Quantum Search-To-Decision Reductions and the State Synthesis Problem
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.
Search-to-decision
state synthesis
quantum computing
Theory of computation~Quantum computation theory
5:1-5:19
Regular Paper
https://arxiv.org/abs/2111.02999
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.
Sandy
Irani
Sandy Irani
Department of Computer Science, University of California, Irvine, CA, USA
https://www.ics.uci.edu/~irani/
https://orcid.org/0000-0002-0642-9436
Anand
Natarajan
Anand Natarajan
CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
http://www.mit.edu/~anandn/
Chinmay
Nirkhe
Chinmay Nirkhe
Department of Computer Science, University of California, Berkeley, CA, USA
Challenge Institute for Quantum Computation, University of California, Berkeley, CA, USA
http://www.cs.berkeley.edu/~nirkhe/
https://orcid.org/0000-0002-5808-4994
Supported by NSF Quantum Leap Challenges Institute Grant number OMA2016245 and an IBM Quantum PhD internship.
Sujit
Rao
Sujit Rao
CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
https://cqe.mit.edu/sujit-rao/
Henry
Yuen
Henry Yuen
Department of Computer Science, Columbia University, New York, NY, USA
https://www.henryyuen.net/
https://orcid.org/0000-0002-2684-1129
Supported by AFOSR award FA9550-21-1-0040 and NSF CAREER award CCF-2144219.
10.4230/LIPIcs.CCC.2022.5
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Sandy Irani, Anand Natarajan, Chinmay Nirkhe, Sujit Rao, and Henry Yuen
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