Quantum Error Mitigation in Optimized Circuits for Particle-Density Correlations in Real-Time Dynamics of the Schwinger Model

Abstract

Quantum computing gives direct access to the study of the real-time dynamics of quantum many-body systems. In principle, it is possible to directly calculate non-equal-time correlation functions, from which one can detect interesting phenomena, such as the presence of quantum scars or dynamical quantum phase transitions. In practice, these calculations are strongly affected by noise, due to the complexity of the required quantum circuits. As a testbed for the evaluation of the real-time evolution of observables and correlations, the dynamics of the ${\mathbb{Z}}_n$ Schwinger model in a one-dimensional lattice is considered. To control the computational cost, we adopt a quantum–classical strategy that reduces the dimensionality of the system by restricting the dynamics to the Dirac vacuum sector and optimizes the embedding into a qubit model by minimizing the number of three-qubit gates. The time evolution of particle-density operators in a non-equilibrium quench protocol is both simulated in a bare noisy condition and implemented on a physical IBM quantum device. In either case, the convergence towards a maximally mixed state is targeted by means of different error mitigation techniques. The evaluation of the particle-density correlation shows a well-performing post-processing error mitigation for properly chosen coupling regimes.

1. Introduction

Investigations of lattice gauge theories constitute an interface among fundamental physics, the characterization of quantum many-body systems, and quantum computing implementations by means of properly engineered coding strategies [1,2]. The availability of noisy intermediate-scale quantum (NISQ) devices in cloud access platforms opens interesting possibilities from both academic and industrial perspectives. Nonetheless, the operating quantum systems are critically affected by noise, thus preventing current NISQ devices from actually outperforming classical computing capabilities [3]. This statement holds true in a diversified way in the cases of different hardware setups engineered for quantum computing purposes, each characterized by different advantages in terms of gate fidelity and experimental realization.

As a benchmark, in this article, we concentrate on the lattice version of quantum electrodynamics (QED) in one dimension, the so-called Schwinger model [4]. A rich variety of phenomena emerge, driven by the intrinsic nature of lattice QED as a kinetically constrained model, due to the existence of conditions that limit the space of physical states [5,6,7]. This characterization leads to the observation of quantum scarring, preserved through linear gauge protection [8,9,10,11] and detected by means of out-of-time-ordered correlators [12]. Other relevant observables are represented by non-equal-time correlation functions, used to detect non-analyticities that point out dynamical quantum phase transitions (DQPTs) in quantum quenches [13,14,15,16,17,18,19,20]. In this context, particle density is specifically relevant, as it can represent the most intuitive quantity to describe in the continuum limit, the decay of an initial Dirac vacuum state after a quench [21,22].

A first attempt to simulate the dynamics of one-dimensional QED was implemented with ion traps [21,23], targeting the observation of pair production starting from the Dirac vacuum. Further developments adopted quantum–classical algorithms in superconducting circuits, based on embedding the dynamics in a specific vacuum sector [24]. Larger system sizes were investigated through analog simulators in optical lattices that host ultracold atoms [25,26]. In setups accessible via cloud platforms, noise that affects computation [27,28,29,30,31,32,33,34] can limit the observation of targeted phenomena [35,36], and error mitigation techniques must be implemented to recover physically meaningful results [37,38,39,40,41,42,43,44].

Our investigation directly targets the possibility to recover the expectation values and correlation functions of physical quantities in spite of a fast convergence towards a maximally mixed state, when working in digital mode. To face the increased experimental complexity measured in terms of circuit depth, we propose an algorithm that makes use of a classical–quantum procedure to optimize the embedding of our model into a qubit system. Specific implementations of Trotter evolution are further used to evaluate particle-density correlation functions for the Schwinger model, based on specifically engineered circuits exploiting an ancilla qubit as proposed for general retarded Green functions [45,46,47]. This lattice QED use case scenario constitutes a properly suited test for error mitigation strategies [48,49] and the related post-processing capabilities in managing noise to evaluate physically meaningful quantities. A non-equilibrium quench protocol is described to study time evolutions of the initial Dirac vacuum characterized by pair production, signaling the emergence of a new ground state, eventually leading to a new phase for a proper choice of parameters [50]. The study of lattice size scaling is beyond the scope of this article, while we focus on possible methodological and circuit tools that are useful in NISQ devices in the characterization of non-equilibrium dynamics, including critical conditions which are hardly captured experimentally [35].

