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arXiv:2501.15137

Quantum Algorithms for Matrix Operations Based on Unitary Transformations and Ancillary State Measurements

Yuhang Liu, Yuan-Hong Tao, Lan, jing-Run, Shao-Ming Fei

Published Jan 25, 2025• 0 citations
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Abstract

MQuantum algorithms of matrix operations are of great significance in many fields in science and technology. In this paper, by leveraging multi-qubit Toffoli gates and basic single-qubit operations, the quantum algorithms of matrix operations of row addition, row swapping, trace calculation and transpose are obtained. In particular, the complexities of these quantum algorithms are presented, too.

DOI
10.48550/arxiv.2501.15137

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MQuantum algorithms of matrix operations are of great significance in many fields in science and technology. In this paper, by leveraging multi-qubit Toffoli gates and basic single-qubit operations, the quantum algorithms of matrix operations of row addition, row swapping, trace calculation and transpose are obtained. In particular, the complexities of these quantum algorithms are presented, too.
We propose protocols for calculating inner product, matrix addition and matrix multiplication based on multiqubit Toffoli-type and the simplest one-qubit operations and employ ancilla-measurements to remove all garbage of calculations. The depth (runtime) of addition protocol is O(1)O(1)O(1) and that of other protocols logarithmically increases with the dimensionality of the considered matrices.
This paper focuses on quantum algorithms for three key matrix operations: Hadamard (Schur) product, Kronecker (tensor) product, and elementary column transformations each. By designing specific unitary transformations and auxiliary quantum measurement, efficient quantum schemes with circuit diagrams are proposed. Their computational depths are: O(1) for Kronecker product; O(max(m,n)) for Hadamard product (linked to matrix dimensions); and O(m) for elementary column transformations of (2^n X 2^m) matrices (dependent only on column count).Notably, compared to traditional column transformation via matrix transposition and row transformations, this scheme reduces computation steps and quantum gate usage, lowering quantum computing energy costs.
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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).
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.