Efficient Variational Quantum Linear Solver for Structured Sparse Matrices

TL;DR

This work tackles efficient evaluation of variational quantum linear solvers for structured sparse matrices by introducing a sigma-basis tensor-product decomposition, which reduces the number of LCU terms from quadratic to poly-logarithmic in matrix size. By employing a unitary completion approach, non-unitary sigma-basis operators are embedded into shallow quantum circuits, enabling efficient computation of global and local VQLS cost functions for PDE-discretized systems such as the Heat equation. The authors demonstrate how to construct circuits for key terms ($\beta_{ij}$, $\gamma_{ij}$, $\delta_{ijk}$) and show that unitary-dilation-based perspectives can yield shallower implementations than standard dilation, while also clarifying the relationship to Pauli-basis methods and Bell-measurement schemes. Overall, the paper provides a concrete, scalable route to leverage PDE structure in VQLS, with implications for faster quantum-assisted simulations in science and engineering.

Abstract

We develop a novel approach for efficiently applying variational quantum linear solver (VQLS) in context of structured sparse matrices. Such matrices frequently arise during numerical solution of partial differential equations which are ubiquitous in science and engineering. Conventionally, Pauli basis is used for linear combination of unitary (LCU) decomposition of the underlying matrix to facilitate the evaluation the global/local VQLS cost functions. However, Pauli basis in worst case can result in number of LCU terms that scale quadratically with respect to the matrix size. We show that by using an alternate basis one can better exploit the sparsity and underlying structure of matrix leading to number of tensor product terms which scale only logarithmically with respect to the matrix size. Given this new basis is comprised of non-unitary operators, we employ the concept of unitary completion to design efficient quantum circuits for computing the global/local VQLS cost functions. We compare our approach with other related concepts in the literature including unitary dilation and measurement in Bell basis, and discuss its pros/cons while using VQLS applied to Heat equation as an example.

Figures (2)

• Figure 1: Hadamard test for computing the terms in the global and local cost functions. The circuit in (a) can be simplified by removing the blocks corresponding to $\mathbf{U} (\mathbf{Z}_k\otimes \mathbf{I}_{\tilde{k}})\mathbf{U}^*$ to compute $\beta_{ij}$.
• Figure 2: Circuits for implementing (a) $\tilde{\mathbf{U}}_l$ and (b) $\bar{\mathbf{U}}$ when $\mathbf{A}_l = \sigma_-$.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Theorem 1
• proof
• Corollary 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof