BQP, meet NP: Search-to-decision reductions and approximate counting

• URL: http://arxiv.org/abs/2401.03943v1
• Date: Mon, 8 Jan 2024 14:59:48 GMT
• Title: BQP, meet NP: Search-to-decision reductions and approximate counting
• Authors: Sevag Gharibian and Jonas Kamminga
• Abstract summary: We focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires $Theta(n)$ queries to an NP oracle, quantumly $Theta(log n)$ queries suffice. Moving to approximate counting, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires $\Theta(n)$ queries to an NP oracle to compute a witness to a given SAT formula, quantumly $\Theta(\log n)$ queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires $\Omega(\log n)$ queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires $\Omega(\log n)$ queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making $o(log n)$ NP queries, then $\text{BPP}^{\text{NP}[o(n)]}$ contains a $\text{P}^{\text{NP}}$-complete problem if the algorithm is classical and $\text{FBQP}^{\text{NP}[o(n)]}$ contains an $\text{FP}^{\text{NP}}$-complete problem if the algorithm is quantum.

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