End-to-end complexity for simulating the Schwinger model on quantum computers

TL;DR

This paper estimates the quantum computational resources needed to simulate the Schwinger model, a simple gauge theory, demonstrating the feasibility and challenges of quantum simulations for complex physical systems.

Contribution

It introduces an efficient block-encoding method for the Schwinger model Hamiltonian and provides resource estimates for computing the vacuum persistence amplitude on quantum computers.

Findings

For N=128, about 10^13 T gates are needed for the simulation.

The block-encoding implementation scales as O(N + log^2(N/ε)) T gates.

Insights into quantum computer performance for complex gauge theories.

Abstract

The Schwinger model is one of the simplest gauge theories. It is known that a topological term of the model leads to the infamous sign problem in the classical Monte Carlo method. In contrast to this, recently, quantum computing in Hamiltonian formalism has gained attention. In this work, we estimate the resources needed for quantum computers to compute physical quantities that are challenging to compute on classical computers. Specifically, we propose an efficient implementation of block-encoding of the Schwinger model Hamiltonian. Considering the structure of the Hamiltonian, this block-encoding with a normalization factor of $\mathcal{O}(N^3)$ can be implemented using $\mathcal{O}(N+\log^2(N/\varepsilon))$ T gates. As an end-to-end application, we compute the vacuum persistence amplitude. As a result, we found that for a system size $N=128$ and an additive error $\varepsilon=0.01$, with an evolution time $t$ and a lattice spacing a satisfying $t/2a=10$, the vacuum persistence amplitude can be calculated using about $10^{13}$ T gates. Our results provide insights into predictions about the performance of quantum computers in the FTQC and early FTQC era, clarifying the challenges in solving meaningful problems within a realistic timeframe.