 Brief Report
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Simulating \(Z_{2}\) lattice gauge theory with the variational quantum thermalizer
EPJ Quantum Technology volume 11, Article number: 20 (2024)
Abstract
The properties of stronglycoupled lattice gauge theories at finite density as well as in real time have largely eluded firstprinciples studies on the lattice. This is due to the failure of importance sampling for systems with a complex action. An alternative to evade the sign problem is quantum simulation. Although still in its infancy, a lot of progress has been made in devising algorithms to address these problems. In particular, recent efforts have addressed the question of how to produce thermal Gibbs states on a quantum computer. In this study, we apply a variational quantum algorithm to a lowdimensional model which has a local abelian gauge symmetry. We demonstrate how this approach can be applied to obtain information regarding the phase diagram as well as unequaltime correlation functions at nonzero temperature.
1 Introduction
Much interest has been generated recently regarding the potential of quantum computing and quantum simulation to address longstanding problems in highenergy physics [1, 2]. Traditionally, lattice gauge theory has been applied in the Euclidean formulation, where advances in computing power have led to great success in studying properties of stronglyinteracting matter in thermal equilibrium. However, due to the notorious sign problem, there has been a growing interest in the Hamiltonian formulation and its application to address nonzero chemical potential and realtime dynamics. For digital quantum computers in the medium term, one problem that has arisen is how to produce thermal states.
In recent years, there has already been an explosion of progress in tackling this problem. Each approach to producing a thermal state has had to address the issue of how to produce a mixed state. This can typically be done by enlarging the system size and then entangling the ancillary system with the original system. One idea which takes this approach to create a Gibbs state via purification is the socalled thermofield double states method [3]. Another interesting method is known as active cooling [4]. This takes a general initial state, couples it to a thermal heat bath, and uses the concept of the “Maxwell demon” to obtain a thermal state at the desired temperature. Alternatively, as the Boltzmann operator is nonunitary, one can also attempt to directly approximate its action on quantum states. Using judiciously chosen initial states produced by Haarrandom circuits, one can use this approximation of the Boltzmann operator to obtain Gibbs states for spin models [5] as well as gauge theories [6].
One can also study the problem of producing thermal states using variational methods. One such method, the variational quantum thermalizer (VQT), introduced in [7], serves as a natural extension of the variational quantum eigensolver (VQE) [8, 9] to the case of nonzero temperature. The latter uses the variational principle at \(T = 0\) to approximate the groundstate of a Hamiltonian Ĥ by a parametrized ansatz \(\psi _{\boldsymbol{\xi}}\rangle \), subject to
Similarily, for the case of \(T\neq 0\), VQT addresses the problem of finding a variational approximation \(\hat{\rho}_{var}\) to the thermal state, \(\hat{\rho}_{Gibbs} \sim e^{\beta \hat{H}}\) for inverse temperature \(\beta = 1/T\). Similar to its zero temperature analogue in Eq. (1), this problem can also be reformulated as an optimization problem for the free energy with respect to variational parameters ξ, such that
where the second term on the righthand side involves the von Neumann entropy \(S = \mathrm{tr}(\hat{\rho}\log{\hat{\rho}})\). As the density operator appears nonlinearly in the expression for the entropy and thus is not simply determined by a quantummechanical expectation value, several avenues have been taken in the literature: The original formulation of VQT [7] employs classical sampling of the distribution corresponding to \(\rho _{\boldsymbol{\xi}}\) which necessitates a model ansatz. Through this ansatz, the entropy is then given classically by a closedform expression. To prepare the latent distribution corresponding to \(\rho _{var}\) in a quantum circuit, the noiseassisted variational quantum thermalizer [10] (NAVQT) uses (simulated) parameterized depolarization gates that allow for the preparation of a mixed state, again with a closedform, approximate expression for the entropy in terms of the noise level λ, which thus becomes a variational parameter. Of particular interest to us, due to both its simplicity and versatility, is a variant of the VQT introduced in [11], see Fig. 1. While the first two examples used a classical sampling of input states [7] or stochastic mixtures of unitary circuits [10], respectively, to create mixed quantum states, the approach in [11] utilizes intermediate projective measurements. As measurements are nonunitary operations, they act as a source of entropy, ultimately leading to a variational approximation of the thermal state.^{Footnote 1}
The subject of our study is the Hamiltonian formulation of the \(\mathbb{Z}_{2}\) lattice gauge theory (LGT) in \(1+1d\) (\(\mathbb{Z}_{2}^{1+1}\)), in which we investigate the application of the VQT to this theory. This is performed in its formulation with a gaugeredundant Hilbert space [13] and its recent, resourceefficient form derived systematically in [14, 15]. In addition, we investigate several observables of interest at both nonzero temperature T and nonzero chemical potential μ.
