Fast Convex Optimization with Quantum Gradient Methods

• URL: http://arxiv.org/abs/2503.17356v2
• Date: Wed, 04 Jun 2025 15:55:57 GMT
• Title: Fast Convex Optimization with Quantum Gradient Methods
• Authors: Brandon Augustino, Dylan Herman, Enrico Fontana, Junhyung Lyle Kim, Jacob Watkins, Shouvanik Chakrabarti, Marco Pistoia,
• Abstract summary: We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles.<n>We demonstrate the first dimension-independent query complexities for zeroth-order convex optimization in both smooth and nonsmooth settings.<n>By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games.
• Score: 2.5094874597551913
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.

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