Analog quantum simulation of parabolic partial differential equations using Jaynes-Cummings-like models

TL;DR

This paper introduces an efficient analog quantum simulation method using Jaynes-Cummings-like models to solve parabolic partial differential equations, leveraging simple Hamiltonian interactions and qudits for improved resource efficiency.

Contribution

It proposes a novel quantum simulation protocol for PDEs using hyperbolic approximations and Jaynes-Cummings-like interactions, optimizing resource use with qudits instead of qubits.

Findings

Efficient simulation of heat, Black-Scholes, and Fokker-Planck equations.

Use of a single d-level system (qudit) for d-dimensional problems.

Potential implementation in cavity and circuit QED systems.

Abstract

We present a simplified analog quantum simulation protocol for preparing quantum states that embed solutions of parabolic partial differential equations, including the heat, Black-Scholes and Fokker-Planck equations. The key idea is to approximate the heat equations by a system of hyperbolic heat equations that involve only first-order differential operators. This scheme requires relatively simple interaction terms in the Hamiltonian, which are the electric and magnetic dipole moment-like interaction terms that would be present in a Jaynes-Cummings-like model. For a d-dimensional problem, we show that it is much more appropriate to use a single d-level quantum system - a qudit - instead of its qubit counterpart, and d+1 qumodes. The total resource cost is efficient in d and precision error, and has potential for realisability for instance in cavity and circuit QED systems.