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arXiv:2409.13510

Simulating the Schwinger Model with a Regularized Variational Quantum Imaginary Time Evolution

Xiao-Wei Li, Fei Li, Jiapei Zhuang, Man‐Hong Yung

Published Sep 20, 2024• 0 citations

Abstract

The Schwinger model serves as a benchmark for testing non-perturbative algorithms in quantum chromodynamics (QCD), emphasizing its similarities to QCD in strong coupling regimes, primarily due to the phenomena such as confinement and charge screening. However, classical algorithms encounter challenges when simulating the Schwinger model, such as the "sign problem" and the difficulty in handling large-scale systems. These limitations motivate the exploration of alternative simulation approaches, including quantum computing techniques, to overcome the obstacles. While existing variational quantum algorithms (VQAs) methods for simulating the Schwinger model primarily rely on mathematical gradient-based optimization, which sometimes fail to provide intuitive and physically-guided optimization pathways. In contrast, the Variational Quantum Imaginary Time Evolution (VQITE) method offers a physically-inspired optimization approach. Therefore, we introduce that VQITE holds promise as a potent tool for simulating the Schwinger model. However, the standard VQITE method is not sufficiently stable, as it encounters difficulties with the non-invertible matrix problem. To address this issue, we have proposed a regularized version of the VQITE, which we have named the Regularized-VQITE (rVQITE) method, as it incorporates a truncation-based approach. Through numerical simulations, we demonstrate that our proposed rVQITE approach achieves better performance and exhibits faster convergence compared to other related techniques. We employ the rVQITE method to simulate the phase diagrams of various physical observables in the Schwinger model, and the resulting phase boundaries are in agreement with those obtained from an exact computational approach.

DOI: 10.48550/arxiv.2409.13510

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The Schwinger model serves as a benchmark for testing non-perturbative algorithms in quantum chromodynamics (QCD), emphasizing its similarities to QCD in strong coupling regimes, primarily due to the phenomena such as confinement and charge screening. However, classical algorithms encounter challenges when simulating the Schwinger model, such as the "sign problem" and the difficulty in handling large-scale systems. These limitations motivate the exploration of alternative simulation approaches, including quantum computing techniques, to overcome the obstacles. While existing variational quantum algorithms (VQAs) methods for simulating the Schwinger model primarily rely on mathematical gradient-based optimization, which sometimes fail to provide intuitive and physically-guided optimization pathways. In contrast, the Variational Quantum Imaginary Time Evolution (VQITE) method offers a physically-inspired optimization approach. Therefore, we introduce that VQITE holds promise as a potent tool for simulating the Schwinger model. However, the standard VQITE method is not sufficiently stable, as it encounters difficulties with the non-invertible matrix problem. To address this issue, we have proposed a regularized version of the VQITE, which we have named the Regularized-VQITE (rVQITE) method, as it incorporates a truncation-based approach. Through numerical simulations, we demonstrate that our proposed rVQITE approach achieves better performance and exhibits faster convergence compared to other related techniques. We employ the rVQITE method to simulate the phase diagrams of various physical observables in the Schwinger model, and the resulting phase boundaries are in agreement with those obtained from an exact computational approach.

Abstract Variational quantum imaginary time evolution (VQITE) is a leading technique for ground state preparation on quantum computers. A significant computational challenge of VQITE is the determination of the quantum geometric tensor. We show that requiring the imaginary-time evolution to be correct only when projected onto a chosen set of operators allows to achieve a twofold reduction in circuit depth by bypassing fidelity estimations, and reduces measurement complexity from quadratic to linear in the number of parameters. We demonstrate by a simulation of the transverse-field Ising model that our algorithm achieves a several orders of magnitude improvement in the number of measurements required for the same accuracy.

