Optimizing Parameterized Quantum Circuits With Free-Axis Single-Qubit Gates

Variational quantum algorithms, which utilize parameterized quantum circuits (PQCs), are promising tools to achieve quantum advantage for optimization problems on near-term quantum devices. Their PQCs have been conventionally constructed from parameterized rotational angles of single-qubit gates around a predetermined set of axes and two-qubit entangling gates, such as
cnot
gates. We propose a method to construct a PQC by continuous parameterization of both the angles and the axes of its single-qubit rotation gates. The method is based on the observation that when rotational angles are fixed, optimal axes of rotations can be computed by solving a system of linear equations whose coefficients can be determined from the PQC with small computational overhead. The method can be further simplified to select axes freely from continuous parameters with rotational angles fixed to
$\pi$
. We show the simplified free-axis selection method has better expressibility against other structural optimization methods when measured with Kullback–Leibler divergence. We also demonstrate PQCs with free-axis selection are more effective to search the ground states of Hamiltonians for condensed matter physics, quantum chemistry, and combinatorial optimization. Because free-axis selection allows designing PQCs without specifying their single-qubit rotational axes, it may significantly improve the handiness of PQCs.