Explicación de mi Paper "P=NP in Polylogarithmic Time with Holographic Proofs and Search-to-Decision Reduction"

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.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.

arXiv:2401.03943Leibniz international proceedings in informaticsJan 1, 20240 citations

P=NP in O(1) Time via Infinite Holography

Kaoru Aguilera Katayama

Jan 1, 20260 citations

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.