The content is as follows. In Section 2, we give a brief introduction to the Schwinger model on a one-dimensional lattice and with a discretized gauge symmetry group, which is mapped into a quantum spin system via a Jordan–Wigner transformation. In Section 3, we adopt a quantum–classical embedding of the considered dynamics by first restricting the dynamics in the Dirac vacuum sector by means of translation and charge conjugation symmetries and then choosing the optimal permutation of states that minimizes the number of three-qubit interactions in the Hamiltonian. Then, in Section 4, we describe the observables we are interested in, namely particle-density operators and their correlation functions, and derive the digital circuits that implement their real-time evolution. Section 5 presents our results, run for a lattice composed by $N=4$ sites, comparing exact evolution, bare noisy simulations and simulations with error mitigation techniques, such as twirled readout error extinction (T-REx) [48] and zero noise extrapolation (ZNE) [49]. For the evolution of the particle-density operators, we also perform runs on a physical IBM device. Finally, we draw our conclusions in Section 6 and collect some details of our calculations in Appendix A, Appendix B, Appendix C, Appendix D, Appendix E and Appendix F.

2. Lattice QED in One Spatial Dimension

Lattice QED in $(1+1)$ dimensions, representing the spatial discretization of the Schwinger model, is a prototypical case of a gauge theory formulated in reduced spatial dimensionality. In this model, gauge degrees of freedom consist of a single (longitudinal) component of the electric field, interacting with spinless fermions of mass m and charge g: canonically anticommuting operators ${\psi }_x,{\psi }_x^{†}$ that represent the matter field live on sites x, while each edge with length a connecting sites x and $x+1$ hosts the electric field operator $E_{x,x+1}$ and the gauge connection $U_{x,x+1}={\mathrm{e}}^{\mathrm{i}a{A}_{x,x+1}}$, where $A_{x,x+1}$ is the vector potential, canonically commuting with the electric field. Since we consider a finite lattice with N sites, labeled by $x\in \{0,\dots ,N-1\}$, we require periodic boundary conditions and identify the fields corresponding to the index $x=N$ to those at the $x=0$ boundary. The system evolution is determined by the Hamiltonian

$$\mathcal{H}=-{\displaystyle \frac{\mathrm{i}J}{2}}\sum _{x=0}^{N-1}\left({\psi }_x^{†}{U}_{x,x+1}{\psi }_{x+1}-H.c.\right)+m\sum _{x=0}^{N-1}{(-1)}^x{\psi }_x^{†}{\psi }_x+{\displaystyle \frac{g^2}{2J}}\sum _{x=0}^{N-1}{E}_{x,x+1}^2.$$

(1)

The first term represents the nearest-neighbor hopping of fermions accompanied by the corresponding transformation of the electric field on the involved lattice edge [21,23,35] with $J=a^{-1}$, the second term is the staggered (Kogut–Susskind) mass term, which is able to solve the doubling problem associated to lattice discretization [51], while the last term gives the standard electric contribution to energy. The Dirac vacuum state, which is realized by creating one fermion in each of the negative-mass (odd-x) sites, coincides with the ground state in the limit of infinite mass [16,17]. We adopt a quench protocol based on an initial Dirac vacuum with a time evolution generated by a Hamiltonian with finite mass such that the initial state is no longer a ground state [13,14,15,16,17,18,19,20]. We notice that Hamiltonian (1) is invariant under charge conjugation and translation by two lattice sites, which, for a lattice with even number N of sites [50], read as

$$\begin{array}{cc}\hfill {\mathcal{C}}_{+}=& \left\{\begin{array}{cc}{\psi }_x\to {(-1)}^{x+1}{\psi }_{x+1}^{†},\hfill & {\psi }_x^{†}\to {(-1)}^{x+1}{\psi }_{x+1},\hfill \\ E_{x,x+1}\to -E_{x+1,x+2},\hfill & U_{x,x+1}\to U_{x+1,x+2}^{†},\hfill \end{array}\right.\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\mathcal{T}}_2=& \left\{\begin{array}{cc}{\psi }_x\to {\psi }_{x+2},\hfill & {\psi }_x^{†}\to {\psi }_{x+2}^{†},\hfill \\ E_{x,x+1}\to E_{x+2,x+3},\hfill & U_{x,x+1}\to U_{x+2,x+3},\hfill \end{array}\right.\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \lambda \left({\mathcal{C}}_{+}\right)=& \{+1,-1,+\mathrm{i},-\mathrm{i}\},\lambda \left({\mathcal{T}}_2\right)=\{+1,-1\},\hfill \end{array}$$

