Quantum Approximate Counting with Additive Error: Hardness and Optimality

• URL: http://arxiv.org/abs/2411.02602v2
• Date: Thu, 13 Mar 2025 21:27:02 GMT
• Title: Quantum Approximate Counting with Additive Error: Hardness and Optimality
• Authors: Mason L. Rhodes, Sam Slezak, Anirban Chowdhury, Yiğit Subaşı,
• Abstract summary: Quantum counting is the task of determining the dimension of the subspace of states that are accepted by a quantum verifier circuit.<n>The complexity of solving the class #BQP of quantum counting problems, either exactly or within suitable approximations, is related to the hardness of computing many-body physics quantities.
• Score: 0.34089646689382486
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum counting is the task of determining the dimension of the subspace of states that are accepted by a quantum verifier circuit. It is the quantum analog of counting the number of valid solutions to NP problems -- a problem well-studied in theoretical computer science with far-reaching implications in computational complexity. The complexity of solving the class #BQP of quantum counting problems, either exactly or within suitable approximations, is related to the hardness of computing many-body physics quantities arising in algebraic combinatorics. Here, we address the complexity of quantum approximate counting under additive error. First, we show that computing additive approximations to #BQP problems to within an error exponential in the number of witness qubits in the corresponding verifier circuit is as powerful as polynomial-time quantum computation. Next, we show that returning an estimate within error that is any smaller is #BQP-hard. Finally, we show that additive approximations to a restricted class of #BQP problems are equivalent in computational hardness to the class DQC1. Our work parallels results on additively approximating #P and GapP functions.

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