient under DoS-like stress; systems need mitigation/containment actions that slow degradation long enough for higher-layer controls to respond. Phase-first selection can be paired with monitoring and anomaly-detection triggers (classical or ML-based) that initiate re-optimization at an appropriate cadence [29,30]. Within the receiver-centric viewpoint emphasized here [6], this supports a practical pipeline: observe receiver-facing degradation, select a phase that reduces a centered degradation proxy, and iterate as required by nonstationarity.

5.3. Limitations

The results in this study establish a hardware-feasible proof-of-concept for phase-only offline phase-first selection under DoS-like receiver-observable stress. The following limitations bound the scope and interpretation of the present investigation.

5.3.1. Hardware-Imposed Control and Readout Constraints (X8_01)

The primary experimental constraints are set by the user-facing capabilities of the X8_01 workflow. Programmable phase operations enable systematic sweeps over $\theta$; however, the demonstrated access mode does not expose (i) dynamic squeezing actuation during a sweep or (ii) an integrated low-latency measurement-to-actuation feedback loop. Accordingly, the experimental validation is restricted to phase-only control instantiated as offline selection via one-dimensional grid/line search. Therefore, simulation-only results that explore joint $(\theta ,r)$ behavior should be interpreted as prospective upper-bound references rather than as experimentally demonstrated gains.

5.3.2. Offline (Not Real-Time) Adaptation and Stationarity Requirements

Although the phase-first framework is adaptive in principle, the validated instantiation does not implement real-time closed-loop updates. Offline phase selection is meaningful when the effective mapping $\theta \mapsto \mathcal{J}\left(\theta \right)$ is approximately stationary over the acquisition window used to evaluate the offline phase-grid evaluation. If the effective disturbance/measurement behavior drifts on timescales comparable to or shorter than the sweep, then the selected ${\theta }^{*}$ can become stale; this motivates periodic re-optimization, interleaved (round-robin) sampling across angles, or closed-loop tracking in future architectures. Such extensions would introduce standard estimation–control tradeoffs (shot budget, latency, estimator variance at shorter windows) that are not optimized here.

Therefore, a concrete next-step roadmap is first to repeat the one-dimensional hardware phase-angle scan with interleaved angles and bootstrap uncertainty bands, second to use those results in calibrating a coarse-to-fine search policy that reduces K while preserving the minimum, and third to extend the simulation track to structured multi-parameter controls $(\theta ,r)$ or low-dimensional multi-mode phase patterns before attempting a higher-bandwidth hardware implementation.

5.3.3. Experimental Observable Scope (Aggregated Scalar Hardware Proxy)

The hardware study is constrained by the native readout, namely, Fock sampling (MeasureFock). Therefore, we report a tomography-light scalar proxy, ${\sigma }_N\left(\theta \right)=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)}$, rather than reconstructing mode-resolved quadrature covariances or a full $2m\times 2m$ covariance matrix. In the reported X8_01 workflow, the shot-level count $N^{\left(s\right)}\left(\theta \right)$ is formed by aggregating the returned Fock counts before postprocessing. As a result, the present proof-of-concept demonstrates that phase-angle scans can reveal non-intuitive settings that reduce a directly observed receiver-side fluctuation proxy under fixed stress, but does not diagnose mode-resolved structure or correlated pathways. Mode-resolved extensions are left to future work.

5.3.4. Metric Scope (Implementation-Layer Resilience, Not a Protocol Security Proof)

This work is framed explicitly as an implementation-layer resilience study, not as a protocol-level security proof. Reported improvements correspond to reductions in centered receiver-observable fluctuation proxies under the platform’s effective stressed configuration (quadrature-variance proxies in simulation, photon-count dispersion in hardware) and should not be interpreted as universal noise cancellation or guaranteed improvements in protocol-level figures of merit. Mapping phase-first gains to protocol-specific quantities (e.g., CV-QKD excess-noise bounds, abort probability, and secret-key-rate impact under finite-size effects) would require a workflow that measures (or reliably infers) the relevant quadrature-level parameters and specifies the protocol, SNU conventions, reconciliation efficiency, and threat model, which is outside the scope of the present hardware proof-of-concept.

