1,979 research outputs found

    A Note on the Probability of Rectangles for Correlated Binary Strings

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    Consider two sequences of nn independent and identically distributed fair coin tosses, X=(X1,,Xn)X=(X_1,\ldots,X_n) and Y=(Y1,,Yn)Y=(Y_1,\ldots,Y_n), which are ρ\rho-correlated for each jj, i.e. P[Xj=Yj]=1+ρ2\mathbb{P}[X_j=Y_j] = {1+\rho\over 2}. We study the question of how large (small) the probability P[XA,YB]\mathbb{P}[X \in A, Y\in B] can be among all sets A,B{0,1}nA,B\subset\{0,1\}^n of a given cardinality. For sets A,B=Θ(2n)|A|,|B| = \Theta(2^n) it is well known that the largest (smallest) probability is approximately attained by concentric (anti-concentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of A,B=2Θ(n)|A|,|B| = 2^{\Theta(n)}. By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize P[XA,YB]\mathbb{P}[X \in A, Y\in B] in the regime of ρ1\rho \to 1. We also prove a similar tight lower bound, i.e. show that for ρ0\rho\to 0 the pair of opposite Hamming balls approximately minimizes the probability P[XA,YB]\mathbb{P}[X \in A, Y\in B]

    The Role of Correlated Noise in Quantum Computing

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    This paper aims to give an overview of the current state of fault-tolerant quantum computing, by surveying a number of results in the field. We show that thresholds can be obtained for a simple noise model as first proved in [AB97, Kit97, KLZ98], by presenting a proof for statistically independent noise, following the presentation of Aliferis, Gottesman and Preskill [AGP06]. We also present a result by Terhal and Burkard [TB05] and later improved upon by Aliferis, Gottesman and Preskill [AGP06] that shows a threshold can still be obtained for local non-Markovian noise, where we allow the noise to be weakly correlated in space and time. We then turn to negative results, presenting work by Ben-Aroya and Ta-Shma [BT11] who showed conditional errors cannot be perfectly corrected. We end our survey by briefly mentioning some more speculative objections, as put forth by Kalai [Kal08, Kal09, Kal11]

    An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance

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    We prove an optimal Ω(n)\Omega(n) lower bound on the randomized communication complexity of the much-studied Gap-Hamming-Distance problem. As a consequence, we obtain essentially optimal multi-pass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The Gap-Hamming-Distance problem is a communication problem, wherein Alice and Bob receive nn-bit strings xx and yy, respectively. They are promised that the Hamming distance between xx and yy is either at least n/2+nn/2+\sqrt{n} or at most n/2nn/2-\sqrt{n}, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS, 2003), it had been conjectured that the naive protocol, which uses nn bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic, or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of C. Borell (1985). To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables

    Quantum states cannot be transmitted efficiently classically

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    We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of nn qubits (held by another), up to constant accuracy, must transmit at least Ω(2n)\Omega(2^n) bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an ϵ\epsilon-net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires Ω(N)\Omega(N) classical queries for an input of size NN. Second, a nonlocal task which can be solved using nn Bell pairs, but for which any approximate classical solution must communicate Ω(2n)\Omega(2^n) bits.Comment: 24 pages; v3: accepted version incorporating many minor corrections and clarification

    Simulation Theorems via Pseudorandom Properties

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    We generalize the deterministic simulation theorem of Raz and McKenzie [RM99], to any gadget which satisfies certain hitting property. We prove that inner-product and gap-Hamming satisfy this property, and as a corollary we obtain deterministic simulation theorem for these gadgets, where the gadget's input-size is logarithmic in the input-size of the outer function. This answers an open question posed by G\"{o}\"{o}s, Pitassi and Watson [GPW15]. Our result also implies the previous results for the Indexing gadget, with better parameters than was previously known. A preliminary version of the results obtained in this work appeared in [CKL+17]

    Rotation-invariant features for multi-oriented text detection in natural images.

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    Texts in natural scenes carry rich semantic information, which can be used to assist a wide range of applications, such as object recognition, image/video retrieval, mapping/navigation, and human computer interaction. However, most existing systems are designed to detect and recognize horizontal (or near-horizontal) texts. Due to the increasing popularity of mobile-computing devices and applications, detecting texts of varying orientations from natural images under less controlled conditions has become an important but challenging task. In this paper, we propose a new algorithm to detect texts of varying orientations. Our algorithm is based on a two-level classification scheme and two sets of features specially designed for capturing the intrinsic characteristics of texts. To better evaluate the proposed method and compare it with the competing algorithms, we generate a comprehensive dataset with various types of texts in diverse real-world scenes. We also propose a new evaluation protocol, which is more suitable for benchmarking algorithms for detecting texts in varying orientations. Experiments on benchmark datasets demonstrate that our system compares favorably with the state-of-the-art algorithms when handling horizontal texts and achieves significantly enhanced performance on variant texts in complex natural scenes
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