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arXiv:2302.10933 · Physical review. D/Physical review. D.

Quench dynamics of the Schwinger model via variational quantum algorithms

Lento Nagano, Aniruddha Bapat, C. Bauer

Published Aug 2, 2023• 26 citations
arXivHTMLPDF

Abstract

We investigate the real-time dynamics of the ($1+1$)-dimensional U(1) gauge theory known as the Schwinger model via variational quantum algorithms. Specifically, we simulate quench dynamics in the presence of an external electric field. First, we use a variational quantum eigensolver to obtain the ground state of the system in the absence of an external field. With this as the initial state, we perform real-time evolution under an external field via a fixed-depth, parameterized circuit whose parameters are updated using McLachlan's variational principle. We use the same ansatz for initial-state preparation and time evolution, by which we are able to reduce the overall circuit depth. We test our method with a classical simulator and confirm that the results agree well with exact diagonalization.

DOI
10.1103/physrevd.108.034501

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In this article we investigate the real-time dynamics in the (1+1)(1+1)(1+1)-dimensional U(1)U(1)U(1) gauge theory called the Schwinger model by using variational quantum algorithms. Specifically, we first prepare the ground state of the Hamiltonian without external electric field via the variational quantum eigensolver, and then perform real-time evolution under the Hamiltonian in the presence of the external field using the variational quantum simulation method. The same ansatz is used for both algorithms which reduces the overall depth of the quantum circuit. We test our protocol by using a noiseless statevector simulator and confirm that results from the quantum algorithms are consistent with those obtained by exact diagonalization. This article summarizes our previous work [1].
We use the (1+1)-dimensional Schwinger model to investigate the nonequilibrium dynamics of a finite lattice system under a constant external electric field. The lattice Hamiltonian is constructed under open boundary conditions. The vacuum state is prepared using the variational quantum eigensolver (VQE). Scans over the external field strength show the flip of the vacuum state at several field strengths. The critical field strengths agree with theoretical predictions. We further investigate the real-time evolution of the zero-field vacuum under an external electric field using a second-order Trotter-Suzuki decomposition. By comparison with exact diagonalization (ED), we verify that the quantum-simulation protocol reproduces the main features of field-induced boundary charge separation, decay of the vacuum-state fidelity, and quasiperiodic energy redistribution between the electric-field energy term and the fermionic sector. Our results indicate that combining VQE-based state preparation with digital real-time evolution provides a useful approach for studying nonequilibrium dynamics in strong-field lattice gauge theories.
Simulating the real-time dynamics of gauge theories represents a paradigmatic use case to test the hardware capabilities of a quantum computer, since it can involve non-trivial input states' preparation, discretized time evolution, long-distance entanglement, and measurement in a noisy environment. We implemented an algorithm to simulate the real-time dynamics of a few-qubit system that approximates the Schwinger model in the framework of lattice gauge theories, with specific attention to the occurrence of a dynamical quantum phase transition. Limitations in the simulation capabilities on IBM Quantum were imposed by noise affecting the application of single-qubit and two-qubit gates, which combine in the decomposition of Trotter evolution. The experimental results collected in quantum algorithm runs on IBM Quantum were compared with noise models to characterize the performance in the absence of error mitigation.
Abstract We explore the first-order phase transition in the lattice Schwinger model in the presence of a topological θ-term by means of the variational quantum eigensolver (VQE). Using two different fermion discretizations, Wilson and staggered fermions, we develop parametric ansatz circuits suitable for both discretizations, and compare their performance by simulating classically an ideal VQE optimization in the absence of noise. The states obtained by the classical simulation are then prepared on the IBM's superconducting quantum hardware. Applying state-of-the art error-mitigation methods, we show that the electric field density and particle number, observables which reveal the phase structure of the model, can be reliably obtained from the quantum hardware. To investigate the minimum system sizes required for a continuum extrapolation, we study the continuum limit using matrix product states, and compare our results to continuum mass perturbation theory. We demonstrate that taking the additive mass renormalization into account is vital for enhancing the precision that can be obtained with smaller system sizes. Furthermore, for the observables we investigate we observe universality, and both fermion discretizations produce the same continuum limit.
This note discusses a method for computing the energy spectra of quantum field theory utilizing digital quantum simulation. A quantum algorithm, called coherent imaging spectroscopy, quenches the vacuum with a time-oscillating perturbation and then reads off the excited energy levels from the loss in the vacuum-to-vacuum probability following the quench. As a practical demonstration, we apply this algorithm to the (1+1)-dimensional quantum electrodynamics with a topological term known as the Schwinger model, where the conventional Monte Carlo approach is practically inaccessible. In particular, on a classical simulator, we prepare the vacuum of the Schwinger model on a lattice by adiabatic state preparation and then apply various types of quenches to the approximate vacuum through Suzuki-Trotter time evolution. We discuss the dependence of the simulation results on the specific types of quenches and introduce various consistency checks, including the exact diagonalization and the continuum limit extrapolation. The estimation of the computational complexity required to obtain physically reasonable results implies that the method is likely efficient in the coming era of early fault-tolerant quantum computers.
Abstract Variational Quantum Eigensolver (VQE) addresses challenges that conventional computation struggles to overcome. However, VQE lacks the ability to autonomously trend towards the direction of quantum advantage, leading to unnecessary consumption of time and resources during parameter optimization. In this work, we present a Simulating Dynamic VQE (D-VQE) that incorporates time evolution for achieving desired simulation accuracy and parameter optimization. Introducing causal cones for Parameterized Quantum Circuits (PQC) and establishing dynamic judgment rules. Consequently, we can reduce the overstacking of components in PQC ansatz. Based on the requirements of precision and circuit scale, a reasonable operator pool is set to deal with the parameters. Through numerical experiments, our method effectively simulates the search for the ground state in the transverse-field Ising model, achieving a relative energy error below 10^(-3) and improving ground state approximation capability. Moreover, in the context of optimized Boltzmann machine learning, the Kullback-Leibler divergence (KLD) value converges to 0.03. These results prove that our work provides an effective and feasible consideration for the improvement of variational quantum algorithms.