5.3.5. Dependence on Platform-Specific Effective Channels

The existence, location, and magnitude of improvement at ${\theta }^{*}$ are device- and condition-dependent, since they reflect the effective disturbance/measurement channel at the receiver interface (e.g., anisotropy, quadrature/mode mixing, LO/phase-reference coupling, calibration drift, or interface nonlinearities). While the phase-agility principle is expected to transfer to other photonic CV platforms with programmable phase control, quantitative gains should be re-estimated per platform and operating regime.

These limitations clarify what is demonstrated here (offline phase-only containment using receiver-observable centered proxies under strong stress) and what requires additional measurement primitives and control bandwidth in future platforms.

5.4. Future Directions

The present work establishes a hardware-feasible baseline for phase-first mitigation (offline, phase-only) under receiver-observable stress. Several follow-on directions can strengthen both the scientific interpretation and the deployment relevance of the approach:

1.

Multi-mode and networked validation with richer observables. A priority next step is to extend phase-first selection beyond the current proof-of-concept by evaluating mode-resolved and correlation-sensitive statistics in multi-mode settings (cf. Section 2.6). When quadrature monitoring is available, this includes tracking not only trace-based inflation summaries but also selected cross-covariance blocks or low-rank structure in the full 2 m × 2 m covariance. When the native readout is photon-number sampling, this includes mode-wise $Var\left(n_i\right)$ and cross-correlations $\mathrm{Cov}(n_i,n_j)$ in addition to the global total-count proxy. These extensions help to distinguish “global inflation” from coupling-dominated failure modes.

2.

From offline sweeps to tracking under nonstationarity. Because offline phase selection relies on approximate stationarity over the acquisition window (Section 2.5.2), an important direction is to study update policies for $\theta \left(t\right)$ when the effective channel drifts or is adversarially nonstationary. Practical candidates include (i) periodic re-optimization on a moving window, (ii) interleaved (round-robin) sampling of a small set of candidate phases to reduce drift bias, and (iii) maintaining a safe default (or randomized) phase schedule when uncertainty is high, with opportunistic re-optimization when sufficient fresh statistics are available.

3.

Sample-efficient phase selection and uncertainty-aware decisions. The demonstrated procedure uses a uniform grid sweep; future work should quantify the tradeoff between angular resolution, shot budget, and confidence in identifying a near-optimal phase (cf. Section 2.5.3). Concrete improvements include coarse-to-fine sweeps, local interpolation/model fits around the empirical minimum, and decision rules that select among statistically indistinguishable candidates using robustness margins (e.g., preferring flatter minima or lower run-to-run variability). Reporting bootstrap confidence intervals on $\mathcal{J}\left(\theta \right)$ and on ${\theta }^{*}$ should become standard in repeated-run studies.

4.

Closing the gap to joint Gaussian control $(\theta ,r)$ under hardware realism. Since squeezing is not actuated in the demonstrated hardware workflow, a key research direction is to quantify when the additional complexity of squeezing becomes cost-effective as a resilience resource. This requires evaluating joint $(\theta ,r)$ control under realistic constraints: finite squeezing, loss, bounded control bandwidth, imperfect phase reference, and platform-specific compilation effects. The goal is to characterize the attainable improvement beyond phase-only operation and identify regimes in which phase-only control is already near-optimal.

5.

Protocol-layer integration and end-to-end impact (e.g., CV-QKD). While this manuscript is explicitly an implementation-layer resilience study and not a security proof, an important next step is to connect phase-first selection to protocol-level quantities. For CV-QKD-like pipelines, this means evaluating how phase selection changes the quadrature statistics used in parameter estimation (finite-size bounds, inferred excess noise) and how that propagates to abort probability, reconciliation efficiency, and achievable key rates. When the experimental workflow does not expose quadrature readout (as in the present X8_01 access mode), additional modeling/validation is needed to relate the measured hardware proxy to the protocol’s quadrature-level estimators.

6.

Monitoring, triggers, and containment workflows. Phase-first mitigation is most useful when embedded in a monitoring-and-response loop. Future work should define explicit trigger logic (thresholds or learned detectors on receiver-observable features) that decides when to (i) re-optimize $\theta$, (ii) switch to conservative operating modes, or (iii) abort/reinitialize under severe DoS-like stress. This enables a containment-oriented workflow rather than detection-only responses, and provides a clear operational interface between physical-layer mitigation and higher-layer controls.

7.

