Study of Error Propagation and Generation in Harrow-Hassidim-Lloyd (HHL) Quantum Algorithm

In this paper, we study the error propagation and generation in the Harrow-Hassidim-Lloyd (HHL) quantum algorithm runs on IBM-Q hardware with the help of a MATLAB simulator. HHL is a quantum algorithm that can provide exponential speedup over the fastest classical algorithm (conjugate gradient method) in solving systems of linear equations (SLE). However, without error correction, it cannot give correct results even in a 2-variable system due to its complexity. In this study, an HHL quantum circuit for a 2-variable SLE is implemented in IBM-Q and the error is extracted after each stage of the circuit and compared to a MATLAB simulator. We identified three major sources of errors, namely single-qubit flipping, gate infidelity, and error propagation. We also found that at the ancillary bit rotation stage, the error becomes large but the encoded solution still has high fidelity. However, the solution is mostly lost after the inverse quantum phase estimation which is necessary to extract the solution efficiently. Therefore, it is suggested that error correction resources, if limited, should be added to the second half of the circuit.