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arXiv:2408.03986

Finding quantum partial assignments by search-to-decision reductions

Jordi R. Weggemans

Published Aug 7, 2024• 0 citations
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Abstract

In computer science, many search problems are reducible to decision problems, which implies that finding a solution is as hard as deciding whether a solution exists. A quantum analogue of search-to-decision reductions would be to ask whether a quantum algorithm with access to a $\mathsf{QMA}$ oracle can construct $\mathsf{QMA}$ witnesses as quantum states. By a result from Irani, Natarajan, Nirkhe, Rao, and Yuen (CCC '22), it is known that this does not hold relative to a quantum oracle, unlike the cases of $\mathsf{NP}$, $\mathsf{MA}$, and $\mathsf{QCMA}$ where search-to-decision relativizes. We prove that if one is not interested in the quantum witness as a quantum state but only in terms of its partial assignments, i.e. the reduced density matrices, then there exists a classical polynomial-time algorithm with access to a $\mathsf{QMA}$ oracle that outputs approximations of the density matrices of a near-optimal quantum witness, for any desired constant locality and inverse polynomial error. Our construction is based on a circuit-to-Hamiltonian mapping that approximately preserves near-optimal $\mathsf{QMA}$ witnesses and a new $\mathsf{QMA}$-complete problem, Low-energy Density Matrix Verification, which is called by the $\mathsf{QMA}$ oracle to adaptively construct approximately consistent density matrices of a low-energy state.

