A Linear Combination of Unitaries Decomposition for the Laplace Operator

TL;DR

This paper introduces a new linear combination of unitaries decomposition for discrete elliptic operators, enabling efficient quantum algorithms for solving boundary value problems with favorable scaling properties.

Contribution

It presents a novel LCU decomposition for elliptic operators that is independent of grid size and scalable with dimension, including explicit circuit constructions and complexity analysis.

Findings

Unitary term count is independent of grid points.

Circuit depth scales logarithmically with grid size.

Method shows favorable scaling within the Variational Quantum Linear Solver.

Abstract

We provide novel linear combination of unitaries decompositions for a class of discrete elliptic differential operators. Specifically, Poisson problems augmented with periodic, Dirichlet, Neumann, Robin, and mixed boundary conditions are considered on the unit interval and on higher-dimensional rectangular domains. The number of unitary terms required for our decomposition is independent of the number of grid points used in the discretization and scales linearly with the spatial dimension. Explicit circuit constructions for each unitary are given and their complexities analyzed. The worst case depth and elementary gate cost of any such circuit is shown to scale at most logarithmically with respect to number of grid points in the underlying discrete system. We also investigate the cost of using our method within the Variational Quantum Linear Solver algorithm and show favorable scaling. Finally, we extend the proposed decomposition technique to treat problems that include first-order derivative terms with variable coefficients.