Cross-platform benchmarking and generality beyond a single device family. Broader validation across photonic CV platforms with different phase-control resolution, detection chains, and reference implementations is needed in order to separate platform-specific effects from general phase-agility phenomena. A useful deliverable would be a benchmark protocol that reports (i) the observed $\theta$-dependence of a centered proxy, (ii) the stability of ${\theta }^{*}$ across repeats and over time, and (iii) the sensitivity to phase-setting error and shot budget.

8.

Reproducibility and artifact hygiene. To support replication and longitudinal comparisons, future releases should include pinned software versions, fixed random seeds where applicable, complete sweep metadata (grid, shots, timestamps), and well-documented postprocessing scripts. Where possible, sharing anonymized raw samples or summary statistics (subject to platform constraints) will strengthen the evidentiary value of future studies.

Together, these directions move the present proof-of-concept towards an end-to-end resilience capability, from offline identification of beneficial phase settings under receiver-observable stress to robust uncertainty-aware tracking and protocol-layer impact quantification in realistic and potentially adversarial operating environments.

6. Conclusions

This paper re-frames resilience in continuous-variable quantum communication (CVQC) through a receiver-observable lens: under finite data, finite resolution, and realistic hardware interfaces, robustness is governed by what the receiver can reliably estimate and control, rather than by idealized channel models alone. Building on the receiver-centric threat-modeling viewpoint in [6], we introduce and evaluate phase-first Gaussian modulation as a lightweight implementation-layer containment mechanism under DoS-like receiver-observable stress.

The core contribution is an operational phase-selection rule: choose a phase-space rotation angle $\theta$ that minimizes a centered receiver-observable second-moment degradation proxy. Because different platforms expose different native observables, we evaluate this idea using two complementary tracks (Section 2.3). In theory/simulation, we use mean-subtracted quadrature-level second moments (notably, the basis-dependent variance $Var\left({\widehat{X}}_{\theta }\right)$) to illustrate the underlying mechanism as a projection/orientation effect under structured directional disturbances. In the X8_01 workflow, where the native readout is Fock sampling, we quantify phase sensitivity using the tomography-light count-standard-deviation proxy ${\sigma }_N\left(\theta \right)=\sqrt{\widehat{Var}\left(N\left(\theta \right)\right)}$ computed from the shot-level detected count used in the hardware workflow. In the reported experiment, this shot-level count is formed by aggregating the returned Fock counts before postprocessing. In this hardware setting, we demonstrate that an offline one-dimensional phase sweep can identify non-intuitive phase settings that reduce a directly observed receiver-side fluctuation proxy relative to static reference angles, establishing feasibility of phase-first containment under current control and readout constraints.

A central interpretive point, made explicit in Section 2.4, is that under ideal phase-independent additive-noise models, a pure rotation leaves rotation-invariant summaries unchanged, while under ideal photon-number measurement, a terminal phase gate does not change photon-number statistics. Therefore, any empirically observed $\theta$-dependence of the reported proxies is interpreted operationally as evidence of a phase-dependent effective disturbance/measurement channel at the receiver interface (e.g., anisotropy, mixing, LO/phase-reference coupling, calibration dynamics, or interface nonlinearities) rather than as intrinsic “noise cancellation” from rotation alone. Within this receiver-centric framing, the existence of a beneficial ${\theta }^{*}$ is itself deployment-relevant, indicating that phase agility can serve as a practical mitigation knob in phase-sensitive effective environments.

Overall, our results support phase-first modulation as a near-term hardware-compatible component of resilient CVQC operation that requires only low-dimensional tuning (phase rotation), can be implemented as offline selection when feedback bandwidth is limited, and can be composed with monitoring and higher-layer control logic. The most important next steps are to (i) strengthen multi-mode and correlation-aware evaluation with richer observables, (ii) quantify robustness and uncertainty of ${\theta }^{*}$ under drift and finite-shot effects, and (iii) connect phase-first selection to protocol-level quantities (e.g., quadrature parameter estimation in CV-QKD) when the relevant measurement primitives are available. Therefore, the present hardware evidence should be interpreted as demonstrating offline phase-first containment under quasi-static stressed conditions in a phase-sensitive effective receiver interface, rather than as a complete real-time resilience guarantee under fast drift or rapidly adaptive adversarial dynamics. The most important experimental next steps are to benchmark offline search latency, repeat the hardware phase-angle scan across sessions with explicit reference-state/recalibration controls, and place uncertainty bounds on ${\theta }^{*}$ under finite-shot and drift-limited conditions.