(4)

where $\lambda$ denotes the spectrum, ${\mathcal{C}}_{+}$ stands for a transformation translating matter and electric fields of a lattice spacing (from site x to site $x+1$), transforming particles into antiparticles and vice versa, while changing the sign to the electric field. ${\mathcal{C}}_{-}$ acts in the opposite direction. We also set ${\mathcal{C}}_{-}={\mathcal{C}}_{+}^{†}$ with translation by two lattice sites ${\mathcal{T}}_2={\mathcal{C}}_{+}^2$.

A further discretization, involving the local gauge degrees of freedom, consists of replacing the original $U\left(1\right)$ gauge group with the finite cyclic group ${\mathbb{Z}}_n$. Thus, the Hilbert space associated to a given link becomes n-dimensional [52]. In this case, we have the following [50,52]:

• A convenient basis of the edge space is represented by the electric field eigenstates $\left\{|e_k\rangle \right\}$, satisfying $E_{x,x+1}|e_k\rangle =e_k|e_k\rangle$, with $e_k=\sqrt{{\displaystyle \frac{2\pi }{n}}}\left(k-{\displaystyle \frac{n-1}{2}}\right)$ and $k=0,\dots ,n-1$;
• The gauge connection $U_{x,x+1}$ acts on this basis as a cyclic permutation: $U_{x,x+1}|e_k\rangle =|e_{k+1}\rangle$ and $U_{x,x+1}|e_{n-1}\rangle =|e_0\rangle$;
• The lattice counterpart of the Gauss law $G_x|\varphi \rangle =0$ which fixes the admissible states $|\varphi \rangle$ is given by

  $$G_x=\sqrt{{\displaystyle \frac{n}{2\pi }}}\left(E_{x,x+1}-E_{x-1,x}\right)-{\psi }_x^{†}{\psi }_x-{\displaystyle \frac{{(-1)}^x-1}{2}},$$

  (5)

  which must be valid at all sites x and at any time.

To take on quantum computation on the model Hamiltonian (1), it is convenient to perform the Jordan–Wigner transformation

$${\psi }_x={\sigma }_x^{+}\prod _{\ell =0}^{x-1}(-\mathrm{i}{Z}_{\ell }),\mathrm{with}{\sigma }_x^{\pm }={\displaystyle \frac{X_x\pm \mathrm{i}{Y}_x}{2}},$$

(6)

where $X_x$, $Y_x$ and $Z_x$ are Pauli matrices acting on the state of site x in the fermion occupation number basis. The transformation maps the matter field into a collection of N spins (qubits), with the advantage of working with operators that commute at different sites. We also rescale the transformed Hamiltonian [19,20,21,23,24,35] by setting $\xi =J^2/g^2$, $\mu =mJ/g^2$ to finally obtain the Hamiltonian rewritten as

$$\mathcal{H}(\xi ,\mu )=\xi \sum _{x=0}^{N-1}\left({\sigma }_x^{-}{U}_{x,x+1}{\sigma }_{x+1}^{+}+H.c.\right)-\mu \sum _{x=0}^{N-1}{(-1)}^x{Z}_x+\sum _{x=0}^{N-1}{E}_{x,x+1}^2,$$

(7)

where for fermionic hopping terms wrapping around the boundary $-\mathrm{i}{\psi }_{N-1}^{†}{U}_{N-1,0}{\psi }_0={\mathrm{i}}^N\left({\prod }_{\ell =0}^{N-1}{Z}_{\ell }\right){\sigma }_{N-1}^{-}{U}_{N-1,0}{\sigma }_0^{+}$, with the same sign of ${\mathrm{i}}^N$ and ${\prod }_{\ell =0}^{N-1}{Z}_{\ell }$ for any N at half filling.

In the following, we consider a lattice composed by $N=4$ sites with two particles. We also discretize the unitary group with ${\mathbb{Z}}_3$. Appendix A contains a description for the gauge-invariant subspace and the orbits of the charge conjugation and translation operators for this case, which identify the basis required to explicitly highlight the Hilbert space sectors.

