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Quantum algorithms for solving linear systems of equations and implementation in prototype quantum computers

Politis Alexandros-Myron

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Summary

In this thesis we study quantum algorithms for solving linear systems of equations. The first quantum linear solver that we study is the HHL algorithm [1]. HHL was invented at MIT by Harrow, Hassidim and Lloyd in 2008 and is an exact quantum algorithm that uses the quantum phase estimation as a subroutine for eigenvalue finding. HHL scales exponentially well, with respect to number of variables and size of equations, but is qubit demanding and noise sensitive. The implementation of this algorithm in quantum hardware for large problems where one could expect to outperform classical computers, requires more fully functional qubits from what today’s prototype quantum computers can provide. For this reason, the attention has been recently shifted towards developing variational hybrid-quantum classical algorithms that combine prototype quantum hardware with classical optimizers, and are expected to achieve quantum advantage in the next five years. In the second part of this thesis, we analyze such an algorithms, the variational quantum linear solver (VQLS). After studying in detail, the mathematical inner workings of both algorithms and deriving the expected quantum speed ups, we experiment with various test equation sets of different gap sizes, condition numbers and sparsity, running both ideal and noisy simulations as well as real quantum hardware on the IBM Q cloud. The result of these experiments is that for small scale problems, VQLS and HHL can produce the same result in regard to fidelity on perfect simulators but VQLS outperformed HHL while working on a realistic noisy setting as it is the case for current day prototype quantum processors.

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