Quantum algorithms for calculating determinant and inverse of matrix and solving linear algebraic systems

• URL: http://arxiv.org/abs/2401.16619v3
• Date: Mon, 25 Nov 2024 14:51:03 GMT
• Title: Quantum algorithms for calculating determinant and inverse of matrix and solving linear algebraic systems
• Authors: Alexander I. Zenchuk, Georgii A. Bochkin, Wentao Qi, Asutosh Kumar, Junde Wu,
• Abstract summary: We propose quantum algorithms for calculating the determinant and inverse of an $(N-1)times (N-1)$ matrix. The basic idea is to encode each row of the matrix into a pure state of some quantum system.
• Score: 43.53835128052666
• License:
• Abstract: We propose quantum algorithms for calculating the determinant and inverse of an $(N-1)\times (N-1)$ matrix (depth is $O(N^2\log N)$) which is a simple modification of the algorithm for calculating the determinant of an $N\times N$ 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. Combination of this algorithm with the quantum algorithm for matrix multiplication, proposed in our previous work, 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 $\sim 2^{-O(N\log N)}$) 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 size.

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