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IEEE Access

Quantum Subroutine for Efficient Matrix Multiplication

Anna Bernasconi, Alessandro Berti, Gianna M. Del Corso, Alessandro Poggiali

Published Jan 1, 2024• 9 citations
Publisher

Abstract

We propose an efficient quantum subroutine for matrix multiplication that computes a state vector encoding the entries of the product of two matrices in superposition. The subroutine exploits efficient state preparation techniques and shows a potential speed-up with respect to classical methods. The most important benefit of our subroutine is that it encodes the entries of the matrix product directly in the state vector, which can be used for further computations within the same quantum circuit. All scenarios involving the computation of non-homomorphic functions of the product of two matrices can benefit from our technique. As a possible application, we discuss the computation of the variance of the entries of a matrix product, which can be a useful tool for some machine learning algorithms.

DOI
10.1109/access.2024.3446176

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This article presents two quantum algorithms for computing the product of a circulant matrix and a vector. The arithmetic complexity of the first algorithm is O(Nlog2N) in most cases. For the second algorithm, when the entries in the circulant matrix and the vector take values in C or R, the complexity is O(Nlog2N) in most cases. However, when these entries take values from positive real numbers, the complexity is reduced to O(log3N) in most cases, which presents an exponential speedup compared to the classical complexity of O(NlogN) for computing the product of a circulant matrix and vector. We apply this algorithm to the convolution calculation in quantum convolutional neural networks, which effectively accelerates the computation of convolutions. Additionally, we present a concrete quantum circuit structure for quantum convolutional neural networks.
Abstract Efficient implementation of fundamental matrix operations on quantum computers, such as matrix products and Hadamard operations, holds significant potential for accelerating machine learning algorithms. A critical prerequisite for quantum implementations is the effective encoding of classical data into quantum states. We propose two quantum computing frameworks for preparing the distinct encoded states corresponding to matrix operations, including the matrix product, matrix sum, matrix Hadamard product and division. Quantum algorithms based on the digital encoding computing framework are capable of implementing the matrix Hadamard operation with a time complexity of O (poly log( mn / ε )) and the matrix product with a time complexity of O (poly log( mnl / ε )), achieving an exponential speedup in contrast to the classical methods of O ( mn ) and O ( mnl ). Quantum algorithms based on the analog-encoding framework are capable of implementing the matrix Hadamard operation with a time complexity of O ( k 1 m n ⋅ poly log ( m n / ε ) ) and the matrix product with a time complexity of O ( k 2 l ⋅ poly log ( m n l / ε ) ) , where k 1 and k 2 are coefficients correlated with the elements of the matrix, achieving a square speedup in contrast to the classical counterparts. As applications, we construct an oracle that can access the trace of a matrix within logarithmic time, and propose several algorithms to respectively estimate the trace of a matrix, the trace of the product of two matrices, and the trace inner product of two matrices within logarithmic time.
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This paper presents a novel quantum algorithm for matrix multiplication, leveraging quantum superposition and entanglement to achieve significantly enhanced computational speed and quantum advantage compared to classical methods. We propose a unique approach to matrix multiplication based on the quantum state representation of matrices, employing quantum entanglement to dramatically reduce the computational complexity inherent in traditional matrix multiplication. The algorithm demonstrates potential for high-throughput matrix operations, offering a substantial improvement in performance for computationally intensive tasks, particularly within the fields of machine learning and image processing. The proposed method offers a fundamentally new perspective on matrix multiplication, potentially ushering in a new era of quantum computing.
This paper explores a novel approach to accelerating matrix multiplication by leveraging the principles of quantum state superposition. Traditional matrix multiplication is computationally intensive, particularly for large matrices, and is a core component of many scientific and engineering applications. We propose a hardware acceleration strategy that maps the matrix multiplication process onto a quantum circuit exploiting the inherent parallelism offered by quantum superposition. The core idea involves representing matrix elements as quantum states and performing the multiplication operations using controlled quantum gates. This approach aims to potentially achieve exponential speedups compared to classical algorithms. We detail the theoretical framework, outlining the quantum circuit design, and discuss the key challenges and potential improvements associated with this methodology. The primary focus is on the algorithmic design and conceptual framework rather than experimental validation, offering a foundational exploration of this concept.
As the most central and computationally intensive component of deep neural networks, the execution efficiency of matrix multiplication directly determines the training and inference performance of models. Harnessing the parallel processing capabilities afforded by quantum superposition and entanglement to reshape matrix multiplication implementations has become a promising entry point for optimising underlying quantum arithmetic logic and improving the operational efficiency of quantum circuits. This paper proposes a universal quantum matrix multiplication (QMM) framework designed to achieve substantial computational acceleration through an optimised quantum arithmetic logic unit. To circumvent the limitations of multi-register and multi-control gates in conventional quantum arithmetic circuits, we encode classical data directly into parameterised RzR_zRz​ rotation gates using the quantum Fourier transform (QFT), thereby reducing the base gate complexity of the quantum adder to O(n)O(n)O(n). In addition, by adopting the column-wise multiplication principle from classical arithmetic, we optimize the gate complexity of the quantum multiplier to O(n2)O(n^2)O(n2). We further extend this approach to a quantum version of the Strassen algorithm, and experimentally quantify the trade-off between reduced multiplication time and increased overhead in addition resources. This work establishes a reliable technical pathway for constructing general-purpose quantum matrix operations, with the potential to unlock substantial computational power for training modern machine learning models.