Near-term quantum algorithms for linear systems of equations with regression loss functions

Solving linear systems of equations is essential for many problems in science and technology, including problems in machine learning. Existing quantum algorithms have demonstrated the potential for large speedups, but the required quantum resources are not immediately available on near-term quantum...

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Vydáno v:New journal of physics Ročník 23; číslo 11; s. 113021 - 113047
Hlavní autoři: Huang, Hsin-Yuan, Bharti, Kishor, Rebentrost, Patrick
Médium: Journal Article
Jazyk:angličtina
Vydáno: Bristol IOP Publishing 01.11.2021
Témata:
Algorithms
Heuristic methods
Linear systems
Machine learning
Mathematical analysis
near-term quantum algorithms
Optimization
Physics
quantum computing
ISSN:1367-2630, 1367-2630
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Shrnutí:Solving linear systems of equations is essential for many problems in science and technology, including problems in machine learning. Existing quantum algorithms have demonstrated the potential for large speedups, but the required quantum resources are not immediately available on near-term quantum devices. In this work, we study near-term quantum algorithms for linear systems of equations, with a focus on the two-norm and Tikhonov regression settings. We investigate the use of variational algorithms and analyze their optimization landscapes. There exist types of linear systems for which variational algorithms designed to avoid barren plateaus, such as properly-initialized imaginary time evolution and adiabatic-inspired optimization, suffer from a different plateau problem. To circumvent this issue, we design near-term algorithms based on a core idea: the classical combination of variational quantum states (CQS). We exhibit several provable guarantees for these algorithms, supported by the representation of the linear system on a so-called ansatz tree. The CQS approach and the ansatz tree also admit the systematic application of heuristic approaches, including a gradient-based search. We have conducted numerical experiments solving linear systems as large as 2 300 × 2 300 by considering cases where we can simulate the quantum algorithm efficiently on a classical computer. Our methods may provide benefits for solving linear systems within the reach of near-term quantum devices.
Bibliografie:NJP-112941.R2
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ISSN:1367-2630
1367-2630
DOI:10.1088/1367-2630/ac325f