2,506 research outputs found

    Efficient quantum protocols for XOR functions

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    We show that for any Boolean function f on {0,1}^n, the bounded-error quantum communication complexity of XOR functions fβˆ˜βŠ•f\circ \oplus satisfies that QΟ΅(fβˆ˜βŠ•)=O(2d(log⁑βˆ₯f^βˆ₯1,Ο΅+log⁑nΟ΅)log⁑(1/Ο΅))Q_\epsilon(f\circ \oplus) = O(2^d (\log\|\hat f\|_{1,\epsilon} + \log \frac{n}{\epsilon}) \log(1/\epsilon)), where d is the F2-degree of f, and βˆ₯f^βˆ₯1,Ο΅=min⁑g:βˆ₯fβˆ’gβˆ₯βˆžβ‰€Ο΅βˆ₯f^βˆ₯1\|\hat f\|_{1,\epsilon} = \min_{g:\|f-g\|_\infty \leq \epsilon} \|\hat f\|_1. This implies that the previous lower bound QΟ΅(fβˆ˜βŠ•)=Ξ©(log⁑βˆ₯f^βˆ₯1,Ο΅)Q_\epsilon(f\circ \oplus) = \Omega(\log\|\hat f\|_{1,\epsilon}) by Lee and Shraibman \cite{LS09} is tight for f with low F2-degree. The result also confirms the quantum version of the Log-rank Conjecture for low-degree XOR functions. In addition, we show that the exact quantum communication complexity satisfies QE(f)=O(2dlog⁑βˆ₯f^βˆ₯0)Q_E(f) = O(2^d \log \|\hat f\|_0), where βˆ₯f^βˆ₯0\|\hat f\|_0 is the number of nonzero Fourier coefficients of f. This matches the previous lower bound QE(f(x,y))=Ξ©(log⁑rank(Mf))Q_E(f(x,y)) = \Omega(\log rank(M_f)) by Buhrman and de Wolf \cite{BdW01} for low-degree XOR functions.Comment: 11 pages, no figur

    On the communication complexity of XOR functions

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    An XOR function is a function of the form g(x,y) = f(x + y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise one-way communication complexity for all f. We also show that, when f is monotone, g's quantum and classical complexities are quadratically related, and that when f is a linear threshold function, g's quantum complexity is Theta(n). More generally, we make a structural conjecture about the Fourier spectra of boolean functions which, if true, would imply that the quantum and classical exact communication complexities of all XOR functions are asymptotically equivalent. We give two randomised classical protocols for general XOR functions which are efficient for certain functions, and a third protocol for linear threshold functions with high margin. These protocols operate in the symmetric message passing model with shared randomness.Comment: 18 pages; v2: minor correction

    Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

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    The class FORMULA[s]∘GFORMULA[s] \circ \mathcal{G} consists of Boolean functions computable by size-ss de Morgan formulas whose leaves are any Boolean functions from a class G\mathcal{G}. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n1.99]∘GFORMULA[n^{1.99}]\circ \mathcal{G}, for classes G\mathcal{G} of functions with low communication complexity. Let R(k)(G)R^{(k)}(\mathcal{G}) be the maximum kk-party NOF randomized communication complexity of G\mathcal{G}. We show: (1) The Generalized Inner Product function GIPnkGIP^k_n cannot be computed in FORMULA[s]∘GFORMULA[s]\circ \mathcal{G} on more than 1/2+Ξ΅1/2+\varepsilon fraction of inputs for s=o ⁣(n2(kβ‹…4kβ‹…R(k)(G)β‹…log⁑(n/Ξ΅)β‹…log⁑(1/Ξ΅))2). s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right). As a corollary, we get an average-case lower bound for GIPnkGIP^k_n against FORMULA[n1.99]∘PTFkβˆ’1FORMULA[n^{1.99}]\circ PTF^{k-1}. (2) There is a PRG of seed length n/2+O(sβ‹…R(2)(G)β‹…log⁑(s/Ξ΅)β‹…log⁑(1/Ξ΅))n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right) that Ξ΅\varepsilon-fools FORMULA[s]∘GFORMULA[s] \circ \mathcal{G}. For FORMULA[s]∘LTFFORMULA[s] \circ LTF, we get the better seed length O(n1/2β‹…s1/4β‹…log⁑(n)β‹…log⁑(n/Ξ΅))O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right). This gives the first non-trivial PRG (with seed length o(n)o(n)) for intersections of nn half-spaces in the regime where Ρ≀1/n\varepsilon \leq 1/n. (3) There is a randomized 2nβˆ’t2^{n-t}-time #\#SAT algorithm for FORMULA[s]∘GFORMULA[s] \circ \mathcal{G}, where t=Ξ©(nsβ‹…log⁑2(s)β‹…R(2)(G))1/2.t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n1.99]∘LTFFORMULA[n^{1.99}]\circ LTF. (4) The Minimum Circuit Size Problem is not in FORMULA[n1.99]∘XORFORMULA[n^{1.99}]\circ XOR. On the algorithmic side, we show that FORMULA[n1.99]∘XORFORMULA[n^{1.99}] \circ XOR can be PAC-learned in time 2O(n/log⁑n)2^{O(n/\log n)}

    Quantum-enhanced Secure Delegated Classical Computing

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    We present a quantumly-enhanced protocol to achieve unconditionally secure delegated classical computation where the client and the server have both limited classical and quantum computing capacity. We prove the same task cannot be achieved using only classical protocols. This extends the work of Anders and Browne on the computational power of correlations to a security setting. Concretely, we present how a client with access to a non-universal classical gate such as a parity gate could achieve unconditionally secure delegated universal classical computation by exploiting minimal quantum gadgets. In particular, unlike the universal blind quantum computing protocols, the restriction of the task to classical computing removes the need for a full universal quantum machine on the side of the server and makes these new protocols readily implementable with the currently available quantum technology in the lab

    A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs

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    The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode nn classical bits into mm qubits, in such a way that each set of kk bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m<0.7nm<0.7 n, then the success probability is exponentially small in kk. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first ``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the proceedings of the IEEE FOCS 08 conferenc
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