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International Symposium on Lattice Field Theory (LATTICE)

Ground state energy of a two-dimensional pure Z2 lattice gauge theory via sample-based Krylov quantum diagonalization

Lento Nagano, Y. Iiyama, Tomoya Hayata, Koji Terashi, Yukio Kawashima, William Kirby, Mário Motta, Vincent Richard Pascuzzi, Antonio Mezzacapo

Published Jul 5, 2026• 0 citations
Publisher

Abstract

We study the application of sample-based Krylov quantum diagonalization (SKQD) to a two-dimensional pure $\mathbb{Z}_2$ 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 $\texttt{ibm}\_\texttt{kawasaki}$, 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.

DOI
10.22323/1.518.0126

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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.
We apply the recently proposed Sample-based Krylov Quantum Diagonalization (SKQD) method to lattice gauge theories, using the Schwinger model with a θθθ-term as a benchmark. SKQD approximates the ground state of a Hamiltonian, employing a hybrid quantum-classical approach: (i) constructing a Krylov space from bitstrings sampled from time-evolved quantum states, and (ii) classically diagonalizing the Hamiltonian within this subspace. We study the dependence of the ground-state energy and particle number on the value of the θθθ-term, accurately capturing the model's phase structure. The algorithm is implemented on trapped-ion and superconducting quantum processors, demonstrating consistent performance across platforms. We show that SKQD substantially reduces the effective Hilbert space, and although the Krylov space dimension still scales exponentially, the slower growth underscores its promise for simulating lattice gauge theories in larger volumes.
Quantum spin simulations of frustrated lattices remain challenging for both classical and quantum algorithms, particularly in parameter regimes relevant to quantum spin liquid (QSL) phases. In this work, we apply Sample-based Krylov Quantum Diagonalization (SKQD) to estimate the ground state of the antiferromagnetic XXZ Heisenberg model on the J1J_1J1​--J2J_2J2​ square lattice, the Kagome lattice, and a 1D chain, studying system sizes from 12 to 72 spins. In our application of SKQD, we identify a ZZ deformation of Δ=2Δ=2Δ=2 as a sufficiently sparse Hamiltonian and introduce two modifications to the SKQD framework tailored to spin models: a canonical bitstring compression scheme that preserves the effectiveness of configuration recovery under spin-flip degeneracy, and the use of multiple Krylov subspaces to improve ground state coverage without any increase in quantum resources. For the 1D chain and Kagome lattice, SKQD achieves sub-percent ground-state energy errors at system sizes up to 24 spins, including a relative error of 0.002%0.002\%0.002% on the 12-site Kagome lattice, surpassing the best prior VQE result of 0.01%0.01\%0.01% on the same system while requiring no variational optimization. SKQD further extends to system sizes well beyond the reach of prior quantum algorithm studies, reaching 72 spins across all three geometries. Beyond 24 spins, accuracy degrades to relative errors of 19%19\%19%--36%36\%36% at 72 sites, but the gradual scaling of error with system size suggests these limits are set by available shot budgets and circuit depth rather than fundamental algorithmic constraints. Although classical tensor network methods remain state-of-the-art for these models, this work establishes a new benchmark for quantum simulation of the frustrated Heisenberg model and demonstrates SKQD as a scalable, hardware-compatible approach for studying strongly correlated spin systems.
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.
We evaluate the Sample-based Krylov Quantum Diagonalization (SKQD) algorithm on one- and two-dimensional Heisenberg models, including strongly correlated regimes in which the ground state is dense. Using problem-informed initial states and magnetization-sector sweeps, SKQD accurately reproduces ground-state energies and field-dependent magnetization across a range of anisotropies. Benchmarks against DMRG and exact diagonalization show consistent qualitative agreement, with accuracy improving systematically in more anisotropic regimes. We further demonstrate SKQD on quantum hardware by implementing 18- and 30-qubit Heisenberg chains, obtaining magnetization curves that match theoretical expectations. Simulations on small 2D square-lattice systems further demonstrate that the method applies effectively beyond 1D geometries.
Sample-based Krylov quantum diagonalization (SKQD) is a paradigmatic example of a quantum-centric supercomputing workflow that combines the convergence structure of Krylov quantum diagonalization with classical sampling-based post-processing. It provides convergence guarantees assuming that the important computational-basis configurations can be sampled from a set of Krylov states with sufficient probability. We analyze this assumption under depolarizing noise, deriving shot-count resource estimates that expose an exponential depth penalty, and perform experiments on current noisy hardware with the one-dimensional single-impurity Anderson model on a 20-site (40-qubit) instance. Device noise breaks the practical convergence predicted by the noiseless SKQD analysis, but approximate compilation techniques can compress the Krylov time-evolution circuits before execution. The compressed circuits accumulate less hardware noise, recover the expected energy convergence, and reduce both the quantum shot budget and the classical subspace dimension, remaining beneficial even when classical configuration recovery is applied.
Predicting ground state energies of quantum many-body systems is one of the central computational challenges in quantum chemistry, physics, and materials science. Krylov subspace methods, such as Krylov Quantum Diagonalization and Sample-based Krylov Quantum Diagonalization, are promising approaches for this task on near-term quantum computers. However, both require repeated quantum circuit executions for each Krylov subspace and for every new Hamiltonian, posing a major bottleneck under noisy hardware constraints. We introduce Generative Krylov Subspace Representations (GenKSR), a framework that learns a classical generative representation of the entire Krylov diagonalization process. To enable effective modeling of quantum systems, GenKSR leverages a conditional generative model framework. We investigate two representative backbone architectures, the standard Transformer and the Mamba state-space model. By learning the distribution of measurement outcomes conditioned on Hamiltonian parameters and evolution time, GenKSR generates Krylov subspace samples for unseen Hamiltonians and for larger subspace dimensions than those used in training. This enables full energy reconstruction purely from the classical model, without additional quantum experiments. We validate our approach through simulations of 15-qubit 1D and 16-qubit 2D Heisenberg models, as well as a hardware experiment on a 20-qubit XXZ chain executed on an IBM quantum processor. Our model successfully learns the distribution from experimental data and generates a high-fidelity representation of the quantum process. This representation enables classical reproduction of experimental outcomes, supports reliable energy estimates for unseen Hamiltonians, and significantly reduces the need for further quantum computation.