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Semidefinite programs for completely bounded norms
, 2009
"... The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them. ..."
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The completely bounded trace and spectral norms in finite dimensions are shown to be expressible by semidefinite programs. This provides an efficient method by which these norms may be both calculated and verified, and gives alternate proofs of some known facts about them.
NonIdentity Check Remains QMAComplete for Short Circuits
, 2008
"... The NonIdentity Check problem asks whether a given a quantum circuit is far away from the identity or not. It is well known that this problem is QMAComplete [14]. In this note, it is shown that the NonIdentity Check problem remains QMAComplete for circuits of short depth. Specifically, we prove ..."
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The NonIdentity Check problem asks whether a given a quantum circuit is far away from the identity or not. It is well known that this problem is QMAComplete [14]. In this note, it is shown that the NonIdentity Check problem remains QMAComplete for circuits of short depth. Specifically, we prove that for constant depth quantum circuit in which each gate is given to at least Ω(log n) bits of precision, the NonIdentity Check problem is QMAComplete. It also follows that the hardness of the problem remains for polylogarithmic depth circuit consisting of only gates from any universal gate set and for logarithmic depth circuit using some specific universal gate set. 1
Computational Distinguishability of Quantum Channels
, 2009
"... The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the wellknown satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that h ..."
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The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the wellknown satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic depth, to mixedunitary channels, to degradable channels, or to antidegradable channels. These hardness results are demonstrated by finding reductions between these classes of quantum channels. These techniques have applications outside the distinguishability problem, as the construction for mixedunitary channels is used to prove that the additivity problem for the classical capacity of quantum channels can be equivalently restricted to the mixed unitary channels.