Quantum Linear System Algorithm for General Matrices in System Identification

Solving linear systems of equations is one of the most common and basic problems in classical identification systems. Given a coefficient matrix A and a vector b, the ultimate task is to find the solution x such that Ax=b. Based on the technique of the singular value estimation, the paper proposes a...

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Bibliographic Details
Published in:Entropy (Basel, Switzerland) Vol. 24; no. 7; p. 893
Main Authors: Li, Kai, Zhang, Ming, Liu, Xiaowen, Liu, Yong, Dai, Hongyi, Zhang, Yijun, Dong, Chen
Format: Journal Article
Language:English
Published: Basel MDPI AG 29.06.2022
MDPI
Subjects:
Algorithms
Circuit design
Design parameters
Efficiency
Identification systems
Information processing
Linear equations
Linear systems
linear systems of equations
Machine learning
Mathematical analysis
Matrices (mathematics)
Maximum likelihood method
Optimization techniques
Partial differential equations
quantum algorithm
Quantum computing
Sparsity
System identification
time complexity
ISSN:1099-4300, 1099-4300
Online Access:Get full text
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Summary:Solving linear systems of equations is one of the most common and basic problems in classical identification systems. Given a coefficient matrix A and a vector b, the ultimate task is to find the solution x such that Ax=b. Based on the technique of the singular value estimation, the paper proposes a modified quantum scheme to obtain the quantum state |x⟩ corresponding to the solution of the linear system of equations in O(κ2rpolylog(mn)/ϵ) time for a general m×n dimensional A, which is superior to existing quantum algorithms, where κ is the condition number, r is the rank of matrix A and ϵ is the precision parameter. Meanwhile, we also design a quantum circuit for the homogeneous linear equations and achieve an exponential improvement. The coefficient matrix A in our scheme is a sparsity-independent and non-square matrix, which can be applied in more general situations. Our research provides a universal quantum linear system solver and can enrich the research scope of quantum computation.
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ISSN:1099-4300
1099-4300
DOI:10.3390/e24070893