2 Background
2.1 Variational quantum thermalizer
As displayed in Fig. 1, starting from an \(n_{q}\)qubit initial state \(\psi _{0}\rangle \) with density matrix \(\rho _{0} = \psi _{0}\rangle \langle \psi _{0} \), we set up a concatenation of two separate variational circuits \(\mathrm{VQC}_{1}\) and \(\mathrm{VQC}_{2}\). The first variational circuit, \(\mathrm{VQC}_{1}\), creates the state \(\psi _{\mathrm{VQC}_{1}}\rangle \), which is given by
An intermediate measurement in the computational basis collapses this state and statistically yields the latent density matrix
The subsequent application of the variational circuit \(\mathrm{VQC}_{2}\) transforms this into
after which a measurement of the energy is performed. Here we defined the probabilities \(p_{i}(\boldsymbol{\theta}) \equiv a_{i}(\boldsymbol{\theta})^{2}\) and the states \(\psi _{i}(\boldsymbol{\phi})\rangle \equiv U_{2}(\boldsymbol{\phi}) i\rangle \). The von Neumann entropy is then calculated as \(S = \sum_{i} p_{i} \log{p_{i}}\), with \(p_{i} = p_{i}(\boldsymbol{\theta}) \approx n_{i}/n_{s}\), where \(n_{i}\) represents the counts for state i in the computational basis (c.f. Equation (4)) and \(n_{s}\) represents the number of shots. By varying the parameter set \(\boldsymbol{\xi}= (\boldsymbol{\theta}, \boldsymbol{\phi})\) in an optimal way, i.e. by minimizing the free energy \(F({\boldsymbol{\xi}})\), Eq. (2), one obtains a variational approximation to the Gibbs state \(\hat{\rho}_{Gibbs} \sim e^{\beta \hat{H}}\). The fact that the circuit is broken up into two parts is intended for noisy intermediatescale quantum (NISQ) devices, as the depth of each variational circuit can be kept relatively shallow. This approach, however, will naturally lead to a limitation in the number of states in Eq. (5), as both the number of shots, \(n_{s}\), and the memory necessary to store the histogram describing the counts scales with the dimension of the Hilbert space of the underlying problem. We pursue this discussion in further detail in Sect. 3.2.4. In [11], the algorithm was demonstrated to reach convergence for the transverse field HeisenbergModel in \(d = 1,2\). Applying qVQT to gauge theories will introduce the need for a gaugeinvariant algorithm implementation. We show below for the case of \(\mathbb{Z}_{2}^{1+1}\) how this can be achieved.
2.2 \(\mathbb{Z}_{2}\) LGT in \(1+1d\)
This section briefly outlines our two approaches to \(\mathbb{Z}_{2}\) lattice gauge theory in the Hamiltonian formulation. In the first approach, introduced in [13] and referred to as the gaugeredundant formulation, local gauge invariance must be enforced on the states to define the physical Hilbert space. This can be achieved, for example, by introducing a penalty term as in [6]. In the second approach, used in [14] and referred to as the resourceefficient formulation, the matter states are decoupled, and the constraint imposed by Gauss’ law is built into the Hamiltonian.
2.2.1 Gaugeredundant formulation
In the socalled group basis, \(\mathbb{Z}_{2}^{1+1}\) LGT with a single flavour of staggered fermion is formally given by the Hamiltonian \(H = H_{E} + H_{GM} + H_{M} + H_{\mu}\) with
In the above equations, the fermionic creation and annihilation operators have been replaced by Pauli matrices living at the sites of the lattice via the usual JordanWigner transformation. Here we have adopted the familiar convention of denoting Pauli operators acting on the link Hilbert space of the gauge bosons by \(\{X,Y,Z\}\), and those acting on the fermionic matter space by \(\{\sigma ^{x}, \sigma ^{y}, \sigma ^{z}\}\). The occurrence of both gaugebosonic and fermionic degrees of freedom implies a gauge redundant Hilbert space. With our choice of basis, a gauge transformation at site n is defined by [16]
We illustrate a gaugeinvariant state \(\psi \rangle \) of the lattice system in Fig. 2, where the fermionic states are in the computational basis, and the gauge bosons are in the Xeigenbasis, \(X\pm\rangle = \pm \pm\rangle \). This subspace of the total Hilbert space is needed to compute physical observables. The particular state shown is known as the strong coupling vacuum state of the theory for \(m >0\) and \(\mu = 0\).
