The class QMA (k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA (k) = QMA (2) for k ≥ 2? Can QMA (k) protocols be amplified to exponentially small error?In this paper, we make progress on all of the above questions.• We give a protocol by which a verifier can be convinced that a 3Sat formula of size m is satisfiable, with constant soundness, given O ( √ m) unentangled quantum witnesses with O (log m) qubits each. Our protocol relies on the existence of very short PCPs.• We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA (2) protocol can be amplified to exponentially small error, and QMA (k) = QMA (2) for all k ≥ 2.• We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one. The class QMA, or Quantum Merlin-Arthur, consists of all languages that admit a proof protocol in which Merlin sends Arthur a polynomial-size quantum state |ψ , and then Arthur decides whether to accept or reject in quantum polynomial time. This class was introduced by Knill [17], Kitaev [15], and Watrous [29] as a quantum analogue of NP. By now we know a reasonable amount about QMA: for example, it allows amplification of success probabilities, is contained in PP, and has natural complete promise problems. (See Aharonov and Naveh [2] for a survey.) In 2003, Kobayashi, Matsumoto, and Yamakami [19] defined a generalization of QMA called QMA (k). Here there are k Merlins, who send Arthur k quantum proofs |ψ 1 , . . . , |ψ k respectively that are guaranteed to be unentangled with each other. (Thus QMA (1) = QMA.) Notice that in the classical case, this generalization is completely uninteresting: we have MA (k) = MA for all k, since we can always simulate k Merlins by a single Merlin who sends Arthur a concatenation of the k proofs. In the quantum case, however, a single Merlin could cheat by entangling the k proofs, and we know of no general way to detect such entanglement.When we try to understand QMA (k), we encounter at least three basic questions. First, do multiple quantum proofs ever actually help? That is, can we find some sort of evidence that QMA (k) = QMA (1) for some k? Second, can QMA (k) protocols be amplified to exponentially small error? Third, are two Merlins the most we ever need? That is, does QMA (k) = QMA (2) for all k ≥ 2? 1 We know of three previous results that are relevant to the above questions. First, in their original paper on QMA (k), Kobayashi et al. [19] proved that a positive answer to the second question implies a positive answer to the third. That is, if QMA (k) protocols can be amplified, then QMA (k) = QMA (2) for all k ≥ 2.Second, Liu, Christandl, and Verstraete [21] gave a natural problem from quantum chemistry, called pure state N -representability, which is in QMA (2) but...
We make the case for studying the complexity of approximately simulating (sampling) quantum systems for reasons beyond that of quantum computational supremacy, such as diagnosing phase transitions. We consider the sampling complexity as a function of time t due to evolution generated by spatially local quadratic bosonic Hamiltonians. We obtain an upper bound on the scaling of t with the number of bosons n for which approximate sampling is classically efficient. We also obtain a lower bound on the scaling of t with n for which any instance of the boson sampling problem reduces to this problem and hence implies that the problem is hard, assuming the conjectures of Aaronson and Arkhipov [Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing (ACM Press, New York, New York, USA, 2011), p. 333]. This establishes a dynamical phase transition in sampling complexity. Further, we show that systems in the Anderson-localized phase are always easy to sample from at arbitrarily long times. We view these results in light of classifying phases of physical systems based on parameters in the Hamiltonian. In doing so, we combine ideas from mathematical physics and computational complexity to gain insight into the behavior of condensed matter, atomic, molecular, and optical systems.
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