Block-encoding structured matrices for data input in quantum computing

TL;DR

This paper introduces methods for efficiently constructing block encoding circuits for structured matrices in quantum computing, reducing data loading costs and qubit usage, with applications to matrices like Toeplitz and tridiagonal.

Contribution

It presents novel schemes for block encoding of structured matrices based on their arithmetic properties, optimizing for sparsity and repeated values, and demonstrates exponential improvements in certain cases.

Findings

Reduces qubit count based on matrix sparsity

Decreases data loading costs for repeated values

Achieves exponential improvements for specific matrix families

Abstract

The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via block encoding circuits, the input model for the quantum singular value transform and related algorithms. We demonstrate how to construct block encoding circuits based on an arithmetic description of the sparsity and pattern of repeated values of a matrix. We present schemes yielding different subnormalisations of the block encoding; a comparison shows that the best choice depends on the specific matrix. The resulting circuits reduce flag qubit number according to sparsity, and data loading cost according to repeated values, leading to an exponential improvement for certain matrices. We give examples of applying our block encoding schemes to a few families of matrices, including Toeplitz and tridiagonal matrices.