On Encoding Matrices using Quantum Circuits

2025-10-22

AI Summary

This work addresses the efficient encoding of classical matrices into quantum circuits, focusing on two fundamental representations: block encodings and state preparation circuits. We propose a general construction method for block encodings and establish low-overhead bidirectional conversion algorithms between the two representations, providing the first rigorous proof of their functional equivalence. Furthermore, we design a higher-order Pauli-basis transformation algorithm based on constant-depth multiplexers, enabling efficient conversion between standard-basis and Pauli-basis matrix expansions. Our methods significantly reduce the quantum resource overhead—particularly qubit count and circuit depth—associated with data encoding in quantum linear algebra algorithms. They also enhance flexibility and practicality in input representation, thereby facilitating the real-world deployment of quantum algorithms. (149 words)

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Efficiently constructing block encodings from classical matrices
• Converting between block encodings and state preparation circuits
• Developing quantum multiplexers for Pauli matrix expansions

Innovation

• Constructing block encodings from classical matrix data
• Converting between block encodings and state preparation circuits
• Using multiplexers for Pauli matrix expansions conversion