An Expressive Ansatz for Low-Depth Quantum Approximate Optimisation

Abstract

The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum-classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz comprising a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on NISQ devices, physical limitations can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA), a classical-relaxed algorithm, and the state-of-the-art Goemans-Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans-Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, XQAOA achieves a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on NISQ devices, offering competitive performance at low depths.