DOI
10.48550/arxiv.2408.03986

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In computer science, many search problems are reducible to decision problems, which implies that finding a solution is as hard as deciding whether a solution exists. A quantum analogue of search-to-decision reductions would be to ask whether a quantum algorithm with access to a QMA\mathsf{QMA}QMA oracle can construct QMA\mathsf{QMA}QMA witnesses as quantum states. By a result from Irani, Natarajan, Nirkhe, Rao, and Yuen (CCC '22), it is known that this does not hold relative to a quantum oracle, unlike the cases of NP\mathsf{NP}NP, MA\mathsf{MA}MA, and QCMA\mathsf{QCMA}QCMA where search-to-decision relativizes. We prove that if one is not interested in the quantum witness as a quantum state but only in terms of its partial assignments, i.e. the reduced density matrices, then there exists a classical polynomial-time algorithm with access to a QMA\mathsf{QMA}QMA oracle that outputs approximations of the density matrices of a near-optimal quantum witness, for any desired constant locality and inverse polynomial error. Our construction is based on a circuit-to-Hamiltonian mapping that approximately preserves near-optimal QMA\mathsf{QMA}QMA witnesses and a new QMA\mathsf{QMA}QMA-complete problem, Low-energy Density Matrix Verification, which is called by the QMA\mathsf{QMA}QMA oracle to adaptively construct approximately consistent density matrices of a low-energy state.
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\textit{de facto}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\mathsf{QMA}QMA, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a QMA\mathsf{QMA}QMA problem up to inverse polynomial precision by making one query to a PP\mathsf{PP}PP decision oracle. We complement this result by showing that QMA\mathsf{QMA}QMA-search does not\textit{not}not reduce to QMA\mathsf{QMA}QMA-decision in polynomial-time, relative to a quantum oracle. We also explore the more general state synthesis problem\textit{state synthesis problem}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)O(n)O(n) queries to a classical oracle to prepare an nnn-qubit state, and asked if the query complexity could be made sublinear.
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)Θ(n)Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(log⁡n)Θ(\log n)Θ(logn) 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)Ω(\log n)Ω(logn) 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)Ω(\log n)Ω(logn) 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(logn)o(log n)o(logn) NP queries, then BPPNP[o(n)]\text{BPP}^{\text{NP}[o(n)]}BPPNP[o(n)] contains a PNP\text{P}^{\text{NP}}PNP-complete problem if the algorithm is classical and FBQPNP[o(n)]\text{FBQP}^{\text{NP}[o(n)]}FBQPNP[o(n)] contains an FPNP\text{FP}^{\text{NP}}FPNP-complete problem if the algorithm is quantum.
We provide several advances to the understanding of the class of quantum Merlin--Arthur (QMA) proof systems, the quantum analogue of NP. Our central contribution is proving a longstanding conjecture that the consistency of local density matrices (CLDM) problem is QMA-hard under Karp reductions. The input of CLDM consists of local reduced density matrices on sets of at most kkk qubits, and the problem asks if there is an nnn-qubit global quantum state that is locally consistent with all of the kkk-qubit local density matrices. The containment of this problem in QMA and the QMA-hardness under Turing reductions were proved by Liu [in Proceeding of APPROX-RANDOM, 2006, pp. 438--449]. Liu also conjectured that CLDM is QMA-hard under Karp reductions, which is desirable for applications, and we finally prove this conjecture. We establish this result using the techniques of simulatable codes of Grilo, Slofstra, and Yuen [in Proceeding of FOCS, 2019, pp. 611--635], simplifying their proofs and tailoring them to the context of QMA. In order to develop applications of CLDM, we propose a framework that we call locally simulatable proofs for QMA: this provides QMA proofs that can be efficiently verified by probing only kkk qubits and, furthermore, the reduced density matrix of any kkk-qubit subsystem of an accepting witness can be computed in polynomial time, independently of the witness. Within this framework, we show several advances in zero-knowledge in the quantum setting. We show for the first time a commit-and-open computational zero-knowledge proof system for all of QMA, as a quantum analogue of a “sigma” protocol. We then define a proof of quantum knowledge, which guarantees that a prover is effectively in possession of a quantum witness in an interactive proof and show that our zero-knowledge proof system satisfies this definition. Finally, we show that our proof system can be used to establish that QMA has a quantum noninteractive zero-knowledge proof system in the secret parameter setting.
We provide several advances to the understanding of the class of Quantum Merlin-Arthur proof systems (QMA), the quantum analogue of NP. Our central contribution is proving a longstanding conjecture that the Consistency of Local Density Matrices (CLDM) problem is QMA-hard under Karp reductions. The input of CLDM consists of local reduced density matrices on sets of at most k qubits, and the problem asks if there is an n-qubit global quantum state that is locally consistent with all of the k-qubit local density matrices. The containment of this problem in QMA and the QMA-hardness under Turing reductions were proved by Liu [APPROX-RANDOM 2006]. Liu also conjectured that CLDM is QMA-hard under Karp reductions, which is desirable for applications, and we finally prove this conjecture. We establish this result using the techniques of simulatable codes of Grilo, Slofstra, and Yuen [FOCS 2019], simplifying their proofs and tailoring them to the context of OMA. In order to develop applications of CLDM, we propose a framework that we call locally simulatable proofs for QMA: this provides QMA proofs that can be efficiently verified by probing only k qubits and, furthermore, the reduced density matrix of any k-qubit subsystem of a good witness can be computed in polynomial time, independently of the witness. Within this framework, we show several advances in zero-knowledge in the quantum setting. We show for the first time a commit-and-open computational zero-knowledge proof system for all of QMA, as a quantum analogue of a “sigma” protocol. We then define a Proof of Quantum Knowledge, which guarantees that a prover is effectively in possession of a quantum witness in an interactive proof, and show that our zero-knowledge proof system satisfies this definition. Finally, we show that our proof system can be used to establish that QMA has a quantum non-interactive zero-knowledge proof system in the secret parameter setting.11The full version of this work can be found in https://arxiv.org/abs/1911.07782.
In this work we define a new classical constraint satisfaction problem that shares many of the properties of the quantum local Hamiltonian problem, distinguishing it from the usual classical k-SAT problem. The problem consists of minimizing the number of violated local constraints over a restricted set of distributions of assignments. We show that these distributions can be 1-to-1 mapped to a superset of the quantum states, which we call k-local quasi-quantum states. Nevertheless, we claim that our optimization problem is essentially classical, by proving that it is an NP-complete problem. Interestingly, the optimal distribution shares many of the properties of quantum states. In particular, it is not determined straightforwardly by its local marginals, and consequently, it can be used as a classical toy model to study several aspects of Hamiltonian complexity that are different from their classical counter parts. These include the complexity of 1D systems (which is in P for classical CSPs, but is QMA-hard for quantum systems), and the lack of an easy search-to-decision reduction. Finally, we believe that our model can be used to gain insights into the quantum PCP conjecture. Indeed, while we have shown that approximating the minimal number of unsatisfiable constraints to within an Θ(1) is NP-hard, it is not clear if the problem remains hard if we want to approximate the minimal fraction of unsatisfiable constraints to within an Θ(1); as in the quantum PCP conjecture, naive quantization of the classical proofs does not seem to work.
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