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arXiv:2202.04888 · Communications in Theoretical Physics

Quantum algorithms for matrix operations and linear systems of equations

Wentao Qi, Alexander Ivanovich Zenchuk, Asutosh Kumar, Junde Wu

Published Jan 29, 2024• 13 citations
arXivHTMLPDF

Abstract

Abstract Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations. Using the ‘sender-receiver’ model, we propose quantum algorithms for matrix operations such as matrix-vector product, matrix-matrix product, the sum of two matrices, and the calculation of determinant and inverse matrix. We encode the matrix entries into the probability amplitudes of the pure initial states of senders. After applying proper unitary transformation to the complete quantum system, the desired result can be found in certain blocks of the receiver’s density matrix. These quantum protocols can be used as subroutines in other quantum schemes. Furthermore, we present an alternative quantum algorithm for solving linear systems of equations.

DOI
10.1088/1572-9494/ad2366

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Abstract Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations. Using the ‘sender-receiver’ model, we propose quantum algorithms for matrix operations such as matrix-vector product, matrix-matrix product, the sum of two matrices, and the calculation of determinant and inverse matrix. We encode the matrix entries into the probability amplitudes of the pure initial states of senders. After applying proper unitary transformation to the complete quantum system, the desired result can be found in certain blocks of the receiver’s density matrix. These quantum protocols can be used as subroutines in other quantum schemes. Furthermore, we present an alternative quantum algorithm for solving linear systems of equations.
Abstract We propose quantum algorithms, purely quantum in nature, for calculating the determinant and inverse of an ( N – 1) × ( N – 1) matrix (depth is O ( N 2 log N )) which is a simple modification of the algorithm for calculating the determinant of an N × N matrix (depth is O ( N log 2 N )). The basic idea is to encode each row of the matrix into a pure state of some quantum system. In addition, we use the representation of the elements of the inverse matrix in terms of algebraic complements. This algorithm together with that for matrix multiplication proposed earlier yields the algorithm for solving systems of linear algebraic equations (depth is O ( N log 2 N )). Measurement of the ancilla state with output 1 (probability is ~ 2 − O ( N log N ) ) removes the garbage acquired during calculation. Appropriate circuits for all three algorithms are presented and have the same estimation O ( N log N ) for the space (number of qubits in the circuit).
The solution of sets of linear simultaneous equations can be accomplished using matrices and we show how this is done. In quantum computation, matrices ‘operate’ or act on vectors to move a system from one quantum state to another as it evolves with time. The solution of sets of linear simultaneous equations can be accomplished using matrices and we show how this is done. Matrices can also be used to perform transformations of points in two- and three-dimensional space through rotations and reflections. Self-adjoint matrices play an important role in quantum computation because, as we shall see, they have real eigenvalues which correspond to the measured values of quantum observables. The procedure for calculating the inverse of a general n×n matrix necessitates the introduction of some new terminology. The rows of a unitary matrix are pairwise orthogonal so that the inner product of two distinct rows will be zero.
Using the tool of unitary transformations of the extended receiver we perform simple operations with the non-diagonal elements of the initial sender's density matrix after their transferring to the receiver. These operations are following: turning some matrix elements to zero, rearranging the matrix elements and preparing their linear combinations with required coefficients. We also propose a protocol for solving a system of linear algebraic equations with a system of two equations as an example. Such operations are numerically simulated on the basis of 42-node spin-1/2 chain with the two-qubit sender and receiver.
Operations on matrices are at the heart of a vast number of optimization algorithms. Quantum computers can only apply unitary matrices directly, but there exists a vast toolbox of quantum algorithms to perform complex operations on nonunitary matrices as well. In this chapter we discuss two aspects of nonunitary matrix manipulation that have been featured prominently in existing quantum algorithms for optimization: algorithms for linear systems (i.e., matrix inversion), and the block-encoding framework. Matrix inversion can also be performed within the block-encoding framework, and it is actually more efficient in that framework than in the phaseestimation-based approach that we initially present. Because the phase estimation approach to matrix inversion is historically important, elegant, and showcases a number of powerful ideas for the design of quantum algorithms, we discuss it anyway, and in fact we begin our discussion in this chapter with it.
With the rapid development of Quantum Computers (QC) and QC Simulators, there will be an increased demand for functioning Quantum Algorithms in the near future. Some of the most ubiquitously useful algorithms are solvers for linear systems of equations. Since the conception of the Quantum Linear Solver Algorithm (QLSA) by Harrow, Hassidim and Lloyd (HHL) in 2009, many improvements have been made, although a generic implementation for arbitrary matrices and vectors is still not available. In this thesis a variant of the HHL QLSA is studied, and the open challenges are investigated. Solutions for two of the challenges, namely the Eigenvalue Inversion subroutine and the Higher-Order Ancilla Rotation subroutine, are discussed. As part of the thesis project, these subroutines have been implemented in the QX Quantum Computer Simulator, and the subroutines are combined to form a complete Quantum Linear Solver (QLS), with the restraint that the implementation for the vector and Hamiltonian of the matrix must be provided by the user. A proof-of-concept QLS by Cao et al. is also implemented in the QX simulator, and using the implementation of the vector and Hamiltonian of Cao et al. the complete solver is tested. In the process of this thesis, a framework for basic Quantum Arithmetic is built providing three variants of Integer Adders, two variants of Integer Subtracters, one Integer Multiplier and one Integer Divider. In addition, gates not natively available in the QX simulator are implemented, and a number of improvements and extensions of algorithms presented in the literature are given, making the described algorithms function on the QX simulator and extending features.
Linear equations play a pivotal role in many areas of science and engineering, making efficient solutions to linear systems highly desirable. The development of quantum algorithms for solving linear systems has been a significant breakthrough, with such algorithms rapidly becoming some of the most influential in quantum computing. Subsequent advances have focused on improving the efficiency of quantum linear solvers and extending their core techniques to address various challenges, such as least-squares data fitting. In this article, we introduce a new quantum algorithm for solving linear systems based on the gradient descent method. Inspired by the vector/density state formalism, we represent a point, or vector, as a density state-like entity, enabling us to leverage recent advancements in quantum algorithmic frameworks, such as block encoding, to directly manipulate operators and carry out the gradient descent method in a quantum setting. The operator corresponding to the intermediate solution is updated iteratively, with a provable guarantee of convergence. The quantum state representing the final solution to the linear system can then be extracted by further measurement. We provide a detailed complexity analysis and compare our approach with existing methods, demonstrating significant improvement in certain aspects.