3. Classical–Quantum Embedding

In this section, we present the algorithm we use to simulate the dynamics of the model described by the Hamiltonian (7). To have an efficient protocol, it is necessary to reduce the number of quantum resources needed for the simulation. We do so by adopting a two-step scheme: first we restrict the computation to the subspace containing the Dirac vacuum, and then we consider a qubit embedding in which the Hamiltonian contains the smallest number of three-body terms.

To reach the first goal, we split the Hilbert space by considering the translation by two lattice sites and charge conjugation symmetry sectors, which are labeled by their eigenvalues $T_2=+1$ and $C=\pm 1$, $T_2=-1$ and $C=\pm \mathrm{i}$ listed in Equation (4). It is thus possible to partition the physical Hilbert space into four sectors such that the Hamiltonian (7) is expressed as $U\mathcal{H}{U}^{†}={⨁}_{T_2,C}{\mathcal{H}}^{(T_2,C)}$. We exploit the block structure described in Appendix A to focus on the $(+,+)$ sector, containing the initial Dirac vacuum. The restriction of the Hamiltonian to this seven-dimensional subspace reads

$${\mathcal{H}}^{(+,+)}=\left(\begin{array}{ccccccc}-2\mu & \xi & & & & & \\ \xi & {\displaystyle \frac{\pi }{3}}& {\displaystyle \frac{\xi }{\sqrt{2}}}& & & & \\ & {\displaystyle \frac{\xi }{\sqrt{2}}}& 2\mu +{\displaystyle \frac{2\pi }{3}}& {\displaystyle \frac{\xi }{\sqrt{2}}}& & & \\ & & {\displaystyle \frac{\xi }{\sqrt{2}}}& \pi & {\displaystyle \frac{\xi }{\sqrt{2}}}& & \\ & & & {\displaystyle \frac{\xi }{\sqrt{2}}}& -2\mu +{\displaystyle \frac{4\pi }{3}}& {\displaystyle \frac{\xi }{\sqrt{2}}}& \\ & & & & {\displaystyle \frac{\xi }{\sqrt{2}}}& {\displaystyle \frac{4\pi }{3}}& \xi \\ & & & & & \xi & 2\mu +{\displaystyle \frac{4\pi }{3}}\end{array}\right),$$

(8)

according to the ordered basis given in Appendix A.

As for the embedding, we notice that seven states can be described by means of three qubits, the elementary approach relying on adding a last column and row filled with zeros to the Hamiltonian matrix to keep the eighth unphysical state idle. The direct approach consists in identifying the first seven states of a three qubit system with the seven states (A1)–(A7) listed in Appendix A: doing so, the Hamiltonian (8) can be written as

$${\tilde{\mathcal{H}}}^{(+,+)}={\displaystyle \frac{1}{8}}\sum _{i,j,k=0}^3\mathrm{Tr}\left\{{\sigma }_i\otimes {\sigma }_j\otimes {\sigma }_k{\tilde{\mathcal{H}}}^{(+,+)}\right\}{\sigma }_i\otimes {\sigma }_j\otimes {\sigma }_k=\sum _{i,j,k=0}^3{c}_{(i,j,k)}{\sigma }_i\otimes {\sigma }_j\otimes {\sigma }_k,$$

(9)

where ${\sigma }_0$ is the $2\times 2$ identity matrix, and ${\sigma }_j$ with $j\in \{1,2,3\}$ are the Pauli matrices. Here and in the following, we use the tilde notation to emphasize the specific embedding into the three-qubit system. A straightforward but lengthy calculation shows that such a decomposition accounts for 19 non-zero terms, of which 7 correspond to a product of 3 Pauli matrices.

In the implementation of the Trotter evolution for $U\left(t\right)=\exp \left(-\mathrm{i}{\tilde{\mathcal{H}}}^{(+,+)}t\right)$, the latter terms require a higher number of $CNOT$ gates. Thus, in order to reduce the computational time, we aim at finding the embedding in which the Hamiltonian contains the smallest number of three-body terms, i.e., we want to minimize the number of coefficients $c_{(i,j,k)}$ with $i,j,k\in \{1,2,3\}$ by considering all possible permutations $\pi \in S_8$ of the eight basis states. For the chosen problem, we find that the optimal permutation is ${\pi }_o=(7,6,1,2,4,5,8,3)$, leading to a Hamiltonian ${\tilde{\mathcal{H}}}_{{\pi }_o(\ell ),{\pi }_o\left(m\right)}^{(+,+)}$ which contains 15 non-zero terms, of which only 3 are triple products of Pauli matrices gates. In Appendix B, we list all coefficients for both the simple and the optimal embedding in the case we consider here, while in Appendix C, we propose an algorithm to find such an optimal permutation in the general case. Finally, to reduce the usage of $SWAP$ gates, circuit optimization can also rely on a properly chosen topology of superconducting qubit connections [35].

