Efficient Quantum Access Model for Sparse Structured Matrices using Linear Combination of Things

TL;DR

This paper introduces the Sigma basis, a new decomposition method for structured sparse matrices that significantly reduces the number of terms needed in quantum algorithms, enabling more efficient quantum simulations of PDEs.

Contribution

The paper proposes the Sigma basis for LCU decompositions, achieving polylogarithmic scaling and improved efficiency over traditional methods for structured sparse matrices.

Findings

Sigma basis enables exponential reduction in decomposition size.

Efficient quantum circuits are constructed using unitary completion.

Demonstrated effectiveness on PDE examples with improved scalability.

Abstract

We present a novel framework for Linear Combination of Unitaries (LCU)-style decomposition tailored to structured sparse matrices, which frequently arise in the numerical solution of partial differential equations (PDEs). While LCU is a foundational primitive in both variational and fault-tolerant quantum algorithms, conventional approaches based on the Pauli basis can require a number of terms that scales quadratically with matrix size. We introduce the Sigma basis, a compact set of simple, non-unitary operators that can better capture sparsity and structure, enabling decompositions with only polylogarithmic scaling in the number of terms. We develop both numerical and semi-analytical methods for computing Sigma basis decompositions of arbitrary matrices. Given this new basis is comprised of non-unitary operators, we leverage the concept of unitary completion to design efficient quantum circuits for evaluating observables in variational quantum algorithms and for constructing block encodings in fault-tolerant quantum algorithms. We compare our method to related techniques like unitary dilation, and demonstrate its effectiveness through several PDE examples, showing exponential improvements in decomposition size while retaining circuit efficiency.