Preconditioned Block Encodings for Quantum Linear Systems

TL;DR

This paper explores quantum preconditioning techniques for block encodings in quantum linear system algorithms, demonstrating methods to improve condition numbers and reduce circuit complexity for practical CFD matrices.

Contribution

It introduces quantum preconditioning strategies, a new matrix filtering technique, and demonstrates significant reductions in QSVT phase factors for large matrices.

Findings

Quantum multiplication increases subnormalisation, negating benefits.

Classical product encoding improves effective condition number.

Matrix filtering reduces circuit depth without harming solutions.

Abstract

Quantum linear system solvers like the Quantum Singular Value Transformation (QSVT) require a block encoding of the system matrix $A$ within a unitary operator $U_A$. Unfortunately, block encoding often results in significant subnormalisation and increase in the matrix's effective condition number $\kappa$, affecting the efficiency of solvers. Matrix preconditioning is a well-established classical technique to reduce $\kappa$ by multiplying $A$ by a preconditioner $P$. Here, we study quantum preconditioning for block encodings. We consider four preconditioners and two encoding approaches: (a) separately encoding $A$ and its preconditioner $P$, followed by quantum multiplication, and (b) classically multiplying $A$ and $P$ before encoding the product in $U_{PA}$. Their impact on subnormalisation factors and condition number $\kappa$ are analysed using practical matrices from Computational Fluid Dynamics (CFD). Our results show that (a) quantum multiplication introduces excessive subnormalisation factors, negating improvements in $\kappa$. We introduce preamplified quantum multiplication to reduce subnormalisation, which is of independent interest. Conversely, we see that (b) encoding of the classical product can significantly improve the effective condition number using the Sparse Approximate Inverse preconditioner with infill. Further, we introduce a new matrix filtering technique that reduces the circuit depth without adversely affecting the matrix solution. We apply these methods to reduce the number of QSVT phase factors by a factor of 25 for an example CFD matrix of size 1024x1024.