Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints

TL;DR

This paper presents a quantum algorithm for semidefinite programming, specifically improving the Goemans-Williamson algorithm for combinatorial problems like MaxCut, using fewer qubits and expectation values.

Contribution

It introduces a variational quantum approach utilizing the Hadamard Test to efficiently approximate semidefinite programs with fewer resources than classical methods.

Findings

Outperforms classical algorithms on MaxCut problems from GSet

Uses only n+1 qubits and polynomial expectation values

Enforces constraints via Hadamard Test and Pauli string amplitude constraints

Abstract

Semidefinite programs are optimization methods with a wide array of applications, such as approximating difficult combinatorial problems. One such semidefinite program is the Goemans-Williamson algorithm, a popular integer relaxation technique. We introduce a variational quantum algorithm for the Goemans-Williamson algorithm that uses only $n{+}1$ qubits, a constant number of circuit preparations, and $\text{poly}(n)$ expectation values in order to approximately solve semidefinite programs with up to $N=2^n$ variables and $M \sim O(N)$ constraints. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit, a technique known as the Hadamard Test. The Hadamard Test enables us to optimize the objective function by estimating only a single expectation value of the ancilla qubit, rather than separately estimating exponentially many expectation values. Similarly, we illustrate that the semidefinite programming constraints can be effectively enforced by implementing a second Hadamard Test, as well as imposing a polynomial number of Pauli string amplitude constraints. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems, including MaxCut. Our method exceeds the performance of analogous classical methods on a diverse subset of well-studied MaxCut problems from the GSet library.