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arXiv:2604.17874 · Physical review. D/Physical review. D.

Ground state preparation in two-dimensional pure $\mathbb{Z}_2$ lattice gauge theory via deterministic quantum imaginary time evolution

Minoru Sekiyama, Lento Nagano

hep-lathep-thquant-phSubmitted Apr 20, 2026• Announced Apr 21, 2026• 0 citations
v1 · Apr 20, 2026v2 · Apr 28, 2026v3 · Jul 22, 2026
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Abstract

In this paper, we apply the deterministic quantum imaginary time evolution (QITE) algorithm to obtain the ground state of a two-dimensional pure $\mathbb{Z}_2$ lattice gauge theory. We first construct the set of Pauli operators commuting with Gauss's law constraints, generalizing a previous result. This makes the deterministic QITE gauge-invariant and reduces both the measurement and gate costs significantly without adding extra algorithm errors in the QITE. Then, the classical numerical simulation of the deterministic QITE using tensor networks is performed, and the results are compared with the density matrix renormalization group (DMRG) to evaluate the accuracy of the algorithm. Specifically, we investigate the coupling and system size dependence, and find that the deterministic QITE can achieve a relative error of less than $0.1\%$ up to a twelve-plaquette system and coupling values in a regime that we study. Furthermore, the error dependence on the number of time steps is studied and discussed.