4. Digital Circuit for Unitary Evolution and Green Functions

Once we have obtained the optimal decomposition for the Hamiltonian, the time-evolution operator $U\left(t\right)$ is implemented via a Trotter decomposition as described in Appendix D. Each Trotter step is therefore represented by the circuit shown in Figure 1.

Let us look now at observables of interest. The ones we want to consider are the time evolution of particle density and its associated correlation functions. The formal expression of the former is [21]

$$\nu ={\displaystyle \frac{1}{N}}\sum _x{(-1)}^x{\psi }_x^{†}{\psi }_x,$$

(10)

admitting the sector decomposition $U\nu U^{†}={⨁}_{T_2,C}{\nu }^{(T_2,C)}$, once we restrict it to the physical subspace. In particular, in the $(+,+)$ sector, we have

$${\tilde{\nu }}^{(+,+)}=\sum _{i,j,k=0}^3{d}_{(i,j,k)}{\sigma }_i\otimes {\sigma }_j\otimes {\sigma }_k,$$

(11)

with a number of non-vanishing coefficients that depends on the specific permutation we choose for the embedding. In Appendix B, we list the non-vanishing coefficients for the optimal permutation described above and give the explicit expression (A20) that we use in the following calculations.

The real-time correlation function is defined as

$$G(t,s)=G^{<}(t,s)-G^{>}(t,s)=-\mathrm{i}\mathsf{\Theta }(t-s)\langle \mathrm{vac}\left|\right\{{\tilde{\nu }}^{(+,+)}\left(t\right),{\tilde{\nu }}^{(+,+)}\left(s\right)\left\}\right|\mathrm{vac}\rangle ,$$

(12)

where, for any two operators, $\{A,B\}=AB+BA$ and $\mathsf{\Theta }\left(t\right)$ is the Heaviside step function. We can choose two arbitrary times $s,t$ since the correlation function vanishes for $s=0$ because of ${\tilde{\nu }}^{(+,+)}\left(0\right)|\mathrm{vac}\rangle =0$. $G(t,s)$ is composed of two terms, named the lesser $G^{<}(t,s)$ and the greater term $G^{>}(t,s)$. The former quantifies for an initial time s how long the particle-density property persists in the future, while the latter measures this memory effect backward in time.

Here, we focus on the lesser term $G^{<}(t,s)$ in order to achieve a trade-off between hardware complexity and physical interpretability. To be precise, the main source of error is the sum of individual decoherence effects imposed by the large number of controlled double-qubit gates as presented in Appendix D. Adding the greater term $G^{>}(t,s)$ would greatly affect the mitigating action by introducing too many noise sources, while not adding considerably to the physical insights gained from the simulation.

The lesser term is defined as

$$G^{<}(t,s)=-\mathrm{i}\mathsf{\Theta }(t-s)\langle \mathrm{vac}|{\tilde{\nu }}^{(+,+)}\left(t\right){\tilde{\nu }}^{(+,+)}\left(s\right)|\mathrm{vac}\rangle ,$$

(13)

which, using the definition of $P_{\alpha }$ in Equation (A20) and the explicit value of the coefficient $d_{(0,0,0)}=7/16$, can be rewritten as

$$\begin{array}{cc}\hfill G^{<}(t,s)=-\mathrm{i}\mathsf{\Theta }(t-s)& \left[{\displaystyle \frac{7}{16}}\left(\langle \mathrm{vac}|{\tilde{\nu }}^{(+,+)}\left(t\right)|\mathrm{vac}\rangle +\langle \mathrm{vac}|{\tilde{\nu }}^{(+,+)}\left(s\right)|\mathrm{vac}\rangle \right)-{\displaystyle \frac{49}{256}}\right.\hfill \\ \hfill & \left.+\sum _{\alpha ,\beta =1}^7\langle \mathrm{vac}|U^{†}\left(t\right)P_{\alpha }U(t-s)P_{\beta }U\left(s\right)|\mathrm{vac}\rangle \right].\hfill \end{array}$$

