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.
Highlights
Quantum computing promises speedups for problems such as integer factoring and search
Such applications are believed to be in quantum chemistry, optimization and machine learning, and possible algorithmic candidates are the variational quantum eigensolver (VQE) [18,19,20] and quantum approximate optimization (QAOA) [21, 22]
We analyze various efforts that circumvent the known barren plateau issue [38], such as properly-initialized imaginary time evolution or the adiabatic-inspired optimization. We show that these attempts may still exhibit the plateau effect, which motivates further efforts and alternative ideas to solve large-scale linear systems with near-term quantum devices and achieve quantum advantage
Summary
Quantum computing promises speedups for problems such as integer factoring and search. We analyze various efforts that circumvent the known barren plateau issue [38], such as properly-initialized imaginary time evolution or the adiabatic-inspired optimization We show that these attempts may still exhibit the plateau effect, which motivates further efforts and alternative ideas to solve large-scale linear systems with near-term quantum devices and achieve quantum advantage. To provide a potential solution to the aforementioned problems, we pursue a different route and propose a different class of algorithms for solving linear systems on near-term quantum devices. These algorithms use the two-norm regression and Tikhonov loss functions, the obtained solutions may not agree with the Hamiltonian loss function. While the overall gate count would increase, the ability to execute a much shorter circuit for each repetition could allow the method to run on a more near-term quantum device
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