Explicit block encodings of boundary value problems for many-body elliptic operators

TL;DR

This paper develops explicit quantum circuit methods for block encoding elliptic operators in boundary value problems, enabling efficient quantum simulation of many-body physical systems with various boundary conditions.

Contribution

It introduces explicit circuit constructions for block encoding many-body elliptic operators, including high-precision schemes and arbitrary domain implementations, advancing quantum simulation capabilities.

Findings

Explicit circuits for many-body Laplacian with various boundary conditions.

High-precision schemes with minimal circuit depth increase.

Block encoding of many-body convective operators for interacting particles.

Abstract

Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.