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First-order phase transition of the Schwinger model with a quantum computer

Takis Angelides, Pranay Naredi, Arianna Crippa, Karl Jansen, Stefan Kühn, Ivano Tavernelli, Derek S. Wang

Published Jan 17, 2025• 21 citations
Publisher

Abstract

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.

DOI
10.1038/s41534-024-00950-6

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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.
We explore the first-order phase transition in the lattice Schwinger model in the presence of a topological θ\thetaθ-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.
Variational methods have long been used to study strongly-correlated quantum systems. Despite the successes of classical protocols, these are limited in the study of quantum systems, either due to numerical errors like the sign problem or the massive memory requirement to represent large quantum systems. Variational quantum eigensolvers (VQEs) overcome these problems and as a result have been an exciting area of research for the past decade. VQEs use a hybrid classical-quantum set-up, where a cost function (usually the expectation value of an operator) is produced by the quantum processor and then optimized on a classical computer. VQEs have been applied to fields such as quantum chemistry, condensed matter physics, high-energy physics, and classical optimization problems. In this thesis, we investigate VQEs to study gauge theories, which describe fundamental physical interactions and are hard to simulate classically. We use two different approaches in implementing a VQE to study two classes of problems. In the first half of the thesis, we design a VQE to study the U(1) Higgs gauge theory in one spatial dimension. This model is particularly interesting due to the presence of a topological term, which is a candidate to explain the observed CP violation in the universe. In order to perform a quantum simulation of the U(1) Higgs model, the theory is cast on a lattice. In the limit where the lattice size is large and the lattice spacing is small, the continuum limit can be reached and the results correctly resemble the original theory. However, when the lattice is too small or the spacing too big, results are spoiled by finite size effects. In this thesis, we investigate the effects of a finite lattice and determine the parameters to correctly reproduce a first-order phase transition of the continuous theory. Importantly, to resemble a realistic experiment, we include the statistical and intrinsic quantum noise in our VQE. We design the protocol to be applied on a microwave-photonic platform, which is well-suited due to the bosonic nature of the theory's electrical modes. \nIn the second half of the thesis, we design a new class of VQEs, based on a measurement-based quantum computer (MBQC) instead of the typical circuit-based set-up. Rather than performing gates on an initial state, a MBQC transforms an initial state by entangling it with a number of auxiliary qubits which are subsequently measured, resulting in a modification of the initial state. In this part of the thesis, we compare the standard, circuit-based VQE with our newly-designed measurement-based VQE (MB-VQE) through two meaningful examples. First, we introduce a novel way to construct variational state families, using measurement-based techniques, that is more costly to access with circuit-based VQEs. We apply this technique to find the ground state of a two-dimensional, periodic Z2 matter-free lattice gauge theory. In the second approach, we directly translate \nthe circuit-based VQE to a MB-VQE and apply this to the U(1) Schwinger lattice gauge theory, which is used as a benchmarking problem for VQEs. These examples demonstrate that there are specific problems which are better suited for a MB-VQE.
We numerically analyze the critical behaviors of the lattice Schwinger model by applying the variational uniform matrix product state (VUMPS) algorithm combined with a gauge-invariant matrix product ansatz that locally enforces the Gauss law constraint. Both the continuum and lattice versions of the Schwinger model with θ=π\theta=\piθ=π are known to exhibit first-order phase transitions for the values of the fermion mass above a critical value, where a second-order phase transition occurs. Our algorithm enables a precise determination of the critical point in the continuum theory. We further analyze the scaling in the simultaneous critical and continuum limits and confirm that the data collapse aligns with the Ising universality class with high precision.
Simulations of lattice gauge theories with tensor networks and quantum computing have so far mainly focused on staggered fermions. In this paper, we use matrix product states to study Wilson fermions in the Hamiltonian formulation and present a novel method to determine the additive mass renormalization. Focusing on the single-flavor Schwinger model as a benchmark model, we investigate the regime of a nonvanishing topological θ-term, which is inaccessible to conventional Monte Carlo methods. We systematically explore the dependence of the mass shift on the volume, the lattice spacing, the θ-parameter, and the Wilson parameter. This allows us to follow lines of constant renormalized mass, and therefore to substantially improve the continuum extrapolation of the mass gap and electric field density. For small values of the mass, our continuum results agree with the theoretical prediction from mass perturbation theory. Going beyond Wilson fermions, our technique can also be applied to staggered fermions, and we demonstrate that the results of our approach agree with a recent theoretical prediction for the mass shift at sufficiently large volumes.
We study the lattice Schwinger model by combining the variational uniform matrix product state (VUMPS) algorithm with a gauge-invariant matrix product ansatz that locally enforces the Gauss law constraint. Both the continuum and lattice versions of the Schwinger model with θ = π are known to exhibit first-order phase transitions for the values of the fermion mass above a critical value, where a second-order phase transition occurs. Our algorithm enables a precise determination of the critical point in the continuum theory. We further analyze the scaling in the simultaneous critical and continuum limits and confirm that the data collapse aligns with the Ising universality class to remarkable precision.
We perform a digital quantum simulation of a gauge theory with a topological term in Minkowski spacetime, which is practically inaccessible by standard lattice Monte Carlo simulations. We focus on 1+11+11+1 dimensional quantum electrodynamics with the θ\thetaθ-term known as the Schwinger model. We construct the true vacuum state of a lattice Schwinger model using adiabatic state preparation which, in turn, allows us to compute an expectation value of the fermion mass operator with respect to the vacuum. Upon taking a continuum limit we find that our result in massless case agrees with the known exact result. In massive case, we find an agreement with mass perturbation theory in small mass regime and deviations in large mass regime. We estimate computational costs required to take a reasonable continuum limit. Our results imply that digital quantum simulation is already useful tool to explore non-perturbative aspects of gauge theories with real time and topological terms.