On Encoding Matrices using Quantum Circuits

TL;DR

This paper systematically studies quantum circuit representations for matrices, introducing efficient methods for constructing block encodings and bidirectional conversions between block encodings and state preparation circuits, demonstrating their equivalence.

Contribution

It presents a general efficient method for constructing block encodings from classical matrices and algorithms for converting between encoding models, establishing their fundamental equivalence.

Findings

Efficient construction of block encodings from classical data.

Low-overhead algorithms for converting between circuit representations.

Demonstration of the equivalence of block encodings and state preparation circuits.

Abstract

Over a decade ago, it was demonstrated that quantum computing has the potential to revolutionize numerical linear algebra by enabling algorithms with complexity superior to what is classically achievable, e.g., the seminal HHL algorithm for solving linear systems. Efficient execution of such algorithms critically depends on representing inputs (matrices and vectors) as quantum circuits that encode or implement these inputs. For that task, two common circuit representations emerged in the literature: block encodings and state preparation circuits. In this paper, we systematically study encodings matrices in the form of block encodings and state preparation circuits. We examine methods for constructing these representations from matrices given in classical form, as well as quantum two-way conversions between circuit representations. Two key results we establish (among others) are: (a) a general method for efficiently constructing a block encoding of an arbitrary matrix given in classical form (entries stored in classical random access memory); and (b) low-overhead, bidirectional conversion algorithms between block encodings and state preparation circuits, showing that these models are essentially equivalent. From a technical perspective, two central components of our constructions are: (i) a special constant-depth multiplexer that simultaneously multiplexes all higher-order Pauli matrices of a given size, and (ii) an algorithm for performing a quantum conversion between a matrix's expansion in the standard basis and its expansion in the basis of higher-order Pauli matrices.