# Electric-circuit simulation of the Schr\"{o}dinger equation and non-Hermitian quantum walks

## TL;DR

This paper demonstrates how electric circuits can simulate the Schrödinger equation and quantum walks, including topological and non-Hermitian effects, by reformulating Kirchhoff's law and analyzing circuit dynamics.

## Contribution

It introduces a method to simulate Schrödinger dynamics and quantum walks in electric circuits, including topological, dissipative, and nonreciprocal effects, expanding circuit-based quantum simulation capabilities.

## Key findings

- Electric circuits can simulate Schrödinger equation dynamics.
- Quantum walks differ in topological and trivial phases in circuits.
- Non-Hermitian effects are incorporated via resistors, enabling complex simulations.

## Abstract

Recent progress has witnessed that various topological physics can be simulated by electric circuits under alternating current. However, it is still a nontrivial problem if it is possible to simulate the dynamics subject to the Schr\"{o}dinger equation based on electric circuits. In this work, we reformulate the Kirchhoff law in one dimension in the form of the Schr\"{o}dinger equation. As a typical example, we investigate quantum walks in $LC$ circuits. We also investigate how quantum walks are different in topological and trivial phases by simulating the Su-Schrieffer-Heeger model in electric circuits. We then generalize them to include dissipation and nonreciprocity by introducing resistors, which produce non-Hermitian effects. We point out that the time evolution of one-dimensional quantum walks is exactly solvable with the use of the generating function made of the Bessel functions.