A Quantum-Walk-Unitary HHL Matrix Equation Solver and its Challenges in the NISQ Era

The Harrow/Hassidim/Lloyd algorithm is a celebrated quantum matrix equation solver. Its Hamiltonian simulation involves a quantum walk process. A newly developed Quantum Walk Unitary HHL (QWU-HHL) leverages the spectral relationship between the quantum walk operator and the system matrix, and uses the quantum walk operator as its unitary in the phase estimation directly, allowing Hamiltonian simulation in the classical HHL to be removed, hence improving its efficiency. Despite the potential of being exponentially faster than classical matrix equation solvers, HHL is not feasible in the noisy intermediate-scale quantum era because its quantum circuit is too deep to preserve an accurate solution. In this work, we investigate the error behavior of QWU-HHL on a 7-qubit quantum system. Instead of executing the entire circuit, the behaviors of its sub-circuits which each contain only an initialization and a single gate are recorded. The error caused by isolated initialization and operation are analyzed.