Grover Mixers for QAOA: Shifting Complexity from Mixer Design to State Preparation

TL;DR

GM-QAOA introduces a novel quantum algorithm that shifts complexity from mixer design to state preparation, enabling exact implementation and improved performance on constrained NP optimization problems.

Contribution

It proposes GM-QAOA, a new variation of QAOA that uses Grover-like operators, enhancing efficiency and accuracy for constrained optimization problems.

Findings

Efficient superposition preparation for permutation problems on O(n^2) qubits

GM-QAOA outperforms existing QAOA methods on certain problems

Operators can be implemented exactly without Trotterization errors

Abstract

We propose GM-QAOA, a variation of the Quantum Alternating Operator Ansatz (QAOA) that uses Grover-like selective phase shift mixing operators. GM-QAOA works on any NP optimization problem for which it is possible to efficiently prepare an equal superposition of all feasible solutions; it is designed to perform particularly well for constraint optimization problems, where not all possible variable assignments are feasible solutions. GM-QAOA has the following features: (i) It is not susceptible to Hamiltonian Simulation error (such as Trotterization errors) as its operators can be implemented exactly using standard gate sets and (ii) Solutions with the same objective value are always sampled with the same amplitude. We illustrate the potential of GM-QAOA on several optimization problem classes: for permutation-based optimization problems such as the Traveling Salesperson Problem, we present an efficient algorithm to prepare a superposition of all possible permutations of $n$ numbers, defined on $O(n^2)$ qubits; for the hard constraint $k$-Vertex-Cover problem, and for an application to Discrete Portfolio Rebalancing, we show that GM-QAOA outperforms existing QAOA approaches.