# Quantum Sensing of Noisy and Complex Systems under Dynamical Control

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## Abstract

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## 1. Introduction

## 2. Bath-Optimized Task-Oriented Control (BOTOC)

#### 2.1. General Formulation of BOTOC

**Γ**with elements

**Γ**to define the probe spectral matrix, dubbed the “filter” functional

**Γ**, depending on whether our control is meant to maximize or minimize P [34,35].

#### 2.2. Dynamical Control of Qubit Dephasing (Decoherence)

#### 2.3. Modulation Forms

#### 2.4. Optimal Decoherence Control of a Qubit

## 3. Control for Bath Diagnostics

## 4. Maximized Information on Bath’s Parameters by Dynamical Control

## 5. Dynamical Control of Quantum Transfer as Means of Diagnosing a Bath in Hybrid Systems

#### Optimized Transfer from Noisy to Quiet Qubits

## 6. Dynamical Control of State Transfer via Noisy Quantum Channels and the Implications for Sensing Diagnostics

#### 6.1. Boundary Qubit Probes and State Transfer through a Quantum Channel

#### 6.2. Optimized Filter Design for Channel Diagnostics

#### 6.3. Diagnosing Noise Effects in Quantum Channels

- (i)
- Dynamical boundary-control can make the channel most robust against static noise, because it filters out the bath-energies that damage the transfer. In Figure 3c $p=2$ control is shown to be advantageous compared with the static control case $p=0$, at the expense of increasing the transfer time by only a factor of 2.
- (ii)
- Modulation is not helpful against Markovian noise. Remarkably, arbitrarily high fidelities can then be achieved by slowing down the transfer time, i.e., by decreasing ${V}_{0}$, because in a Markovian bath, the fast coupling fluctuations suppress disorder-localization effects that hamper the transfer fidelity.
- (iii)
- For non-Markovian fluctuating noise ${V}_{i}+{V}_{i}{\Delta}_{i}\left(t\right),$ that randomly varies with correlation time $\tau $ (Figure 3b), in contrast to Markovian noise with $\tau \to 0$, optimized dynamical control can strongly reduce the infidelity that lies between the static and Markovian limits, provided the bath-spectrum is gapped.

## 7. Dynamical Control of Multipartite Probes for Bath Diagnostics

#### 7.1. Multipartite Decoherence Matrix

**,**where

**I**is the identity operator), becomes a $\delta $-function centered at the qubit energy separation, ${\omega}_{a}$, resulting in a coupling between the qubit probe and a particular mode of the bath, with coupling strength $G\left({\omega}_{a}\right)$, in accordance with Fermi’s Golden Rule.

#### 7.2. Multiqubit Probe Modulations for Reconstructing Coupling Spectra

## 8. Conclusions

- Local modulation can effectively decorrelate the different dephasings of the multiple qubits, i.e., eliminate their cross-decoherence, resulting in their equal dephasing rates. For two qubits, the singlet and triplet Bell-states acquire the same dynamically modified decoherence rate.
- For different couplings to a bath, one can better preserve any initial state by local modulation, which can reduce the mixing with other states, than by global modulation. Local modulation which eliminates the cross-decoherence terms, increases the fidelity more than the global modulation alternative. For two qubits, local modulation better preserves an initial Bell-state, whether a singlet or a triplet, compared to global π-phase “parity kicks.”

#### 8.1. Comparison of Bath-Optimized Task-Oriented Control (BOTOC) to Dynamical Decoupling (DD)

- (i)
- BOTOC relaxes the DD assumption that the control fields must be either very short or very strong. In our formalism, the control fields are considered concurrently with the coupling to the bath, hence allowing a much wider variety of pulse sequences, ranging from continuous modulation all the way to DD sequences.
- (ii)
- Dynamical decoupling suggests using the same pulse sequence (be it periodic, optimized or concatenated), regardless of the shape of the bath spectrum. By contrast, BOTOC explicitly considers the bath spectrum and allows optimal tailoring of the modulation to a given bath spectrum. In many cases, the standard π-phase “bang-bang” is then found to be inadequate or non-optimal compared to dynamic control based on the optimization of the universal formula. Whereas our BOTOC approach reduces to the DD method in the particular case of proper dephasing or decay via coupling to spectrally symmetric (e.g., Lorentzian or Gaussian) noise baths with limited spectral width, phase modulation advocated for the suppression of coupling to baths with frequency cutoff or other non-monotonic spectra is, however, drastically different from the DD method which may fail for multipeak spectra.
- (iii)
- Our BOTOC universal strategy has far broader applicability than DD: It can simultaneously control, unlike DD [10,11], both decay and decoherence (proper dephasing) by either pulsed or continuous wave (CW) modulation of the system-bath coupling governed by a simple universal formula. BOTOC has been generalized by us to finite temperatures and to qubits driven by an arbitrary time-dependent field, which may cause the failure of the rotating-wave approximation [11]. It has also been extended to the analysis of multi-level systems, where quantum interference between the levels may either inhibit or accelerate the decay [19].
- (iv)
- Even if DD is adequate for independently decohering qubits, its extension to correlated multipartite systems is highly nontrivial. By contrast, BOTOC modulations with low energy decorrelate the different proper dephasings of the multiple two-level systems (TLS), resulting in equal dephasing rates for all states. For two TLS, we have shown that the singlet and triplet Bell-states acquire the same dynamically modified dephasing rate. This should be beneficial compared to standard DD based on global “bang-bang” (π-phase flips) if both the triplet and the singlet states are used (intermittently) for information transmission or storage.

