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Explicit Block Encodings of Discrete Laplacians with Mixed Boundary Conditions

Alexandre Boutot, Viraj Dsouza

Abstract

Discrete Laplacian operators arise ubiquitously in scientific computing and frequently appear in quantum algorithms for tasks such as linear algebra, Hamiltonian simulation, and partial differential equations. Block encoding provides the standard method for accessing matrix data within quantum circuits. Efficient implementations of such algorithms require efficient block encodings of the discretized operator. While several general-purpose techniques exist for block encoding arbitrary matrices, they usually require deep quantum circuits. Moreover, existing efficient constructions that exploit Laplacian structure are limited in scope, typically assuming fixed boundary conditions or uniform grid resolutions. In this work, we present a unified framework for efficiently block encoding finite-difference discretizations of the Laplacian that supports Dirichlet, periodic, and Neumann boundary conditions in arbitrary spatial dimensions. Our construction allows different boundary conditions and grid sizes to be specified independently along each coordinate axis, enabling mixed-boundary and anisotropic discretizations within a single modular circuit architecture. We provide analytical gate-complexity estimates and perform circuit-level benchmarks after transpilation to an IBM hardware gate set. Across one-, two-, and three-dimensional examples, the resulting circuits exhibit substantially lower gate counts and higher success probabilities when compared to certain existing approaches.

Explicit Block Encodings of Discrete Laplacians with Mixed Boundary Conditions

Abstract

Discrete Laplacian operators arise ubiquitously in scientific computing and frequently appear in quantum algorithms for tasks such as linear algebra, Hamiltonian simulation, and partial differential equations. Block encoding provides the standard method for accessing matrix data within quantum circuits. Efficient implementations of such algorithms require efficient block encodings of the discretized operator. While several general-purpose techniques exist for block encoding arbitrary matrices, they usually require deep quantum circuits. Moreover, existing efficient constructions that exploit Laplacian structure are limited in scope, typically assuming fixed boundary conditions or uniform grid resolutions. In this work, we present a unified framework for efficiently block encoding finite-difference discretizations of the Laplacian that supports Dirichlet, periodic, and Neumann boundary conditions in arbitrary spatial dimensions. Our construction allows different boundary conditions and grid sizes to be specified independently along each coordinate axis, enabling mixed-boundary and anisotropic discretizations within a single modular circuit architecture. We provide analytical gate-complexity estimates and perform circuit-level benchmarks after transpilation to an IBM hardware gate set. Across one-, two-, and three-dimensional examples, the resulting circuits exhibit substantially lower gate counts and higher success probabilities when compared to certain existing approaches.
Paper Structure (19 sections, 4 theorems, 54 equations, 21 figures)

This paper contains 19 sections, 4 theorems, 54 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Discrete Laplacians and Block Encoding Preliminaries
  3. Finite-Difference Laplacian Operators
  4. Block-Encoding
  5. Generalized Block Encoding of Multi-Dimensional Laplacians
  6. Block encoding a $1D$ Laplacian
  7. Periodic boundary conditions
  8. Dirichlet boundary conditions
  9. Von Neumann boundary conditions
  10. Block encoding N-dimensional Laplacian operators
  11. Resource Estimates and Comparisons
  12. Theoretical resource estimates
  13. Comparative circuit implementation benchmarks
  14. $1$D Laplacians
  15. $2$D Laplacians
  16. ...and 4 more sections

Key Result

Theorem 1

Let $U_p$ denote the unitary implemented by the quantum circuit shown in Fig.fig:1DP_circuit, acting on $n$ system qubits and $m=2$ ancilla qubits. Then $U_p$ is an exact $(1,2,0)$-block encoding of the scaled one-dimensional periodic Laplacian $\widetilde{L}^{(1)}_{\mathrm{p}}$, i.e., Equivalently, $U_p$ has the block form Consequently, when the ancilla register is initialized in $\ket{0}^{\oti

Figures (21)

  • Figure 1: Block encoding circuit for the 1D Laplacian with periodic boundary, as provided in sturm2025efficientexplicitblockencoding.
  • Figure 2: Block encoding circuit for the 1D Laplacian with Dirichlet boundary.
  • Figure 3: Block encoding circuit for the 1D Laplacian with von Neumann boundary.
  • Figure 4: Block encoding circuit for the $D-$dimensional Laplacian with boundaries $b_1, b_2, \cdots b_{D}$ where each $b_i$ can be Dirichlet, periodic or von Neumann. The circuits for each $U_{b_i}$ can be inferred from the 1D constructions provided earlier. For example, if a dimension $m$ has a Dirichlet boundary $U_{b_m}$ is just the Identity matrix.
  • Figure 5: State preparation circuit ($U_{prep\_k}$) for $D=3,4$.
  • ...and 16 more figures

Theorems & Definitions (7)

  • Theorem 1: Block encoding of the 1D periodic Laplacian
  • Theorem 2: Block encoding of the 1D Dirichlet Laplacian
  • proof
  • Theorem 3: Block encoding of the 1D Neumann Laplacian
  • proof
  • Theorem 4: Block encoding of $D-$dimensional Laplacian
  • proof