2.2.2 Resourceefficient formulation
It turns out that one can eliminate the fermionic matter fields while simultaneously removing the Hilbert space redundancy of the theory. This is accomplished in a twostep process whereby the fermions are first mapped to hardcore bosons by a unitary transformation and are then decoupled from the gauge bosons by a second unitary transformation [14, 15]. This method is, in fact, a very powerful formalism which can also be applied in arbitrary dimensions to continuous gauge groups such as \(SU(N)\) and \(U(N)\) [17]. For \(\mathbb{Z}_{2}\) LGT in \(1+1\) dimensions, the Hamiltonian in this formulation is as follows
One notices that the Hamiltonian obtained by performing the abovementioned transformations only vaguely resembles the original \(1+1\)dimensional KogutSusskind Hamiltonian.^{Footnote 2} For example, the gaugematter interaction in Eq. (12) couples sets of three adjacent links. Similarly, for the term originating from the staggered mass in Eq. (13), one obtains a coupling between two adjacent links. With this resourceefficient formulation, one can simulate larger systems without worrying about the elements of the variational circuit preserving gauge invariance.
3 Implementation and results
The \(\mathbb{Z}_{2}\) lattice gauge theory has a discrete chiral symmetry in the massless limit [18] which can break spontaneously at low T. At large temperatures as well as at large values of the chemical potential, this symmetry is restored. To benchmark our approximation to the Gibbs state, we calculate the chiral condensate, \(\langle \bar{\psi }\psi \rangle \), which is sensitive to restoring chiral symmetry. As this is a local, timeindependent observable, one would like a second, timedependent observable that measures correlations in the system. We chose to study the unequaltime densitydensity correlator, which provides information about charge dynamics in the system.
3.1 Gauge redundant formulation
In this formulation of the theory, gauge invariance has to be considered explicitly. Apart from the gaugeinvariant initial state (c.f. Figure 2), this not only applies to the variational circuits \(\mathrm{VQC_{1/2}}\) but also to the intermediate measurement during which qubits representing gauge bosons will be measured in the Xeigenbasis, whereas fermionic qubits are measured in the computational basis. A schematic depiction of the circuit illustrating the abovementioned components is shown in Fig. 1. For \(\mathrm{VQC_{1/2}}\), a gaugeinvariant gate set can be constructed by following the general considerations laid out in [19]. This consists of simply taking the Pauli strings in our Hamiltonian Eq. (6) as elementary building blocks. Due to the simplicity of the model, the gate set will hence decompose into parameterdependent single qubit \(R_{x}\) and \(R_{z}\) gates, together with a 3qubit gate \(\sim e^{i\sigma ^{x} Z \sigma ^{x} + i\sigma ^{y}Z\sigma ^{y}}\) stemming from the hopping term in Eq. (6). Following the procedure presented in [20], the latter can be easily decomposed into singlequbit and entangling gates.^{Footnote 3} In Fig. 3, one observes the resulting circuit layer consisting of a sublayer of singlequbit gates along with a sublayer of the decomposed multiqubit gate for a threequbit system where the \(\Delta _{i}\) represent generic variational parameters. A schematic of the variational circuit layer is also shown in Fig. 4 (left). Several such layers then yield the \(\mathrm{VQC}_{i}\), with parameter sets θ and ϕ for \(i =1,2\), respectively (Eq. (5)). In practice, we find that using \(N/2\) layers per variational circuit \(\mathrm{VQC}_{i}\) yields a good variational approximation for a theory with N matter sites.
