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Polyakov S.V.,U.S. National Institute of Standards and Technology | Polyakov S.V.,University of South Florida | Muller A.,U.S. National Institute of Standards and Technology | Flagg E.B.,U.S. National Institute of Standards and Technology | And 8 more authors.
Physical Review Letters | Year: 2011

Single photons produced by fundamentally dissimilar physical processes will in general not be indistinguishable. We show how photons produced from a quantum dot and by parametric down-conversion in a nonlinear crystal can be manipulated to be indistinguishable. The measured two-photon coalescence probability is 16%, and is limited by quantum-dot decoherence. Temporal filtering to the quantum-dot coherence time and accounting for detector time response increases this to 61% while retaining 25% of the events. This technique can connect different elements in a scalable quantum network. © 2011 American Physical Society.

Kulkarni R.,Center for Quantum Technologies
ITCS 2013 - Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science | Year: 2013

A function f : {0, 1}n → {0, 1} is called evasive if its decision tree complexity is maximal, i.e., D(f) = n. The long-standing Anderaa-Rosenberg-Karp (ARK) Conjecture asserts that every non-trivial monotone graph property is evasive. The Evasiveness Conjecture (EC) is a generalization of ARK Conjecture from monotone graph properties to arbitrary monotone transitive Boolean functions. In this paper we study a weakening of the Evasiveness Conjecture called Weak Evasivenss Conjecture (weak-EC). The weak-EC asserts that every non-trivial monotone transitive Boolean function must have D(f) ≥ n1-∈, for every ∈ > 0. The purpose of this note is to make some remarks on weak-EC that hint towards a plausible attack on EC. First we observe that weak-EC is equivalent to EC. Further we observe that ruling out only certain simple (monotone-NC1) counter-examples to weak-EC suffices to confirm EC in its whole generality. Finally we rule out some simple counter-examples to weak-EC (AC0 : unconditionally; and monotone-TC0 : under a conjecture of Benjamini, Kalai, and Schramm on their noise stability). We also investigate an analogue of weak-EC for the stronger model of parity decision trees and provide a counter-example to this seemingly stronger version under a conjecture of Montanaro and Osborne. © 2013 ACM.

Lee T.,Center for Quantum Technologies | Mittal R.,University of Waterloo | Reichardt B.W.,University of Waterloo | Spalek R.,Google | Szegedy M.,Rutgers University
Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS | Year: 2011

State conversion generalizes query complexity to the problem of converting between two input-dependent quantum states by making queries to the input. We characterize the complexity of this problem by introducing a natural information-theoretic norm that extends the Schur product operator norm. The complexity of converting between two systems of states is given by the distance between them, as measured by this norm. In the special case of function evaluation, the norm is closely related to the general adversary bound, a semi-definite program that lower-bounds the number of input queries needed by a quantum algorithm to evaluate a function. We thus obtain that the general adversary bound characterizes the quantum query complexity of any function whatsoever. This generalizes and simplifies the proof of the same result in the case of boolean input and output. Also in the case of function evaluation, we show that our norm satisfies a remarkable composition property, implying that the quantum query complexity of the composition of two functions is at most the product of the query complexities of the functions, up to a constant. Finally, our result implies that discrete and continuous-time query models are equivalent in the bounded-error setting, even for the general state-conversion problem. © 2011 IEEE.

Degorre J.,Center for Quantum Technologies | Kaplan M.,University of Montreal | Laplante S.,University Paris - Sud | Roland J.,NEC Laboratories America Inc.
Quantum Information and Computation | Year: 2011

We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a, b distributed according to some pre-specified joint distribution p(a, b(x, y). Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa. By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication. © Rinton Press.

Lee T.,Center for Quantum Technologies | Roland J.,University of America
Proceedings of the Annual IEEE Conference on Computational Complexity | Year: 2012

We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with less than k times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. For a boolean function f we also show an XOR lemma - computing the parity of k copies of f with less than k times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, characterizes bounded-error quantum query complexity. In particular, we show that the multiplicative adversary bound is always at least as large as the additive adversary bound, which is known to characterize bounded-error quantum query complexity. © 2012 IEEE.

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