(14)

The first two terms of this expression are the expectation values of the density operator evaluated on the time-evolved vacuum state and can be evaluated using the Trotter decomposition of the evolution operator and the circuit of Figure 1. The last term can be instead calculated via the circuit [46]

(15)

which uses a measurement over an ancilla qubit A (initialized in the state $|0\rangle$). Indeed, denoting with R the register encoding the system (initialized in the vacuum state $|\mathrm{vac}\rangle$) and with ${\varrho }_{\mathrm{out}}$ the output state of the total circuit, it is not difficult to prove that the measurement on the ancilla yields

$${\mathrm{Tr}}_{AR}\left\{(Z\otimes \mathbb{1}){\varrho }_{\mathrm{out}}\right\}=\mathrm{Re}\left\{{\mathrm{e}}^{-\mathrm{i}\varphi }\langle \mathrm{vac}|U^{†}\left(t\right)P_{\alpha }U(t-s)P_{\beta }U\left(s\right)|\mathrm{vac}\rangle \right\}.$$

(16)

In Appendix E, we discuss how this circuit can be realized by means of a Mach–Zender interferometer [53,54].

5. Implementation on IBM Quantum Platform

The algorithm described in the previous section is implemented on the IBM Quantum platform [55], more specifically both on the device ibmq_quito and by means of noise models available in the Python package qiskit (https://pypi.org/project/qiskit/, accessed on 10 April 2025), with error rates updated from the aforementioned device. To limit noise sources in state preparation, gates and measurements, error mitigation techniques are applied in both cases using tools of the package qiskit-ibm-runtime as discussed in Appendix F. In the following, we present the results for the problem of $N=4$ sites, a chosen number of 1000 shots, whose collected statistics impose the shown error bars, and the available LinearExtrapolator for ZNE.

The real-time evolution of the particle density (11) is shown in Figure 2, with the first column showing the noise-simulated data and the second column giving the output of the actual computation on the device. Since the initial state corresponds to the Dirac vacuum, we can interpret the considered simulation as a quenched dynamics starting from the ground state for $\mu \to \infty$ in (7), followed by an evolution generated for finite $\mu$ and $\xi$. We call the strong (weak) coupling regime a condition in the free parameters space $(\xi ,\mu )$ that causes a slowly (rapidly) deviating evolution from the initial Dirac vacuum. The three rows of Figure 2 correspond to three different coupling regimes that go from strong to weak coupling, that is, $(\xi ,\mu )=\left\{\right(0.6,0.1),(1.5,0.5),(4,1\left)\right\}$. In each graph, we compare four curves: the exact noiseless simulation (in blue), the bare results of noisy simulation/real results (in orange), and the results processed via two error mitigation techniques, T-REx (in green) and ZNE (in red).

The ZNE mitigation is characterized by an almost identical curve with the unmitigated case, while the T-REx mitigation works better during the first time steps, where noise is mainly dominated by readout errors. We observe that the behavior of noisy and mitigated curves are more affected in the strong coupling case. However, in all cases, for longer times, the expectation value approximates the maximally mixed condition $\langle {\tilde{\nu }}^{(+,+)}\left(t\right)\rangle ={\displaystyle \frac{7}{16}}$, showing that noise destroys any coherence. Also, for the real device ibmq_quito, the strong and intermediate couplings converge faster towards the maximally mixed condition than the corresponding noise model, probably because of the missing correlated noise in the assumed standalone gate hypothesis [35], which neglects, for example, the noise propagation in subsequent applications of double qubit gates sharing a circuit line. The resilient behavior of the weak coupling regime may be due to the presence of the $\xi$ parameter in all double and triple qubits gates presented in Appendix B.

