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International Symposium on Lattice Field Theory (LATTICE)

Ground state energy of the two-dimensional pure $\mathbb{Z}_2$ lattice gauge theory via quantum imaginary time evolution.

Minoru Sekiyama, Lento Nagano

Published Jun 23, 2026• 0 citations
Publisher

Abstract

In this paper, we employ the quantum imaginary time evolution (QITE) algorithm to obtain the ground state of a two-dimensional pure $\mathbb{Z}_2$ lattice gauge theory. Specifically, we perform the (classical) numerical simulation of the QITE in the $\mathbb{Z}_2$ 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 $0.1\%$ 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.

DOI
10.22323/1.518.0130

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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.
Ground state preparation in (2+1)(2+1)(2+1)-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 2+12+12+1-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.
We study the application of sample-based Krylov quantum diagonalization (SKQD) to a two-dimensional pure Z2\mathbb{Z}_2Z2​ lattice gauge theory. SKQD samples configurations from states obtained via time-evolution quantum circuits, projects the Hamiltonian into the subspace spanned by the sampled configurations, and solves an eigenvalue problem to estimate the ground-state energy. We consider a model defined on a triangular lattice, which allows for an efficient construction of time-evolution circuits on the IBM Quantum Heron processor. We also propose a configuration recovery method based on Gauss's law and a conditional restricted Boltzmann machine (CRBM), to mitigate the effects of noise. We simulate a lattice of 108 links using the superconducting quantum processor ibm_kawasaki\texttt{ibm}\_\texttt{kawasaki}ibm_kawasaki, which allows us to obtain the ground-state energy within a few percent of tensor network results. We also compare configuration recovery with and without the CRBM, and observe that the CRBM improves the accuracy at each iteration.
Gauge theories describe fundamental interactions in the standard model of particle physics and build effective theories in condensed matter physics. Being notoriously hard to simulate, these theories are commonly regularized as lattice gauge theories which can be evaluated with high precision through Monte Carlo algorithms or variational methods. However, Monte Carlo algorithms are not applicable in all regimes due to the sign problem; while variational methods, quantum and classical alike, only give upper-bounds on the ground-state energy. Their precision crucially depends on the chosen ansatz. Here, we present a framework based on a hierarchy of semidefinite programs to yield increasingly good lower-bounds on the ground-state energy and propagate these bounds to extended observables like Wilson loops and mesonic strings. We demonstrate the numerical capabilities by applying the algorithm to a (1+1)-dimensional Z2\mathbb{Z}_2Z2​ theory with dynamic fermionic matter and its pure-gauge version in (2+1)-dimensions. In both cases, we obtain certified intervals for the ground-state energy. For a one-dimensional system with 129 sites, the certified interval has a relative spread of 0.04%0.04\%0.04%. For a two-dimensional system with 97 sites (corresponding to a 7×87 \times 87×8 lattice), the relative spread is 2.08%2.08\%2.08%. Beyond energy estimates, the proposed method provides rigorous bounds on both short- and long-range observables such as Wilson loops over multiple plaquettes and the magnitude of the mesonic string. The method can be both used as a computational microscope into many-body physics and a way to certify results of quantum simulators.
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.
Efficiently preparing approximate ground states of large, strongly correlated systems on quantum hardware is challenging, and yet, nature is innately adept at this. This has motivated the study of thermodynamically inspired approaches to ground-state preparation that aim to replicate cooling processes via imaginary-time evolution. However, synthesizing quantum circuits that efficiently implement imaginary-time evolution is itself difficult, with prior proposals generally adopting heuristic variational approaches or using deep block encodings. Here, we use the insight that quantum imaginary-time evolution is a solution of Brockett's double-bracket flow and synthesize circuits that implement double-bracket flows coherently on the quantum computer. We prove that our double-bracket quantum imaginary-time evolution (DB-QITE) algorithm inherits the cooling guarantees of imaginary-time evolution. Concretely, each step is guaranteed to (i) decrease the energy of an initial approximate ground state by an amount proportional to the energy fluctuations of the initial state and (ii) increase the fidelity with the ground state. We provide gate counts for DB-QITE through numerical simulations in qrisp that demonstrate scenarios where DB-QITE outperforms quantum phase estimation. Thus, DB-QITE provides a means to systematically improve the approximation of a ground state using shallow circuits.
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.