With the composition of the variational quantum circuits at hand, we still need to define the figures of merit quantifying the quality of the optimization step depicted in the classical block of Fig. 1. In addition to the relative difference in free energy, \(\Delta F = F_{\rho _{var}}/F_{\rho _{\text{Gibbs}}}1\), we take the fidelity [22]
as a distance measure between two quantum states represented by \(\rho _{1}\) and \(\rho _{2}\), respectively. In Fig. 5, one sees the results of the classically simulated variational optimization process for our gaugeredundant theory for \(N = 4\) matter sites corresponding to \(n_{q} = 7\) qubits using open boundary conditions with a twolayer ansatz in the \(\mathrm{VQC}_{i}\). Simulating at \(a\epsilon = a m = 0.5\), \(a\mu = 0\) and a number of shots \(n_{s} = 10^{4}\) for varying \(T/m\), chiral symmetry is explicitly broken in our model. On the left, we plot the performance measures defined above, where each point corresponds to the minimum in free energy over up to 10^{2} samples, resulting in optimal parameter sets \((\boldsymbol{\theta}^{*},\boldsymbol{\phi}^{*})\). The quality of the approximation by the variational ansatz decreases slightly in the crossover region until, with increasing \(T/m\), the entropic term in Eq. (2) becomes dominant, yielding a highfidelity solution. On the right, the chiral condensate \(\langle \bar{\psi}\psi \rangle \) evaluated using \(\rho _{2}(\boldsymbol{\theta}^{*},\boldsymbol{\phi}^{*})\) is shown as a function of \(T/m\). The displayed errors were obtained from the variation of the observable over solutions corresponding to the 20th percentile of the free energy distribution for each set of samples.
3.2 Resource efficient formulation
Turning now to the Hamiltonian Eqs. (11) with matter degrees of freedom eliminated and gauge invariance builtin, the choice of variational ansatz circuits for the \(\mathrm{VQC}_{i}\) is in principle unrestricted, leaving us with the freedom to use, e.g. hardware efficient circuits such as EfficientSU2 from qiskit’s standard circuit library [23]. As for the set of physical gates, by inspecting the Hamiltonian one sees that the single and multiqubit terms can be employed as elementary ansätze. As an example, the decomposed threequbit term is depicted in Fig. 6. An alternation of a singlequbit sublayer containing \(R_{x}\) and \(R_{y}\) gates, followed by a staggered multiqubit sublayer, will hence serve as our variational ansatz for both \(\mathrm{VQC}_{1/2}\), where the circuit schematic is shown in Fig. 4 (right). Using the resourceefficient formulation, for the system size studied in this work (\(N=8\) matter sites), we found that already one such alternation per \(\mathrm{VQC}_{i}\) yielded acceptable results in terms of the above defined figures of merit, Eq. (15).
This should be contrasted by a comparison of the gate depths of our variational ansätze for both, the gaugeredundant formulation and the resource efficient formulation. In Table 1 we list the gate depth of the corresponding sublayers displayed in Fig. 4. Even though our choice of ansatz for the resource efficient formulation leads to a higher gate depth, we expect this to be compensated by the higher number of layers per \({\mathrm{VQC}}_{i}\) required by the gaugeredundant formulation.
3.2.1 Observables at \(T,\mu > 0\)
As was done in the gaugeredundant formulation, we quantitatively investigate the quality of our variational approximation to the Gibbs state and calculate the chiral condensate. This has been done as a function of temperature \(T/m\) and varying chemical potential \(\mu /m\) at \(\epsilon /m = 0.5\) with \(n_{s} = 10^{4}\). Owing to the resource efficiency of the formulation, \(n_{q} = 7\) classically simulated qubits now correspond to twice the system size (\(N = 8\)) when compared to the gaugeredundant formulation. While simulations for varying \(T/m\) across the crossover region, Fig. 7, show good agreement between exact and simulated results, entering the transition region for fixed temperature and increasing \(\mu /m\), we see that our algorithm has difficulty to match the exact result. One notes that as we lower the temperature, the transition becomes more and more discontinuous. In Fig. 8 and Fig. 9, this observation is confirmed by our estimation of the statistical error which measures the variation of \(\langle \bar{\psi}\psi \rangle \) over the top 20th percentile of variational solutions. We expect an increase in performance can be achieved here by changing from the gradientfree optimization algorithm (COBYLA) used in this study to gradientbased methods. This is an approach which we have not pursued further.
