On Encoding Matrices using Quantum Circuits

TL;DR

This work analyzes how matrices can be encoded in quantum circuits for quantum linear algebra, focusing on block encodings and state-preparation circuits as two complementary input representations. It introduces a general, efficient method to construct block encodings from dense classical matrices and proves bidirectional, low-overhead conversions between block encodings and matrix-state preparations, establishing a practical equivalence of the two models. A key technical advance is the PMX_n Pauli-multiplexer, which enables ultra-low-depth multiplexing of all higher-order Pauli matrices and underpins compact, scalable block-encoding constructions from Pauli decompositions. By bridging the two encoding paradigms, the paper enables flexible algorithm design, enhances compatibility with QSVT-based approaches, and clarifies the resource tradeoffs involved in loading and manipulating dense data for quantum linear algebra.

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.

Figures (14)

• Figure 1: Quantum multiplexer constructions. (a) Notation for quantum multiplexer for a family of circuits $\{{\cal U}_{\theta}\}$. (b) Generic construction of multiplexer between ${\cal U}_{0},{\cal U}_{1},{\cal U}_{2},{\cal U}_{3}$. (c) Generic construction of 2D ${\cal R}_{z}$ multiplexer with angles in a matrix ${\bm{\mathrm{\Theta}}}$.
• Figure 2: Illustration of the recursive construction of mottonen2004transformation for 1D ${\cal R}_{z}$-rotation multiplexer of angles $\theta_{0},\theta_{1},\theta_{2},\theta_{3}$. The angles $\hat{\theta}_{0},\hat{\theta}_{1},\hat{\theta}_{2},\hat{\theta}_{3}$ are computed as linear combination of $\theta_{0},\theta_{1},\theta_{2},\theta_{3}$.
• Figure 3: Illustration of the recursive construction of 2D ${\cal R}_{z}$rotation multiplexer of angles ${\bm{\mathrm{\Theta}}}\in\mathbb{R}^{K\times4}$. The angles $\hat{{\bm{\mathrm{\Theta}}}}$ are computed by $\hat{{\bm{\mathrm{\Theta}}}}=\frac{1}{2}{\bm{\mathrm{\Theta}}}{\bm{\mathrm{H}}}_{4}{\bm{\mathrm{G}}}_{4}$.
• Figure 4: The CNOT Comb circuit, $\mathcal{U}_{\mathrm{comb}}^{(q)}$, on $2q$ qubits. The row qubits ($r_{i}$) act as controls on their corresponding target column qubits ($c_{i}$).
• Figure 5: Permutation conjugation property for tensor products of single-qubit operators: $\mathcal{S}_{\sigma^{-1}}\cdot\left(\bigotimes_{i=0}^{M-1}{\cal U}_{i}\right)\cdot\mathcal{S}_{\sigma}=\bigotimes_{i=0}^{M-1}\mathcal{U}_{\sigma(i)}$. The permutation redistributes operators according to the mapping $\sigma$.

Theorems & Definitions (38)

• Definition 1: Block Encoding Circuit
• Definition 2: Hermitian Block Encoding Circuit
• Definition 3: Hypermatrix State Preparation Circuit
• Remark 4
• Remark 5
• Definition 6: Direct sum between matrices
• Definition 7: Multiplexer circuit
• Definition 8
• Definition 10: Pauli matrices
• Definition 12: Higher-order Pauli matrices