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.

中文翻译：

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在量子计算机上模拟施温格模型的端到端复杂性


施温格模型是最简单的规范理论之一。众所周知，模型的拓扑项导致了经典蒙特卡罗方法中臭名昭著的符号问题。与此相反，最近，哈密顿形式主义的量子计算引起了人们的关注。在这项工作中，我们估计了量子计算机计算物理量所需的资源，而这些物理量在经典计算机上计算具有挑战性。具体来说，我们提出了施温格模型哈密顿量块编码的有效实现。考虑到哈密顿量的结构，这种标准化因子为 $\mathcal{O}(N^3)$ 的块编码可以使用 $\mathcal{O}(N+\log^2(N/\varepsilon) 来实现)$ T 门。作为端到端应用程序，我们计算真空持久振幅。结果，我们发现，对于系统大小 $N=128$ 和加性误差 $\varepsilon=0.01$，演化时间 $t$ 和晶格间距 a 满足 $t/2a=10$，真空持久振幅可以使用大约 $10^{13}$ T 门来计算。我们的结果提供了对 FTQC 和早期 FTQC 时代量子计算机性能的预测的见解，阐明了在现实时间范围内解决有意义的问题所面临的挑战。