#### 8.2. Open Issues—Outlook

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) Probing bath-induced decoherence by a system probe that is subject to control. Key bath parameters that define the bath spectrum $G\left(\omega \right)$, such as the correlation time τ and the bath-coupling coupling g, are estimated from measurements of the probe in the (anti-)symmetric superposition states. Spectral Filters ${F}_{t}\left(\omega \right)$ designed by: (

**a**) Free evolution (blue sinc filter function), and Car-Purcell-Meiboom-Gill (CPMG) sequence control (green narrow band pass filter); (

**b**) Modulation with small phase shifts, ϕ ≪ 1; (

**c**) π-pulses (ϕ = π) modulation; (

**d**) Quasi-periodic modulation.

**Figure 2.**(

**a**) Schematic view of the quantum state transfer (QST) process from a “noisy”qubit probe, due to the interaction with a bath, to the “quiet” qubit that may serve as robust storage of the information. Controlling the interaction $V\left(t\right)$ allows optimized QST; (

**b**) Lowest achievable averaged infidelity as a function of the transfer time T normalized to the infidelity at the fastest transfer time ${T}_{min}$ at a given transfer energy, plotted for (solid blue line) $\tau =0$ (Markovian) and (dashed red line) non-Markovian baths $\tau /{T}_{min}=10$. For non-Markovian baths ($\tau \gtrsim {T}_{min}$) two plateaux (regions of insensitivity to $T$) are found. The first plateau is independent of the memory time. The second plateau is lower for longer $\tau $.

**Figure 3.**(

**a**) Top: Controlled qubit probes coupled though a symmetric noisy quantum channel. In frequency space, the probes are resonantly coupled by a fermionic-mode of the channel whose energy is defined to be the energy origin (zero). These 3 degrees of freedom constitute the effective system (green rectangular bars); while the remaining channel modes are taken to be the bath (red lines) with a gapped spectrum ${G}_{\pm}\left(\omega \right)$. Under noise, the perturbed energies change/fluctuate bounded by the (colored gapped) Wigner-semicircle. In the gap, the optimal spectral-filters for the high fidelity state-transfer ${F}_{T,-}\left(\omega \right)$ correspond to dynamical boundary-control with $V\left(t\right)$: $p=0$ (black dotted), $p=2$ (green); (

**b**) Transfer infidelity as a function of the transfer time T for the gapped Wigner-semicircle spectral bath: static control $p=0$ (black dotted) and dynamic control $p=2$ (green); (

**c**) Averaged transfer infidelity $1-\overline{F}\left(\mathit{T}\right)$ as a function of the perturbation strength ${\xi}_{V}$ of the noisy channel. Dynamical $p=2$ control (empty squares) strongly reduced the infidelity obtained under static control $p=0$ (empty circles). A fluctuating noisy channel results in less damage than the static-noisy channel $\mathbf{\tau}\to \infty $, where its fidelity can even approach its unperturbed value in the Markovian limit $\tau \to 0$ ($p=0$, green solid circles) . Here N = 29 channel-qubits and ${V}_{i}\equiv 1$.

**Figure 4.**Overlap of coupling spectrum ${G}_{ij}\left(\omega \right)$, and modulation matrix elements ${F}_{T,ij}\left(\omega \right)=\frac{1}{t}{\epsilon}_{T,i}^{*}\left(\omega \right){\epsilon}_{T,j}\left(\omega \right)$, resulting in modified decoherence matrix elements (overlap), for: (

**a**) cross-decoherence elimination (IIP symmetry); and (

**b**) global modulation (ICP symmetry); (

**c**) Two qubits comprising a probe in a cavity. The qubits are coupled to the cavity modes (thin lines) and subject to local control fields (green lines); (

**d**) Fidelity as a function of the probed time T of the IIP symmetry for the initial entangled two-qubit probe state $|\psi \left(0\right)\rangle =\frac{1}{\sqrt{2}}\left(|\downarrow \uparrow \rangle \pm |\uparrow \downarrow \rangle \right)$, coupled to a zero-temperature bath. The overall fidelity $F\left(T\right)={F}_{p}\left(T\right){F}_{c}\left(T\right)$ is in terms of the correlation preservation ${F}_{c}\left(T\right)$, and the population preservation ${F}_{p}\left(T\right)$. Two qubits comprising a probe in a cavity.

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**MDPI and ACS Style**

Kurizki, G.; Alvarez, G.A.; Zwick, A.
Quantum Sensing of Noisy and Complex Systems under Dynamical Control. *Technologies* **2017**, *5*, 1.
https://doi.org/10.3390/technologies5010001

**AMA Style**

Kurizki G, Alvarez GA, Zwick A.
Quantum Sensing of Noisy and Complex Systems under Dynamical Control. *Technologies*. 2017; 5(1):1.
https://doi.org/10.3390/technologies5010001

**Chicago/Turabian Style**

Kurizki, Gershon, Gonzalo A. Alvarez, and Analia Zwick.
2017. "Quantum Sensing of Noisy and Complex Systems under Dynamical Control" *Technologies* 5, no. 1: 1.
https://doi.org/10.3390/technologies5010001