Comments
15 pages, 7 figures

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Ground state preparation in two-dimensional pure Z2\mathbb{Z}_2Z2​ lattice gauge theory via deterministic quantum imaginary time evolution
In this paper, we apply the deterministic quantum imaginary time evolution (QITE) algorithm to obtain the ground state of a two-dimensional pure Z2\mathbb{Z}_2Z2​ lattice gauge theory. We first construct the set of Pauli operators commuting with Gauss's law constraints, generalizing a previous result. This makes the deterministic QITE gauge-invariant and reduces both the measurement and gate costs significantly without adding extra algorithm errors in the QITE. Then, the classical numerical simulation of the deterministic QITE using tensor networks is performed, and the results are compared with the density matrix renormalization group (DMRG) to evaluate the accuracy of the algorithm. Specifically, we investigate the coupling and system size dependence, and find that the deterministic QITE can achieve a relative error of less than 0.1%0.1\%0.1% up to a twelve-plaquette system and coupling values in a regime that we study. Furthermore, the error dependence on the number of time steps is studied and discussed.
Ground state energy of the two-dimensional pure Z2\mathbb{Z}_2Z2​ lattice gauge theory via quantum imaginary time evolution.
In this paper, we employ the quantum imaginary time evolution (QITE) algorithm to obtain the ground state of a two-dimensional pure Z2\mathbb{Z}_2Z2​ lattice gauge theory. Specifically, we perform the (classical) numerical simulation of the QITE in the Z2\mathbb{Z}_2Z2​ lattice gauge theory using the tensor networks, and the results are compared with the density matrix renormalization group (DMRG) to evaluate the accuracy. We find that the QITE can achieve a relative error of less than 0.1%0.1\%0.1% for a four-plaquette system and coupling values in a regime we study. It is also observed that the algorithmic error in the QITE decreases with increasing the number of time steps, at least for our settings, at the cost of increasing the number of gates. Furthermore, we show that utilizing Gauss's law constraints allows us to reduce both the measurement and gate costs significantly without sacrificing simulation accuracy.
Quantum Imaginary-Time Evolution (QITE) is a powerful method for preparing ground states on quantum hardware. However, executing QITE has costly measurement budgets for general Hamiltonians. Both fidelity and computational cost are strongly dependent on the definition of suitable local domains and Hamiltonian partitions. In this work, we introduce the Multiple-Time QITE algorithm (MT-QITE). We show how using more than one imaginary time substantially improves the fidelity of the resulting ground state as well as the measurement overhead with respect to the previously published QITE algorithm, while preserving its deterministic character and its independence from ad hoc ansatze. Moreover, unlike QITE and other QITE-based algorithms, MT-QITE is parallelizable, and we show that even in Hamiltonians with non-local interactions, partitioning may entail a computational advantage.
Quantum imaginary time evolution (QITE) is one of the promising candidates for finding the eigenvalues and eigenstates of a Hamiltonian on a quantum computer. However, the original proposal suffers from large circuit depth and measurements due to the size of the Pauli operator pool and Trotterization. To alleviate the requirement for deep circuits, we propose a time-dependent drifting scheme inspired by the qDRIFT algorithm [Campbell, E. Phys. Rev. Lett. 2019, 123, 070503]. We show that this drifting scheme removes the depth dependency on the size of the operator pool and converges inversely with respect to the number of steps. We further propose a deterministic algorithm that selects the dominant Pauli term to reduce the fluctuation for the ground state preparation. We also introduce an efficient measurement reduction scheme across Trotter steps that removes its cost dependence on the number of iterations. We analyze the main source of error for our scheme both theoretically and numerically. We numerically test the validity of depth reduction, convergence performance of our algorithms, and the faithfulness of the approximation for our measurement reduction scheme on several benchmark molecules. In particular, the results on the LiH molecule give circuit depths comparable to that of the advanced adaptive variational quantum eigensolver (VQE) methods while requiring much fewer measurements.
Two-dimensional Z2\mathbb{Z}_2Z2​ lattice gauge theory on a near-term quantum simulator: variational quantum optimization, confinement, and topological order
We propose an implementation of a two-dimensional Z2\mathbb{Z}_2Z2​ lattice gauge theory model on a shallow quantum circuit, involving a number of single and two-qubits gates comparable to what can be achieved with present-day and near-future technologies. The ground state preparation is numerically analyzed on a small lattice with a variational quantum algorithm, which requires a small number of parameters to reach high fidelities and can be efficiently scaled up on larger systems. Despite the reduced size of the lattice we consider, a transition between confined and deconfined regimes can be detected by measuring expectation values of Wilson loop operators or the topological entropy. Moreover, if periodic boundary conditions are implemented, the same optimal solution is transferable among all four different topological sectors, without any need for further optimization on the variational parameters. Our work shows that variational quantum algorithms provide a useful technique to be added in the growing toolbox for digital simulations of lattice gauge theories.
Quantum simulation is one of the major applications of quantum devices. In the noisy intermediate-scale quantum era, however, the general quantum simulation is not yet feasible, such as that of lattice gauge theories, which is likely limited due to the violation of the Gauss law constraint and the complexity of the real-time dynamics, especially in the deconfined phase. Inspired by the recent works of Ashkenazi and Zohar [Phys. Rev. A 105, 022431 (2022)] and of Tantivasadakarn, Thorngren, Vishwanath, and Verresen [arXiv:2112.01519], we propose to simulate dynamics of lattice gauge theories by using the Kramers-Wannier transformation via cluster-state-like entanglers, mid-circuit measurements, and feedforwarded corrections, which altogether is a constant-depth deterministic operation. In our scheme, specifically, we first quantum simulate the time evolution under a corresponding symmetric Hamiltonian from an initial symmetric state, and then apply the Kramers-Wannier procedure. This results in a wave function that has time evolved under the corresponding lattice gauge theory from a corresponding initial, gauged wave function. In the presence of noises in time evolution, the procedure succeeds when we are able to pair up magnetic monopoles represented by nontrivial measurement outcomes. Furthermore, given a noise-free Kramers-Wannier transformation, the resulting wave function from a noisy time evolution satisfies the Gauss law constraint. We give explicit examples with the low dimensional pure gauge theories and gauge theories coupled to bosonic or fermionic matters such as the Fradkin-Shenker model.
We develop a hybrid qubit-qumode framework for simulating quantum electrodynamics in 2+1 dimensions. In this approach, fermionic matter fields are represented by qubits, while U(1) gauge fields are encoded in continuous-variable bosonic modes whose canonical quadratures capture the electric and vector-potential components of the theory. To reconcile the non-compact phase space of the qumodes with the compact U(1) gauge symmetry, we introduce and compare two complementary constraint-enforcement strategies: (i) a squeezing-based projection that confines qumode states to the unit circle through an effective modification of the inner product, and (ii) a method that dynamically enforces compactness via a penalty Hamiltonian term. We construct the corresponding hybrid Hamiltonian, derive its decomposition into experimentally accessible qubit-qumode gates, and analyze its spectrum in the analytically tractable single-plaquette limit. The hybrid formulation reproduces the correct gauge-invariant dynamics and provides a scalable route toward simulating Abelian lattice gauge theories coupled to fermionic matter on near-term hybrid quantum architectures. Ground-state preparation and convergence are demonstrated using a continuous-variable extension of the Quantum Imaginary Time Evolution (QITE) algorithm, establishing a general framework for hybrid discrete-continuous quantum simulations of lattice gauge theories.