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

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

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Author Details

Sandy Irani
  • Department of Computer Science, University of California, Irvine, CA, USA
Anand Natarajan
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Chinmay Nirkhe
  • Department of Computer Science, University of California, Berkeley, CA, USA
  • Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
Sujit Rao
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Henry Yuen
  • Department of Computer Science, Columbia University, New York, NY, USA


SI, CN, and HY were participants in the Simons Institute for the Theory of Computing Summer Cluster on Quantum Compuatation. Additionally, we thank Aram Harrow and Zeph Landau for insightful discussions.

Cite AsGet BibTex

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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Search-to-decision
  • state synthesis
  • quantum computing


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