BQP, meet NP: Search-to-decision reductions and approximate counting
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910583305011200 |
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| author | Gharibian, Sevag Kamminga, Jonas |
| author_facet | Gharibian, Sevag Kamminga, Jonas |
| contents | 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 $Θ(n)$ queries to an NP oracle to compute a witness to a given SAT formula, quantumly $Θ(\log n)$ 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 $Ω(\log n)$ 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 $Ω(\log n)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | BQP, meet NP: Search-to-decision reductions and approximate counting Gharibian, Sevag Kamminga, Jonas Quantum Physics Computational Complexity 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 $Θ(n)$ queries to an NP oracle to compute a witness to a given SAT formula, quantumly $Θ(\log n)$ 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 $Ω(\log n)$ 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 $Ω(\log n)$ 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. |
| title | BQP, meet NP: Search-to-decision reductions and approximate counting |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2401.03943 |