Optimized quantum algorithms for simulating the Schwinger effect

TL;DR

This work analyzes the quantum-resource costs required to simulate real-time Schwinger-effect dynamics in a (1+1)D lattice QED model under two regimes: a quench and a scattering setup. It compares two fault-tolerant quantum algorithms—second-order Suzuki-Trotter (PF2) and an interaction-picture (IP) Dyson-series approach—providing optimized circuit constructions, rigorous error bounds, and explicit resource counts across a wide parameter range. The study demonstrates regimes where IP outperforms PF2 (notably for large evolution times and high electric-field cutoffs) and where PF2 is favorable (short times or modest precision), offering a detailed, parameter-driven framework for planning quantum simulations of lattice gauge theories. The methods and subroutines developed here—improved error bounds, optimized circuit designs, and compiled primitives—are broadly applicable to other lattice models in high-energy physics and beyond, potentially informing near- and future fault-tolerant quantum simulations of gauge theories.

Abstract

The Schwinger model, which describes lattice quantum electrodynamics in $1+1$ space-time dimensions, provides a valuable framework to investigate fundamental aspects of quantum field theory, and a stepping stone towards non-Abelian gauge theories. Specifically, it enables the study of physically relevant dynamical processes, such as the nonperturbative particle-antiparticle pair production, known as the Schwinger effect. In this work, we analyze the quantum computational resource requirements associated with simulating the Schwinger effect under two distinct scenarios: (1) a quench process, where the initial state is a simple product state of a non-interacting theory and then interactions are turned on at time $t=0$, and (2) a splitting (or scattering) process where two Gaussian states, peaked at given initial momenta, are shot away from (or towards) each other. We explore different physical regimes in which the Schwinger effect is expected to be observable. These regimes are characterized by initial momenta and coupling strengths, as well as simulation parameters such as lattice size and electric-field cutoffs. Leveraging known rigorous bounds for electric-field cutoffs, we find that a reliable simulation of the Schwinger effect is provably possible at high cutoff scales. Furthermore, we provide optimized circuit implementations of both the second-order Trotter formula and an interaction-picture algorithm based on the Dyson series to implement the time evolution. Our detailed resource estimates show the regimes in which the interaction-picture approach outperforms the Trotter approach, and vice versa. The improved theoretical error bounds, optimized quantum circuit designs, and explicitly compiled subroutines developed in this study are broadly applicable to simulations of other lattice models in high-energy physics and beyond.

Figures (8)

• Figure 1: A schematic representation of the initial state for the second experiment: an electron-positron pair ($e^- - e^+$) is created that are $l$ apart, and moving away from each other with momentum $p_1$ and $p_2$, respectively.
• Figure 2: Resource cost comparison of two different implementations of IP-based algorithms. The blue data points correspond to the implementation without SORT, using $t_0= 0.5$, as described in Ref. low2018hamiltonian. The orange data points correspond to the implementation using the optimal $t_0= \ln(2)$ with SORT, which requires more qubits but fewer T-gates. The comparison is shown in terms of T counts and qubit counts for time parameters $t \in (t_\text{min}, 10t_\text{min})$, and accuracies $\epsilon \in \{0.001, 0.01, 0.1\}$. We fix $N_0 = 8$, $x = 0.1$, $\mu = 1$, $\Lambda_0 = \sqrt{10}\,\mu = \sqrt{10}$, and $t_\text{min} = 0.5/x$. The parameters $N$ and $\Lambda$ are chosen according to Tab. \ref{['tab:Parameters']}.
• Figure 3: Resource cost comparison of two different compilations of the IP-based algorithm with $t_0 = \ln(2)$. The blue data points correspond to the compilations that use the PGA, and the orange data points correspond to the Mult approach. The comparison is shown in terms of T counts and qubit counts for time parameters $t \in (t_\text{min}, 10t_\text{min})$, and accuracies $\epsilon \in \{0.001, 0.01, 0.1\}$. We fix $N_0 = 8$, $x = 0.1$, $\mu = 1$, $\Lambda_0 = \sqrt{10}\,\mu = \sqrt{10}$, and $t_\text{min} = 0.5/x$. The parameters $N$ and $\Lambda$ are chosen according to Tab. \ref{['tab:Parameters']}.
• Figure 4: Resource cost comparison between the best implementations of IP-based algorithm and $\textrm{PF}_2$ approach. The comparison is given in terms of T-counts and qubit counts, for different time parameters $t \in (t_\text{min}, 10t_\text{min})$, and for accuracy $\epsilon= \{0.001, 0.01, 0.1\}$. We fix $N_0=8$, $\mu =1$, $\Lambda_0 = \sqrt{10}\mu = \sqrt{10}$, and $t_\text{min} = 0.5/x$. The parameters $N$ and $\Lambda$ are chosen according to Tab. \ref{['tab:Parameters']}.
• Figure 5: Resource cost of simulating the time evolution of the Schwinger model with a second-order product-formula method, for $x \in \{0.1, 1, 10, 100\}$ (indicated by the colors) and time-independent $\eta \in \{2, 3, 4, 5\}$ (from the smallest to the largest number of qubits). We fix $\mu=1$, $N_0=8$ and $t = t_\text{min}= 0.5/x$. The number of fermions $N$ is chosen according to Tab. \ref{['tab:Parameters']}.

Theorems & Definitions (18)

• Lemma 7.1: Time evolution operator in the Schrödinger and in the interaction picture
• proof
• Definition 7.2: $D_{(K,M)}(t)$: $(K,M)$-truncated-discretized Dyson series
• Definition 7.3: $\tilde{D}_{(K,M)}(t)$: $(K,M)$-truncated-discretized Dyson series with collisions
• Theorem 7.4: Approximating time evolution in the interaction picture with truncated-discretized Dyson series
• Lemma 7.5: Approximation error of truncating the Dyson series
• Lemma 7.6: Approximation error of discretizing the truncated Dyson series
• Lemma 7.7: Approximating time evolution in the interaction picture with the truncated-discretized Dyson series with collisions
• proof
• Theorem 7.8: Approximating time evolution in the interaction picture with truncated-discretized Dyson series without collisions