Testing Product States, Quantum Merlin-Arthur Games and Tensor OptimizationTesting Product States, Quantum Merlin-Arthur Games and Tensor Optimization

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 Author Harrow, Aram W ♦ Montanaro, Ashley Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2013 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Entanglement ♦ Quantum Merlin-Arthur games ♦ Tensor optimization Abstract We give a test that can distinguish efficiently between product states of $\textit{n}$ quantum systems and states that are far from product. If applied to a state $|\textit{ψ}⟩$ whose maximum overlap with a product state is 1 ™ $\textit{ε},$ the test passes with probability 1 ™ $\textit{Θ}(\textit{ε}),$ regardless of $\textit{n}$ or the local dimensions of the individual systems. The test uses two copies of $|\textit{ψ}⟩.$ We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarizing channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: $QMA(\textit{k})$ = QMA(2) for $\textit{k}$ ≥ 2. Building on a previous result of Aaronson et al., this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of $\textit{Õ}(√\textit{n})$ qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalization of classical linearity testing. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2013-02-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 60 Issue Number 1 Page Count 43 Starting Page 1 Ending Page 43

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Source: ACM Digital Library