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 $\textit{de facto}$ 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 $\mathsf{QMA}$ , and show that there exists a quantum polynomial-time algorithm that can generate a witness for a $\mathsf{QMA}$ problem up to inverse polynomial precision by making one query to a $\mathsf{PP}$ decision oracle. We complement this result by showing that $\mathsf{QMA}$ -search does $\textit{not}$ reduce to $\mathsf{QMA}$ -decision in polynomial-time, relative to a quantum oracle. We also explore the more general $\textit{state synthesis problem}$ , 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 from 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.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 $\textit{de facto}$ 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 $\mathsf{QMA}$, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a $\mathsf{QMA}$ problem up to inverse polynomial precision by making one query to a $\mathsf{PP}$ decision oracle. We complement this result by showing that $\mathsf{QMA}$-search does $\textit{not}$ reduce to $\mathsf{QMA}$-decision in polynomial-time, relative to a quantum oracle. We also explore the more general $\textit{state synthesis problem}$, 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 from 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.

arXiv:2111.02999Leibniz international proceedings in informaticsNov 4, 20218 citations

Quantum Computation Relative to Oracles

Christino Tamon, Tomoyuki Yamakami

Springer eBooksJan 1, 20011 citation

Oracle Quantum Algorithms: An Overview

Rafael Pereira da Silva

Jan 1, 20240 citations

Oracle Quantum Computing

Departement Iro

This paper continues the study of the power of oracles to separate quantum com.plexity classes from classical (including probabilistic and nondeterministic) complexity classes, which we initiated in an earlier paper [5]. The new results are that, under appropriate oracles, (1) there is a decision problem that can be solved in polynomial time on the quantum computer, which would classically require exponential time in the worst case even on a nondeterministic computer (in particular QP g NP relative to this oracle), and (2) there is a decision problem that can be solved in exponential time on the quantum computer for which any deterministic classical computer would require double exponential time on all but finitely many instances.This paper continues the study of the power of oracles to separate quantum com.plexity classes from classical (including probabilistic and nondeterministic) complexity classes, which we initiated in an earlier paper [5]. The new results are that, under appropriate oracles, (1) there is a decision problem that can be solved in polynomial time on the quantum computer, which would classically require exponential time in the worst case even on a nondeterministic computer (in particular QP g NP relative to this oracle), and (2) there is a decision problem that can be solved in exponential time on the quantum computer for which any deterministic classical computer would require double exponential time on all but finitely many instances.

Jan 1, 19936 citations

Quantum computation: Algorithms and Applications

Chien‐Hung Cho, Chih-Yu Chen, Kuo-Chin Chen +6 more

Chinese Journal of PhysicsMay 8, 202148 citations

Oracle Quantum Computing

André Berthiaume, Gilles Brassard A

Building on the work of Deutsch and Jozsa, we construct oracles relative to which (1) there is a decision problem that can be solved with certainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynomial time if errors are not tolerated, nor even in nondeterministic polynomial time, and (2) there is a decision problem that can be solved in exponential time on the quantum computer, which requires double exponential time on all but finitely many instances on any classical deterministic computer.Building on the work of Deutsch and Jozsa, we construct oracles relative to which (1) there is a decision problem that can be solved with certainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynomial time if errors are not tolerated, nor even in nondeterministic polynomial time, and (2) there is a decision problem that can be solved in exponential time on the quantum computer, which requires double exponential time on all but finitely many instances on any classical deterministic computer.

Journal of Modern OpticsDec 1, 199499 citations

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

\textit{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

\mathsf{QMA}

, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a

\mathsf{QMA}

problem up to inverse polynomial precision by making one query to a

\mathsf{PP}

decision oracle. We complement this result by showing that

\mathsf{QMA}

-search does

\textit{not}

reduce to

\mathsf{QMA}

-decision in polynomial-time, relative to a quantum oracle. We also explore the more general

\textit{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 from 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.