Quantum-Classical Computation of Schwinger Model Dynamics using Quantum Computers

TL;DR

This work presents a hybrid quantum-classical approach to simulate the Schwinger model on IBM quantum hardware by projecting onto symmetry sectors (momentum and parity) and Gauss's law constraints to drastically shrink the Hilbert space. Ground-state properties are obtained via VQE with Bayesian optimization, while real-time dynamics are explored using both SU(4)-based exact propagators and Trotterized time evolution, complemented by CNOT error mitigation. The results demonstrate meaningful quantum simulations within a reduced physical subspace and highlight hardware limitations that currently constrain longer-time dynamics, outlining a practical path toward more complex lattice gauge theories on NISQ devices. The study argues that symmetry-based reduction, combined with hybrid computation, can enable non-equilibrium and finite-density QFT explorations that are challenging for classical methods and point toward future extensions toward QCD-scale problems.

Abstract

We present a quantum-classical algorithm to study the dynamics of the two-spatial-site Schwinger model on IBM's quantum computers. Using rotational symmetries, total charge, and parity, the number of qubits needed to perform computation is reduced by a factor of $\sim 5$, removing exponentially-large unphysical sectors from the Hilbert space. Our work opens an avenue for exploration of other lattice quantum field theories, such as quantum chromodynamics, where classical computation is used to find symmetry sectors in which the quantum computer evaluates the dynamics of quantum fluctuations.

Figures (12)

• Figure 1: A schematic of the qubit and electric flux link structure of the two-spatial-site lattice Schwinger model. Even sites (marked 0 and 2) represent the electron content with spin up denoting the presence of an electron. Odd sites (marked 1 and 3) represent the positron content with spin down denoting the presence of a positron. The strong-coupling vacuum (unoccupied) state is antiferromagnetic.
• Figure 2: The $H_{\textbf{k=0},+}^{\tilde{\Lambda}=3}$ ground state energy and chiral condensate (purple, blue extrapolated to -1.000(65) and -0.296(13), respectively) expectation values as a function of $r$, the noise parameter. $r-1$ is the number of additional CNOT gates inserted at each location of a CNOT gate in the original VQE circuit. (1200 IBM allocation units and $\sim6.4$ QPU$\cdot$s)
• Figure 3: The probability of finding an $e^+e^-$ pair (blue, lower line) and the expectation value of the energy of the electric field (purple, upper line) in the two-spatial-site Schwinger model following time evolution with $U(\theta_i(t))$ from the initial empty state. The solid curves are exact results while the the data points are quadratic extrapolations obtained with the ibmqx2 quantum computer using a circuit involving 3 CNOT gates VidalDawson2004. (1000 IBM allocation units and $\sim12.3$ QPU$\cdot$s)
• Figure 4: The probability of finding an $e^+e^-$ pair in the two-spatial-site Schwinger model from the initial empty state following time evolution with $U_T(t,\delta t)$. In the unshaded region, the blue points (triangle markers with visible error bars) are quadratic extrapolations to zero noise using the data above each point at increasing values of the noise parameter, $r$. (260 IBM allocation units and $\sim3.6$ QPU$\cdot$s)
• Figure 5: Examples of the action of the parity operators defined by the "electron" axes (blue lines, horizontal arrows and site 0-2 symmetry axis) and "positron" axes (green lines, diagonal arrows and site 1-3 symmetry axis). An $e^-$ or an $e^+$ in one of the squares at a site indicates that the particle is present. An arrow indicates an electric flux link aligned with the arrow, while a dashed link corresponds to the absence of an electric flux link. In the $0+1$ example (upper panel) the only symmetry axis passes through both an electron and positron. In the $1+1$ example (lower panel) there are two symmetry axes, one through the electron sites, and one through the positron sites.