scholarly journals Quantum algorithms for linear systems of equations

2021 ◽  
Author(s):  
Yigit Subasi
Keyword(s):  
Linear Systems ◽  
Quantum Algorithms ◽  
Systems Of Equations ◽  
Linear Systems Of Equations

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

2021 ◽  
Vol 23 (11) ◽  
pp. 113021
Author(s):  
Hsin-Yuan Huang ◽  
Kishor Bharti ◽  
Patrick Rebentrost
Keyword(s):  
Linear Systems ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Systems Of Equations ◽  
Quantum Devices ◽  
Near Term ◽  
Variational Algorithms ◽  
Core Idea ◽  
Classical Combination ◽  
Linear Systems Of Equations

Abstract 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 2300 × 2300 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.

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