3.2.2 Thermal unequaltime correlators
In addition to the study of quantum systems at finite fermion density, quantum simulation in the NISQ era and beyond will allow the investigation of realtime observables. These include thermal unequaltime correlation functions. In general, an unequaltime correlator of two operators takes the form
where H is the Hamiltonian of the system and \(A(t)\) is the operator in the Heisenberg picture. In this study, the expectation value in (16) will be taken with respect to our variational density matrix, \(\rho _{2}(\boldsymbol{\theta}^{*},\boldsymbol{\phi}^{*})\).
The question of how one measures such a correlation function naturally arises when studying lattice gauge theories. Fortunately, a general procedure has already been devised to measure the correlation function of two arbitrary unitary operators A and B on a universal quantum computer [24, 25]. This procedure is commonly referred to as Ramsey interferometry and only involves adding a single ancillary qubit entangled with our system’s original \(n_{q}\) qubits. The time evolution can be performed with a Trotterization of the time evolution operator whose circuit depth increases linearly with the number of steps. A variational approach also exists to obtain the time evolution of an arbitrary quantum state with a fixed circuit depth [15]. In this work, a simple Trotterization has been used.
In Fig. 10 we show the unequaltime thermal densitydensity correlation function \(\langle O_{0}(0) O_{i}(t)\rangle \) at \(T/m = 1\) for \(N = 8\), where \(O_{i} \equiv \varepsilon _{i}Z_{i1}\otimes Z_{i}\). The different data sets represent different spatial separations \(i = 1,\ldots ,5\) of the density operator on the lattice plotted as a function of time with a remarkable agreement between simulated and exact results for the time range displayed. In these classical simulations of our variational quantum circuit, both \(\mathrm{VQC}_{1}\) and \(\mathrm{VQC}_{2}\) contain two layers of alternating single and multiqubit sublayers. For simplicity’s sake, we have assumed infinite statistics in the computation of \(\rho _{2}(\boldsymbol{\theta}^{*},\boldsymbol{\phi}^{*})\), hence excluding shot noise. One is ultimately interested in obtaining transport coefficients from realtime dynamics. As the latter involves time integrals of unequaltime correlators, it remains to be seen if such simulations yield sufficient accuracy.
3.2.3 q̄q meson screening
It is useful to check that our algorithm correctly reproduces physical observables, which are more easily accessible by standard Euclidean methods. As an example, we consider the thermal expectation value of a gaugeinvariant q̄q meson operator, i.e. a quark antiquark pair connected by a flux tube [14, 26],
where the links \(U_{k}\) are taken in the electric basis, \(U_{k} = X_{k}\), used to derive the resourceefficient expression of the observable [14]. As demonstrated recently with the help of DMRG [27], at \(T=0\), this equaltime twopoint function shows an exponential decay for any \(\epsilon > 0\) (see also [28, 29] for recent theoretical work on confinement in \(\mathbb{Z}_{2}\) and [30] for its investigation on quantum hardware). The exponents of a spatial decay are screening masses, which in this case give the energy of the flux tube and its excitations, whose linear increase with separation indicates confinement. We expect this behaviour to persist for small temperatures until the onset of string breaking, when the flux gets increasingly screened and its energy no longer grows with separation.
Using the same variational setup as described in Sect. 3.2.2, Fig. 11 shows \(\langle M_{i}\rangle \) at \(T/m = 0.5\) (left) and \(T/m = 1.25\) (right), respectively. Whereas the lowtemperature correlator exhibits marked similarities to the zerotemperature decay (c.f. [27]), which is accurately captured by the variational Gibbs state, at higher temperatures, the algorithm fails to reproduce the exact correlator accurately, possibly due to the low complexity of the ansatz circuits in our simulations (two layers of variational gates for the \(\mathrm{VQC}_{i}\)). While this precludes a detailed picture of the long distance behavior with increasing temperature, our variational calculation does exhibit two qualitative features known from other techniques: i) the screening masses generally grow with temperature as expected (faster decay); ii) the difference between high and low temperature is small at short distances, but pronounced at large distances, because of the flux screening.
