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arXiv:1911.07782 · SIAM Journal on Computing

QMA-Hardness of Consistency of Local Density Matrices with Applications to Quantum Zero-Knowledge

Anne Broadbent, Alex B. Grilo

Published Aug 1, 2022• 28 citations
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Abstract

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 $k$ 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 [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 $k$ qubits and, furthermore, the reduced density matrix of any $k$-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.

DOI
10.1137/21m140729x

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QMA-hardness of Consistency of Local Density Matrices with Applications to Quantum Zero-Knowledge

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The complexity of the consistency and N-representability problems for quantum states - eScholarship

Yi-Kai Liu

Quantum mechanics has important consequences for machines that store and manipulate information. In particular, quantum computers might be more powerful than classical computers; examples of this include Shor's algorithm for factoring and discrete logarithms, and Grover's algorithm for black-box search. Because of these theoretical results, and the possibility that we may eventually succeed in building scalable quantum computers, it is interesting to study complexity classes based on quantum computation. QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most of which are variants of the Local Hamiltonian problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is Consistency of Local Density Matrices : given a collection of density matrices describing different subsets of an n-qubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, Fermionic Local Hamiltonian and N- representability, are QMA-complete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. N-representability is particularly interesting, as it is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960's, it is only recently that the theory of quantum computation has provided the right tools to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and stoquastic systems. We also give an alternative proof of a result due to Jaynes : whenever local density matrices are consistent, they are consistent with a Gibbs state

•Jan 1, 2007•0 citations
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.
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration. The proof system relies on a new variant of the QMA-complete local Hamiltonian problem in which the local terms are described by Clifford operations and standard basis measurements. We believe that the QMA-completeness of this problem may have other uses in quantum complexity.
In this work we investigate the computational complexity of the pure consistency of local density matrices (PureCLDM) and pure N-representability (Pure-N-Representability; analog of PureCLDM for bosonic or fermionic systems) problems. In these problems the input is a set of reduced density matrices and the task is to determine whether there exists a global pure state consistent with these reduced density matrices. While mixed CLDM, i.e. where the global state can be mixed, was proven to be QMA-complete by Broadbent and Grilo [JoC 2022], almost nothing was known about the complexity of the pure version. Before our work the best upper and lower bounds were QMA(2) and QMA. Our contribution to the understanding of these problems is twofold. Firstly, we define a pure state analogue of the complexity class QMA+ of Aharanov and Regev [FOCS 2003], which we call PureSuperQMA. We prove that both pure-N-Representability and PureCLDM are complete for this new class. Along the way we supplement Broadbent and Grilo by proving hardness for 2-qubit reduced density matrices and showing that mixed N-Representability is QMA-complete. Secondly, we improve the upper bound on PureCLDM. Using methods from algebraic geometry, we prove that PureSuperQMA is in PSPACE. Our methods, and the PSPACE upper bound, are also valid for PureCLDM with exponential or even perfect precision, hence precisePureCLDM is not preciseQMA(2)=NEXP-complete, unless PSPACE=NEXP. We view this as evidence for a negative answer to the longstanding open question whether PureCLDM is QMA(2)-complete. The techniques we develop for our PSPACE upper bound are quite general. We are able to use them for various applications: from proving PSPACE upper bounds on other quantum problems to giving an efficient parallel (NC) algorithm for (non-convex) quadratically constrained quadratic programs with few constraints.
QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most notably the ``Local Hamiltonian'' problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems. The first one is ``Consistency of Local Density Matrices'': given several density matrices describing different (constant-size) subsets of an n-qubit system, decide whether these are consistent with a single global state. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. This uses algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, ``Fermionic Local Hamiltonian'' and ``N-representability,'' are QMA-complete. These problems arise in calculating the ground state energies of molecular systems. N-representability is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960's, it is only recently that the theory of quantum computation has allowed us to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and ``stoquastic'' systems. We also give an alternative proof of a result due to Jaynes: whenever local density matrices are consistent, they are consistent with a Gibbs state.
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration.
Quantum mechanics has important consequences for machines that store and manipulate information. In particular, quantum computers might be more powerful than classical computers; examples of this include Shor's algorithm for factoring and discrete logarithms, and Grover's algorithm for black-box search. Because of these theoretical results, and the possibility that we may eventually succeed in building scalable quantum computers, it is interesting to study complexity classes based on quantum computation. QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most of which are variants of the Local Hamiltonian problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is Consistency of Local Density Matrices : given a collection of density matrices describing different subsets of an n-qubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, Fermionic Local Hamiltonian and N- representability, are QMA-complete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. N-representability is particularly interesting, as it is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960's, it is only recently that the theory of quantum computation has provided the right tools to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and stoquastic systems. We also give an alternative proof of a result due to Jaynes : whenever local density matrices are consistent, they are consistent with a Gibbs state