An Expressive Ansatz for Low-Depth Quantum 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 that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and statepreparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), a modified version of QAOA that assigns more classical parameters to the ansatz to improve the performance of low-depth quantum circuits. XQAOA includes an additional Pauli-Y component in the mixer Hamiltonian, thereby allowing the mixer to effectively implement arbitrary unitary transformations on each qubit. To benchmark the performance of the XQAOA ansatz at low depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, MA-QAOA [Sci Rep 12, 6781 (2022)], 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 and degrees ranging from 3 to 10. Our results show that XQAOA performs better than QAOA, MA-QAOA, and the Classical-Relaxed algorithm on all graph instances and outperforms the Goemans-Williamson algorithm on unweighted regular graphs with degrees greater than 4. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and show analytically that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for implementing quantum combinatorial optimisation on near-term quantum devices, as it can achieve better results with a single iteration, despite requiring additional classical resources.

Amplitude amplification-inspired QAOA: improving the success probability for solving 3SAT

“…To tackle the problem of variational quantum algorithms amplifying suboptimal solutions, Bennett and Wang [18] employ a similar adaption to Variant 1 to the more general framework of the QWOA. They Transverse-field Grover search [19] MAOA [18] ma-QAOA [55] GM-QAOA [16] Th-QAOA [17] GM-Th-QAOA [17] XY-QAOA [56] XQAOA [57] FAM-QAOA [58] RQAOA [59] present the maximum amplification optimization algorithm (MAOA) and apply it to combinatorial optimization problems. They use a similar distinction between good and bad solutions to classify the elements in the set of feasible solutions into ones meeting a certain threshold for the cost function and those that do not.…”

“…ma-QAOA [55]) or that adapt the mixer Hamiltonian to increase the reachable solutions (e.g. X-QAOA [57], FAM-QAOA [58]).…”