3.2.4 Resources and entropy estimation
We now turn to the question of resources in runtime and classical memory. Considering the limited system sizes currently studied in the NISQ era, estimating the von Neumann entropy \(S = \sum_{i} p_{i} \log{p_{i}}\) from the intermediate measurements via \(p_{i} \approx n_{i}/n_{s}\) is a viable approach. Applying the algorithm with the information flow depicted in Fig. 1 for realistic system sizes will necessarily involve a certain level of approximation. This stems from the fact that the number of \(p_{i}\)’s grows with the dimension of the Hilbert space, \(2^{n_{q}}\). One possible way proposed in [11], is the partitioning of the full \(n_{q}\) qubit system into \(n_{q}/n_{ss}\) independent subsystems of size \(n_{ss}\) in the entropic part of the algorithm (\(\mathrm{VQC}_{1}\)) that prepares the latent distribution \(\rho _{1}(\boldsymbol{\theta})\). In the version of the algorithm which employs a classical estimation of the entropy [7] this corresponds to a particular choice of the latent distribution known as factorized latent space models. It remains to be seen if such an approximation is a valid approach for gauge theories where thermalization and subsystem entanglement are nontrivially connected [31].
There are, however, alternative methods that can be used to estimate the entropy efficiently. For example, it turns out that an evaluation of the Taylor series approximation to the entropy can be useful [32], i.e.
where
Using the e.g. qubitefficient, repeated SWAPtest to estimate the trace of various powers of the latent density matrix, \(\mathrm{tr}(\rho ^{n+1}_{1}(\theta ))\), this method requires the addition of N qubits in the resource efficient formulation. It should be noted that the corresponding circuit depth for the largest order \(\mathrm{tr}(\rho ^{K+1}_{1}(\theta ))\) is then given by \(K(\frac{N}{2} d_{L} + N)\), not counting the reset, and thus still scales polynomially in N for a system size of N matter sites, assuming conservatively \(N/2\) layers for \(\mathrm{VQC}_{1}\) with layer depth \(d_{L}\), given in Table 1. The results of this approach for the simple gauge theory used in this study are displayed in Fig. 12. For temperatures up to \(T/m\sim 1\), only a few orders are needed to give reasonable values of the fidelity and we attribute the change in ordering in the limit \(T \rightarrow 0\) to the accuracy of the minimum found by the gradientfree optimization algorithm.
4 Conclusion
We have explored the use of variational quantum algorithms in producing thermal states. These approaches were applied to \(\mathbb{Z}_{2}\) lattice gauge theory in \(1+1\) dimensions using two independent formulations, and several observables of interest were computed. We find this variant of the VQT a suitable algorithm to approximate the thermal states of quantum systems in the NISQ era as estimating the entropy of larger systems will necessarily lead to larger approximation errors due to limited statistics. A particular aspect we have left for future work is the performance of intermediate measurements for mixedstate creation on actual quantum devices compared to the usual approach of entangling ancillary systems followed by a terminal measurement.
A further possible extension of our work is the formulation of the VQT to study nonAbelian gauge theories at finite temperatures. As the latter will generally come with a gaugeredundant Hilbert space, even after reformulating in a resourceefficient manner, this amounts to respecting gauge invariance or equivalent local constraints throughout the variational quantum circuit.
Data availability
Simulation data are available from the corresponding author upon reasonable request.
Notes
For a version of this algorithm using entanglement of the system with \(n_{q}\) ancillary qubits followed by a projective measurement on the latter, see [12].
With the additional note that the starting point to derive the above Hamiltonian in ref. [14] is actually a KogutSusskind Hamiltonian in the electric basis of \(\mathbb{Z}_{2}\) LGT.
We would also like to point to ref. [21] which introduces a shearing approach for gaugeinvariant Trotterization of abelian LGTs.
Abbreviations
 QCD:

Quantum Chromodynamics
 QED:

Quantum Electrodynamics
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Funding
Open Access funding enabled and organized by Projekt DEAL. The work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the grant CRCTR 211 “Stronginteraction matter under extreme conditions” – project number 315477589 – TRR 211 and by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006). M.S. and M.F. acknowledge support by the Munich Institute for Astro, Particle and BioPhysics (MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC2094 – 390783311.
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All authors contributed equally in the conception phase, C.W., M.F. and M.S. carried out the computations and simulations. All authors jointly drafted and reviewed the manuscript and approved the submission.
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Fromm, M., Philipsen, O., Spannowsky, M. et al. Simulating \(Z_{2}\) lattice gauge theory with the variational quantum thermalizer. EPJ Quantum Technol. 11, 20 (2024). https://doi.org/10.1140/epjqt/s40507024002322
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DOI: https://doi.org/10.1140/epjqt/s40507024002322