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We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1}<i><sup>N</sup></i> in the <i>black-box</i> model. We show that the exponential quantum speed-up obtained for <i>partial</i> functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be(More)
Consider a Boolean function χ : X → {0, 1} that partitions set X between its good and bad elements, where x is good if χ(x) = 1 and bad otherwise. Consider also a quantum algorithm A such that A|0 = x∈X αx|x is a quantum superposition of the elements of X, and let a denote the probability that a good element is produced if A|0 is measured. If we repeat the(More)
We examine the number T of oracle calls that a quantum network requires to compute some Boolean function on {0, 1} N in the so-called black-box model, where the input is given as an oracle. We show that the acceptance probability of a network can be written as an N-variate polynomial of the input, having degree at most 2T. Using lower bounds on the degrees(More)
We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method we implemented this algorithm and found factorizations of the commonly used quantum logical operations into elementary gates(More)
In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring Z[ 1 √ 2 , i], in the single-qubit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard and T gates used. We conjecture that the equivalence(More)