Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Linear Combination of Unitaries Decomposition for the Laplace Operator

Thomas Hogancamp, Reuben Demirdjian, Daniel Gunlycke

TL;DR

This work develops explicit linear combination of unitaries (LCU) decompositions for discrete Poisson operators arising from finite-difference discretizations of the Laplacian on unit intervals and box-type domains with periodic, Dirichlet, Neumann, Robin, and mixed boundaries. The decompositions yield a number of unitary terms that is independent of grid size and scales linearly with dimension, with explicit circuit constructions achieving $O(n)$-type gate costs and $O(\log N)$-depth in favorable settings. The authors provide detailed LCUs for 1D Dirichlet (5-term) and Neumann/Robin (up to 10-term) cases and extend the framework to higher dimensions and to first-order derivative terms with variable coefficients, including polynomial data, giving $O(n^k)$ terms for degree-$k$ polynomials. They analyze resource counts and compare with existing approaches, showing significant reductions in Toffoli counts for VQLS iterations and outlining extensions to fault-tolerant solvers and broader PDE classes, with potential for practical quantum PDE solvers on near-term devices.

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.

A Linear Combination of Unitaries Decomposition for the Laplace Operator

TL;DR

This work develops explicit linear combination of unitaries (LCU) decompositions for discrete Poisson operators arising from finite-difference discretizations of the Laplacian on unit intervals and box-type domains with periodic, Dirichlet, Neumann, Robin, and mixed boundaries. The decompositions yield a number of unitary terms that is independent of grid size and scales linearly with dimension, with explicit circuit constructions achieving -type gate costs and -depth in favorable settings. The authors provide detailed LCUs for 1D Dirichlet (5-term) and Neumann/Robin (up to 10-term) cases and extend the framework to higher dimensions and to first-order derivative terms with variable coefficients, including polynomial data, giving terms for degree- polynomials. They analyze resource counts and compare with existing approaches, showing significant reductions in Toffoli counts for VQLS iterations and outlining extensions to fault-tolerant solvers and broader PDE classes, with potential for practical quantum PDE solvers on near-term devices.

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.
Paper Structure (28 sections, 8 theorems, 97 equations, 2 figures, 1 table)

This paper contains 28 sections, 8 theorems, 97 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. VQLS Overview
  5. The Problem
  6. $1$-dimensional Dirichlet
  7. $1$-dimensional Periodic
  8. $1$-dimensional Neumann/Robin
  9. Higher-dimensional problems
  10. Main Results
  11. LCU Decompositions
  12. Periodic case
  13. Dirichlet case
  14. Neumann/Robin case
  15. Higher-Dimensional Problems
  16. ...and 13 more sections

Key Result

Theorem 4.1

(Dirichlet Case) The matrix $A_{n,1}$ admits a decomposition of the form where $c_{1,l} \in \{1, -2,\pm \frac{1}{2} \}$ and each $R_{1,l}$ is unitary. Moreover, if a single ancilla qubit is available, then a given $R_{1,l}$ may be implemented using at most $2$-Hadamard gates and $O(n)$ Toffoli, $X$, and $CNOT$ gates.

Figures (2)

  • Figure 1: Hadamard Test Circuit for $\beta_{ll'}$ (real-valued case).
  • Figure 2: Hadamard Test Circuit for $\delta^{(1)}_{ll'}$ (real-valued case). The target of the controlled $Z$ gate must be modified appropriately for $j\neq 1$.

Theorems & Definitions (19)

  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • Theorem 4.5
  • Remark 4.6
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof
  • ...and 9 more