The real-time correlation function $G^{<}(t,s)$ can be evaluated from Equation (14) only with the aid of the noise models because of the required high circuit depth, implying an execution time longer than the dephasing time of superconducting circuits. To do so, we use the circuit in Equation (15) by collecting measurements for any pair of Pauli string operators $P_{\alpha }{P}_{\beta }$ ($\alpha ,\beta =1,\cdots ,7$) in Equation (14) and by setting the phase $\varphi =0\left({\displaystyle \frac{\pi }{2}}\right)$ to select the imaginary (real) part of lesser contribution. In Figure 3, we collect our results for bare noisy simulations (dashed lines) to be compared to the exact noiseless simulation (solid lines). To interpret the results, we notice that, since particle densities ${\tilde{\nu }}^{(+,+)}\left(t\right)$ and ${\tilde{\nu }}^{(+,+)}\left(s\right)$ are Hermitian observables, they contribute only to the imaginary part of lesser terms, while the 49 terms in Equation (14) related to Pauli strings contribute to both the real and imaginary parts. This might explain the offset between the exact results and bare noise simulations we observe in the imaginary part and not in the real one. Indeed, we can conclude that this offset is caused by the higher deviation between exact and noisy measurements of the density $\langle {\tilde{\nu }}^{(+,+)}\left(t\right)\rangle$ as observed in Figure 2. On the other hand, the 49 Pauli strings pairs, that might contribute with opposite signs, yield a lower deviation. This offset is almost identical for the imaginary part in panels (a,c) for any coupling regime because of the same readout errors and a negligible value of particle density for low values of s. For higher values of s, the deviation decreases in stronger couplings, even if oscillations observed in the noiseless case are damped in the presence of noise.

In Figure 4 and Figure 5, we collect our results for bare noisy simulation data corrected via T-REx and ZNE mitigation, respectively.

As for the former, we notice that real parts of lesser contributions shown in panels (b,d) are not distinguishable from the noiseless case, thus revealing a significant improvement with respect to the unmitigated case. Instead, the imaginary parts in panels (a,c) are almost unchanged because the dominant contribution comes from ${\tilde{\nu }}^{(+,+)}\left(t\right)$ and ${\tilde{\nu }}^{(+,+)}\left(s\right)$, which coincides in the noisy and mitigated scenario. The weak coupling regime oscillation is well captured with respect to its period in the considered time window, but the convergence towards the maximally mixed state causes amplitude dampings already observed in Figure 3.

Similar conclusions can be drawn concerning the real part of lesser contributions corrected by means of ZNE mitigation techniques, as it is shown in panels (b,d) of Figure 5, which show significantly improved behavior. Instead, a non-negligible worsening emerges in the imaginary parts as shown in panels (a,c), with the exception of the weak coupling oscillation in panel (c), which is almost unchanged. These behaviors reveal that the circuit folding procedure used in ZNE and discussed in Appendix F yields a non-trivial noise scaling, beyond the linear one, for the considered evaluation of the correlation function imaginary part. For sufficiently low values assumed by $G^{<}(t,s)$, errors behave as a small perturbation and are efficiently taken under control by mitigation procedures adopting the previously mentioned available LinearExtrapolator. Such effects are instead washed out for higher values of the considered function, with a correspondingly increasing order of magnitude for errors.

In general, the measured offset in imaginary parts is caused by an increasing number of contributions driven by mixed states, as explained by Equation (A26) in Appendix E, which bring additional contributions with respect to the targeted pure initial Dirac vacuum. A more precise description of output states requires an extension in terms of channels corresponding to gates, in order to target the generation of correlated noise [35], but this goes beyond the scope of this paper.

6. Conclusions and Outlook

We considered real-time dynamics of the Schwinger model on a periodic lattice in $(1+1)$ dimensions, implemented on IBM Quantum [55] real devices and examined via different noise models. We proposed a quantum–classical approach to minimize the required computational cost: first we used translation by two lattice sites and charge conjugation symmetries to restrict the dynamics to the sector of the Dirac vacuum, then we reduced the number of triple-qubit gates involved in the Trotter evolution by choosing the optimal permutation of states in the qubits embedding. We examined first the real-time evolution of the particle-density operator in different coupling regimes, showing a resilient behavior with respect to noise in the weak coupling case. In Appendix B, we list the contribution to three qubits gates from the hopping term: in the strong coupling regime, these high-depth Trotter circuits act almost uniquely as a noise source without an effective contribution to the time evolution because of the low value of $\xi$. Then, we calculated the time-dependent correlation functions, which are very much affected by noise. Our results prove that error mitigation works properly for small variations in the measured real parts of lesser Green functions.

Further developments will investigate to what extent decoherence and noise resiliency might affect the analysis of errors occurrence in different coupling regimes [56], by explicitly identifying circuits parts more affected by a specific parameter. Non linear extrapolation is included in the new version of the package qiskit-ibm-runtime, and we will target the exploitation of this error mitigation resource.