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arXiv:1205.2761

Multi-Prover Quantum Merlin-Arthur Proof Systems with Small Gap

Attila Pereszlényi

Published May 12, 2012• 13 citations

Abstract

This paper studies multiple-proof quantum Merlin-Arthur (QMA) proof systems in the setting when the completeness-soundness gap is small. Small means that we only lower-bound the gap with an inverse-exponential function of the input length, or with an even smaller function. Using the protocol of Blier and Tapp [arXiv:0709.0738], we show that in this case the proof system has the same expressive power as non-deterministic exponential time (NEXP). Since single-proof QMA proof systems, with the same bound on the gap, have expressive power at most exponential time (EXP), we get a separation between single and multi-prover proof systems in the 'small-gap setting', under the assumption that EXP is not equal to NEXP. This implies, among others, the nonexistence of certain operators called disentanglers (defined by Aaronson et al. [arXiv:0804.0802]), with good approximation parameters. We also show that in this setting the proof system has the same expressive power if we restrict the verifier to be able to perform only Bell-measurements, i.e., using a BellQMA verifier. This is not known to hold in the usual setting, when the gap is bounded by an inverse-polynomial function of the input length. To show this we use the protocol of Chen and Drucker [arXiv:1011.0716]. The only caveat here is that we need at least a linear amount of proofs to achieve the power of NEXP, while in the previous setting two proofs were enough. We also study the case when the proof-lengths are only logarithmic in the input length and observe that in some cases the expressive power decreases. However, we show that it doesn't decrease further if we make the proof lengths to be even shorter.

DOI: 10.48550/arxiv.1205.2761

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This paper studies multiple-proof quantum Merlin-Arthur (QMA) proof systems in the setting when the completeness-soundness gap is small. Small means that we only lower-bound the gap with an inverse-exponential function of the input length, or with an even smaller function. Using the protocol of Blier and Tapp [arXiv:0709.0738], we show that in this case the proof system has the same expressive power as non-deterministic exponential time (NEXP). Since single-proof QMA proof systems, with the same bound on the gap, have expressive power at most exponential time (EXP), we get a separation between single and multi-prover proof systems in the 'small-gap setting', under the assumption that EXP is not equal to NEXP. This implies, among others, the nonexistence of certain operators called disentanglers (defined by Aaronson et al. [arXiv:0804.0802]), with good approximation parameters. We also show that in this setting the proof system has the same expressive power if we restrict the verifier to be able to perform only Bell-measurements, i.e., using a BellQMA verifier. This is not known to hold in the usual setting, when the gap is bounded by an inverse-polynomial function of the input length. To show this we use the protocol of Chen and Drucker [arXiv:1011.0716]. The only caveat here is that we need at least a linear amount of proofs to achieve the power of NEXP, while in the previous setting two proofs were enough. We also study the case when the proof-lengths are only logarithmic in the input length and observe that in some cases the expressive power decreases. However, we show that it doesn't decrease further if we make the proof lengths to be even shorter.

We prove a deterministic exponential time upper bound for Quantum Merlin-Arthur games with k unentangled provers. This is the first non-trivial upper bound of QMA(k) better than NEXP and can be considered an exponential improvement, unless EXP=NEXP. The key ideas of our proof are to use perturbation theory to reduce the QMA(2)-complete Separable Sparse Hamiltonian problem to a variant of the Separable Local Hamiltonian problem with an exponentially small promise gap, and then to decide this instance using epsilon-net methods. Our results imply an exponential time algorithm for the Pure State N-Representability problem in quantum chemistry, which is in QMA(2), but is not known to be in QMA. We also discuss the implications of our results on the Best Separable State problem.

BellQMA protocols are a subclass of multi-prover quantum Merlin-Arthur protocols in which the verifier is restricted to perform nonadaptive,unentangled measurements on the quantum states received from each Merlin. In this paper, we prove that m-clause 3-SAT instances have BellQMA proofs of satisfiability with constant soundness gap, in which O(sqrt(m)polylog(m)) Merlins each send O(log m) qubits to Arthur. Our result answers a question of Aaronson et al., who gave a protocol with similar parameters that used entangled measurements; the analysis of our protocol is significantly simpler than that of Aaronson et al. Our result also complements recent work of Brandao, Christandl, and Yard, who showed upper bounds on the power of multiprover quantum proofs with unentangled but adaptive (LOCC) measurements.

We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly MQA (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class BellQMA(poly), characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resulting measurement outcomes. We show that if the number of outcomes per nonadaptive measurement is a polynomially-bounded function, then the expressive power of the proof system is exactly QMA. Finally, we study a class equivalent to QMA(m), denoted SepQMA(m), where the verifier's measurement operator corresponding to outcome "accept" is a fully separable operator across the m quantum proofs. Using cone programming duality, we give an alternate proof of a result of Harrow and Montanaro [FOCS, pp. 633--642 (2010)] that shows a perfect parallel repetition theorem for SepQMA(m) for any m.

We study three variants of multi-prover quantum Merlin-Arthur proof systems. We first show that the class of problems that can be efficiently verified using polynomially many quantum proofs, each of logarithmic-size, is exactly \class{MQA} (also known as QCMA), the class of problems which can be efficiently verified via a classical proof and a quantum verifier. We then study the class $\class{BellQMA}(\poly)$, characterized by a verifier who first applies unentangled, nonadaptive measurements to each of the polynomially many proofs, followed by an arbitrary but efficient quantum verification circuit on the resulting measurement outcomes. We show that if the number of outcomes per nonadaptive measurement is a polynomially-bounded function, then the expressive power of the proof system is exactly \class{QMA}. Finally, we study a class equivalent to \class{QMA}(

m

), denoted $\class{SepQMA}(m)$, where the verifier's measurement operator corresponding to outcome {\it accept} is a fully separable operator across the

m

quantum proofs. Using cone programming duality, we give an alternate proof of a result of Harrow and Montanaro [FOCS, pp. 633--642 (2010)] that shows a perfect parallel repetition theorem for $\class{SepQMA}(m)$ for any

m

.

We present three contributions to the understanding of QMA with multiple provers:• We give a tight soundness analysis of the protocol of [Blier and Tapp, ICQNM '09], yielding a soundness gap Ω(N -2 ).Our improvement is achieved without the use of an instance with a constant soundness gap (i.e., without using a "PCP").• We give a tight soundness analysis of the protocol of [Chen and Drucker, ArXiV '10], thereby improving their result from a "monolithic" protocol where Θ( √ N) provers are needed in order to have any soundness gap, to a protocol with a smooth trade-off between the number of provers κ and a soundness gap Ω(κ 2 N -1 ), as long as κ ∈ Ω(log N). (And, when κ ∈ Θ( √ N), we recover the original parameters of Chen and Drucker.)• We make progress towards an open question of [Aaronson et al., ToC '09] about what kinds of NP-complete problems are amenable to "sublinear" multiple-prover QMA protocols, by observing that a large class of such examples can easily be derived from results already in the PCP literature -namely, at least the languages recognized by a non-deterministic RAMs in quasilinear time.

We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [arXiv:1410.2882]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One interpretation is that Merlin's ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2)

\subseteq

NEXP.