16,421 research outputs found

    Black-box Hamiltonian simulation and unitary implementation

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    We present general methods for simulating black-box Hamiltonians using quantum walks. These techniques have two main applications: simulating sparse Hamiltonians and implementing black-box unitary operations. In particular, we give the best known simulation of sparse Hamiltonians with constant precision. Our method has complexity linear in both the sparseness D (the maximum number of nonzero elements in a column) and the evolution time t, whereas previous methods had complexity scaling as D^4 and were superlinear in t. We also consider the task of implementing an arbitrary unitary operation given a black-box description of its matrix elements. Whereas standard methods for performing an explicitly specified N x N unitary operation use O(N^2) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N^{2/3} (log log N)^{4/3}) queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only O(sqrt{N}) queries, which is optimal.Comment: 19 pages, 3 figures, minor correction

    Maximum Inner-Product Search using Tree Data-structures

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    The problem of {\em efficiently} finding the best match for a query in a given set with respect to the Euclidean distance or the cosine similarity has been extensively studied in literature. However, a closely related problem of efficiently finding the best match with respect to the inner product has never been explored in the general setting to the best of our knowledge. In this paper we consider this general problem and contrast it with the existing best-match algorithms. First, we propose a general branch-and-bound algorithm using a tree data structure. Subsequently, we present a dual-tree algorithm for the case where there are multiple queries. Finally we present a new data structure for increasing the efficiency of the dual-tree algorithm. These branch-and-bound algorithms involve novel bounds suited for the purpose of best-matching with inner products. We evaluate our proposed algorithms on a variety of data sets from various applications, and exhibit up to five orders of magnitude improvement in query time over the naive search technique.Comment: Under submission in KDD 201

    Learning and Testing Variable Partitions

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    Let FF be a multivariate function from a product set Σn\Sigma^n to an Abelian group GG. A kk-partition of FF with cost δ\delta is a partition of the set of variables V\mathbf{V} into kk non-empty subsets (X1,…,Xk)(\mathbf{X}_1, \dots, \mathbf{X}_k) such that F(V)F(\mathbf{V}) is δ\delta-close to F1(X1)+⋯+Fk(Xk)F_1(\mathbf{X}_1)+\dots+F_k(\mathbf{X}_k) for some F1,…,FkF_1, \dots, F_k with respect to a given error metric. We study algorithms for agnostically learning kk partitions and testing kk-partitionability over various groups and error metrics given query access to FF. In particular we show that 1.1. Given a function that has a kk-partition of cost δ\delta, a partition of cost O(kn2)(δ+ϵ)\mathcal{O}(k n^2)(\delta + \epsilon) can be learned in time O~(n2poly(1/ϵ))\tilde{\mathcal{O}}(n^2 \mathrm{poly} (1/\epsilon)) for any ϵ>0\epsilon > 0. In contrast, for k=2k = 2 and n=3n = 3 learning a partition of cost δ+ϵ\delta + \epsilon is NP-hard. 2.2. When FF is real-valued and the error metric is the 2-norm, a 2-partition of cost δ2+ϵ\sqrt{\delta^2 + \epsilon} can be learned in time O~(n5/ϵ2)\tilde{\mathcal{O}}(n^5/\epsilon^2). 3.3. When FF is Zq\mathbb{Z}_q-valued and the error metric is Hamming weight, kk-partitionability is testable with one-sided error and O(kn3/ϵ)\mathcal{O}(kn^3/\epsilon) non-adaptive queries. We also show that even two-sided testers require Ω(n)\Omega(n) queries when k=2k = 2. This work was motivated by reinforcement learning control tasks in which the set of control variables can be partitioned. The partitioning reduces the task into multiple lower-dimensional ones that are relatively easier to learn. Our second algorithm empirically increases the scores attained over previous heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202

    Max-stable sketches: estimation of Lp-norms, dominance norms and point queries for non-negative signals

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    Max-stable random sketches can be computed efficiently on fast streaming positive data sets by using only sequential access to the data. They can be used to answer point and Lp-norm queries for the signal. There is an intriguing connection between the so-called p-stable (or sum-stable) and the max-stable sketches. Rigorous performance guarantees through error-probability estimates are derived and the algorithmic implementation is discussed
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