Quantum speedups in solving near-symmetric optimization problems by low-depth QAOA

• URL: http://arxiv.org/abs/2411.04979v2
• Date: Mon, 24 Feb 2025 10:25:45 GMT
• Title: Quantum speedups in solving near-symmetric optimization problems by low-depth QAOA
• Authors: Ashley Montanaro, Leo Zhou,
• Abstract summary: We focus on families of optimization problems that exhibit symmetry and contain planted solutions.<n>We rigorously prove that the 1-step Quantum Approximate Optimization Algorithm (QAOA) can achieve a success probability of $Omega (1/sqrtn)$.<n>Finally, we construct various families of near-symmetric Max-SAT problems and benchmark state-of-the-art classical solvers.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present new advances towards achieving exponential quantum speedups for solving optimization problems by low-depth quantum algorithms. Specifically, we focus on families of combinatorial optimization problems that exhibit symmetry and contain planted solutions. We rigorously prove that the 1-step Quantum Approximate Optimization Algorithm (QAOA) can achieve a success probability of $\Omega(1/\sqrt{n})$, and sometimes $\Omega(1)$, for finding the exact solution in many cases. This allows us to prove a separation of $O(1)$ quantum queries and $\Omega(n/\log n)$ classical queries required to find the planted solution in the latter setting. Furthermore, we construct near-symmetric optimization problems by randomly sampling the individual clauses of symmetric problems, and prove that the QAOA maintains a strong success probability in this setting even when the symmetry is broken. Finally, we construct various families of near-symmetric Max-SAT problems and benchmark state-of-the-art classical solvers, discovering instances where all known general-purpose classical algorithms require exponential time. Therefore, our results indicate that low-depth QAOA may achieve an exponential quantum speedup for optimization problems.

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