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Quantum query complexity and semidefinite programming
 In Proceedings of the 18th IEEE Annual Conference on Computational Complexity
, 2003
"... We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1. show that the workspace of a quantum computer can be limited to at most n + k qubits (where n and k are the number of input and output bits res ..."
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We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1. show that the workspace of a quantum computer can be limited to at most n + k qubits (where n and k are the number of input and output bits respectively) without reducing the computational power of the model. 2. give an algorithm that on input the truth table of a partial boolean function and an integer t runs in time polynomial in the size of the truth table and estimates, to any desired accuracy, the minimum probability of error that can be attained by a quantum query algorithm attempts to evaluate f in t queries. 3. use semidefinite programming duality to formulate a dual SDP ˆ P (f,t,ɛ) that is feasible if and only if f can not be evaluated within error ɛ by a tstep quantum query algorithm Using this SDP we derive a general lower bound for quantum query complexity that encompasses a lower bound method of Ambainis and its generalizations. 4. Give an interpretation of a generalized form of branching in quantum computation.
selfreducibility, sampling
, 2012
"... Abstract: We give new evidence that quantum computers—moreover, rudimentary quantum computers built entirely out of linearoptical elements—cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a ..."
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Abstract: We give new evidence that quantum computers—moreover, rudimentary quantum computers built entirely out of linearoptical elements—cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linearoptical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomialtime classical algorithm that samples from the same probability distribution as a linearoptical network, then P #P = BPP NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the PermanentofGaussians Conjecture, which says that it is #Phard to approximate the permanent of a matrix A of
cb Licensed under a Creative Commons Attribution License (CCBY)
, 2012
"... Abstract: Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is 1. publickey—meaning that anyone can verify a banknote as genuine, not only ..."
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Abstract: Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is 1. publickey—meaning that anyone can verify a banknote as genuine, not only the bank that printed it, and 2. cryptographically secure, under a “classical ” hardness assumption that has nothing to do with quantum money. Our scheme is based on hidden subspaces, encoded as the zerosets of random multivariate polynomials. A main technical advance is to show that the “blackbox ” version of our scheme, where the polynomials are replaced by classical oracles, is unconditionally secure. Previously, such a result had only been known relative to a quantum oracle (and even there, the proof was never published). Even in Wiesner’s original setting—quantum money that can only be verified by the bank—we are able to use our techniques to patch a major security hole in Wiesner’s scheme. We give the first privatekey quantum money scheme that allows unlimited verifications and that remains unconditionally secure, even if the counterfeiter can interact adaptively with the bank.