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Block encoding of sparse matrices with a periodic diagonal structure

Alessandro Andrea Zecchi, Claudio Sanavio, Luca Cappelli, Simona Perotto, Alessandro Roggero, Sauro Succi

TL;DR

The paper addresses the challenge of efficiently block-encoding sparse matrices with periodic diagonal structure for quantum algorithms. It introduces a Fourier-based LCU framework that builds a diagonal unitary $V(ω)$ from phase gates, extracts real and imaginary components as $C(ω)$ and $S(ω)$, and combines them with shift operators $L$ and $R$ to block-encode a periodic matrix $M = α_0 C(ω) + α_1 L + α_2 R + α_3 I$. This approach generalizes to multi-frequency decompositions and is demonstrated on discretized elliptic PDEs and ADR-type systems, with numerical validation showing favorable scaling compared to dense encodings. The method enables efficient quantum simulation and linear-system solving via QSVT, offering a practical path for exploiting periodic structure in classical PDEs on quantum hardware, while also outlining open questions about ancilla requirements, α-optimization, and frequency-selection strategies.

Abstract

Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency $ω$ multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.

Block encoding of sparse matrices with a periodic diagonal structure

TL;DR

The paper addresses the challenge of efficiently block-encoding sparse matrices with periodic diagonal structure for quantum algorithms. It introduces a Fourier-based LCU framework that builds a diagonal unitary from phase gates, extracts real and imaginary components as and , and combines them with shift operators and to block-encode a periodic matrix . This approach generalizes to multi-frequency decompositions and is demonstrated on discretized elliptic PDEs and ADR-type systems, with numerical validation showing favorable scaling compared to dense encodings. The method enables efficient quantum simulation and linear-system solving via QSVT, offering a practical path for exploiting periodic structure in classical PDEs on quantum hardware, while also outlining open questions about ancilla requirements, α-optimization, and frequency-selection strategies.

Abstract

Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a gate complexity, where is the number of qubits, in the worst-case scenario used for banded matrices, and when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.
Paper Structure (12 sections, 1 theorem, 34 equations, 11 figures)

This paper contains 12 sections, 1 theorem, 34 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Quantum circuit
  3. Diagonal sinusoidal matrix
  4. Shift permutation matrices
  5. Block encoding of sparse matrices with a periodic structure
  6. Fourier decomposition: periodic structure with more than one frequency
  7. Applications
  8. Elliptic problem with periodically varying reaction
  9. ADR system with periodically varying reaction term
  10. ADR system with periodically varying velocity
  11. Numerical simulations
  12. Conclusion

Key Result

Theorem 1

Let $\omega \in \mathbb{R}$ and let $V(\omega)=V(\omega,0)$ be the diagonal unitary operator defined in Equation eq:V. The unitary $U_{C(\omega)}$ acting on $n+1$ qubits, defined as is an exact block encoding of the diagonal matrix $C(\omega)=C(\omega,0)$ defined in Equation eq:cos_matrix with sub-normalization factor $\alpha=1$ and $m=1$ ancilla qubit.

Figures (11)

  • Figure 1: Quantum circuit to implement the diagonal unitary operator \ref{['eq:V']} using $n$ phase gates.
  • Figure 2: Block encoding circuit of the diagonal matrix $C$ defined in Equation \ref{['eq:cos_matrix']}.
  • Figure 3: Probability $p_0$ of successfully applying the proposed block encoding for different values of $\omega$ and different choices of the initial state $\psi$, the uniform distribution (solid), the computational basis case (dashed) and a generic distribution (dotted). The quantum register of $\psi$ consists of 4 qubits.
  • Figure 4: Alternative block encoding circuit of the diagonal matrix $C$ defined in Equation \ref{['eq:cos_matrix']}. The CNOT gates have one control and $n$ targets.
  • Figure 5: Shift circuits for a quantum register of size 4.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Theorem 1