We study the Schwinger model at finite-temperature regime using a quantum-classical hybrid algorithm. The preparation of thermal state on quantum circuit presents significant challenges. To address this, we adopt the Thermal Pure Quantum (TPQ) state approach and apply the Quantum Imaginary Time Evolution (QITE) algorithm to implement the necessary imaginary time evolution. We first compute the chiral condensate in the massless Schwinger model, verifying its consistency with the analytical solution. We then simulate the massive Schwinger model with non-zero topological

\theta

-term to investigate the temperature and

\theta

-dependence of the chiral condensate. Our method works well even at non-zero

\theta

regime, while the conventional lattice Monte Carlo method suffers from the sign problem in this system.

Quantum computing encompasses a variety of computation models, including gate-based (GB), topological, quantum annealing, and measurement-based (MB). MB performs computation through single-qubit measurements on a highly entangled cluster state and is proposed to offer some practical advantages on near-term quantum hardware in terms of circuit depth. In this work, we propose MB-VQITE, which leverages the MB model for solving optimization problems using the variational quantum imaginary time evolution (VQITE) algorithm. We apply MB-VQITE to the molecular docking problem, which is relevant in pharmaceutical research. We perform direct comparisons between the MB and GB computing paradigms on IBM Cleveland quantum hardware and perform optimization procedures on noisy simulators. We find that MB-VQITE outperforms GB-VQITE on current quantum hardware by reducing circuit depth, thus decreasing noise to allow for smoother optimization.

Simulations in high-energy physics are currently emerging as an application of noisy intermediate-scale quantum (NISQ) computers. In this work, we explore the multi-flavor lattice Schwinger model - a toy model inspired by quantum chromodynamics - in one spatial dimension and with nonzero chemical potential by means of variational quantum simulation on a shuttling-based trapped-ion quantum processor. This fermionic problem becomes intractable for classical numerical methods even for small system sizes due to the notorious sign problem. We employ a parametric quantum circuit executed on our quantum processor to identify ground states in different parameter regimes of the model, mapping out a quantum phase transition which is the hallmark feature of the model. The resulting states are analyzed via quantum state tomography, to reveal how characteristic properties such as correlations in the output state change across the phase transition. Moreover, we use the results to determine the phase boundaries of the model.

Preparing ground states of correlated molecules on near-term quantum computers requires algorithms that are both accurate and resource-efficient. We introduce the qubit-adaptive variational quantum imaginary time evolution (QAITE) algorithm, which synthesizes the systematic convergence of imaginary-time evolution with hardware-efficient, adaptive ansatz construction. By integrating the linear-scaling qubit operator pool into the adaptive variational quantum imaginary time evolution framework, QAITE deterministically constructs a compact ansatz without costly classical optimization loops. Numerical simulations for molecular systems (H4, LiH, H2O, BeH2), benchmarked against our own implementations of established adaptive algorithms under identical conditions, show that QAITE achieves chemical accuracy with a 15-45% reduction in CNOT gates compared to the standard adaptive variational quantum imaginary time evolution (AVQITE) method. For hydrogen chains (RHH = 1.5 Å, N = 6-16 qubits), QAITE's circuit complexity grows approximately quadratically and remains the lowest of the four methods at every system size, reaching 31% fewer CNOT gates than AVQITE at 16 qubits─a saving of 1322 gates. This efficiency extends to heterogeneous (LiH)n and (HBe)n chains, where QAITE maintains a CNOT gate ratio below 1 for all sizes studied (e.g., 0.81 for (LiH)2). We further show that the deterministic parameter update is numerically robust: the results are insensitive to the regularization of the quantum geometric tensor over at least 4 orders of magnitude, and chemical accuracy is retained under realistic shot noise applied to both the quantum geometric tensor and the force vector. By outperforming established adaptive algorithms in both absolute gate count and scaling, QAITE establishes a promising, scalable pathway for quantum molecular simulations on noisy intermediate-scale quantum devices.

Abstract While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum--classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy---standard L2 squared-norm regularization---can systematically stabilize hybrid quantum--classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty

(R(\boldsymbol{\theta}) = \lambda\|\boldsymbol{\theta}\|^2)

, without modifying the quantum circuit or measurement process. Across all tested Hamiltonians---H

(_2)

, LiH, and the Random Field Ising Model (RFIM)---we observe improved performance over a broad window of the regularization strength

(\lambda)

. Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.