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

• URL: http://arxiv.org/abs/2206.14999v3
• Date: Sun, 9 Jul 2023 03:51:59 GMT
• Title: Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints
• Authors: Taylor L. Patti, Jean Kossaifi, Anima Anandkumar, and Susanne F. Yelin
• Abstract summary: We introduce a variational quantum algorithm for Goemans-Williamson algorithm that uses only $n+1$ qubits. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems.
• Score: 62.72309460291971
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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