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

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 actual resources needed for quantum computers to compute physical quantities that are challenging to compute on classical computers. Specifically, we propose an implementation of block-encoding of the Schwinger model Hamiltonian, which has a normalization factor of O(N³) and requires O(N+log²(N/ε)) 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=100 and an additive error ε=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 10¹³ 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.