2 Search Results for "Rao, Sujit"

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.70

BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting

Authors: Sevag Gharibian and Jonas Kamminga

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

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 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 BPP^NP[o(n)] contains a 𝖯^NP-complete problem if the algorithm is classical and FBQP^NP[o(n)] contains an FP^NP-complete problem if the algorithm is quantum.

Cite as

Sevag Gharibian and Jonas Kamminga. BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 70:1-70:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gharibian_et_al:LIPIcs.ICALP.2024.70,
  author =	{Gharibian, Sevag and Kamminga, Jonas},
  title =	{{BQP, Meet NP: Search-To-Decision Reductions and Approximate Counting}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{70:1--70:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.70},
  URN =		{urn:nbn:de:0030-drops-202134},
  doi =		{10.4230/LIPIcs.ICALP.2024.70},
  annote =	{Keywords: Approximate Counting, Search to Decision Reduction, BQP, NP, Oracle Complexity Class}
}

Document

DOI: 10.4230/LIPIcs.CCC.2022.5

Quantum Search-To-Decision Reductions and the State Synthesis Problem

Authors: Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract

It is a useful fact in classical computer science that many search problems are reducible to decision problems; this has led to decision problems being regarded as the de facto computational task to study in complexity theory. In this work, we explore search-to-decision reductions for quantum search problems, wherein a quantum algorithm makes queries to a classical decision oracle to output a desired quantum state. In particular, we focus on search-to-decision reductions for QMA, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a QMA problem up to inverse polynomial precision by making one query to a PP decision oracle. We complement this result by showing that QMA-search does not reduce to QMA-decision in polynomial-time, relative to a quantum oracle. We also explore the more general state synthesis problem, in which the goal is to efficiently synthesize a target state by making queries to a classical oracle encoding the state. We prove that there exists a classical oracle with which any quantum state can be synthesized to inverse polynomial precision using only one oracle query and to inverse exponential precision using two oracle queries. This answers an open question of Aaronson [Aaronson, 2016], who presented a state synthesis algorithm that makes O(n) queries to a classical oracle to prepare an n-qubit state, and asked if the query complexity could be made sublinear.

Cite as

Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen. Quantum Search-To-Decision Reductions and the State Synthesis Problem. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{irani_et_al:LIPIcs.CCC.2022.5,
  author =	{Irani, Sandy and Natarajan, Anand and Nirkhe, Chinmay and Rao, Sujit and Yuen, Henry},
  title =	{{Quantum Search-To-Decision Reductions and the State Synthesis Problem}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{5:1--5:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.5},
  URN =		{urn:nbn:de:0030-drops-165674},
  doi =		{10.4230/LIPIcs.CCC.2022.5},
  annote =	{Keywords: Search-to-decision, state synthesis, quantum computing}
}

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1 Gharibian, Sevag
1 Irani, Sandy
1 Kamminga, Jonas
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1 Theory of computation → Quantum complexity theory
1 Theory of computation → Quantum computation theory

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