2,099 research outputs found
Quantum pattern matching fast on average
The -dimensional pattern matching problem is to find an occurrence of a
pattern of length within a text of length , with . This task models various problems in text and
image processing, among other application areas. This work describes a quantum
algorithm which solves the pattern matching problem for random patterns and
texts in time . For
large this is super-polynomially faster than the best possible classical
algorithm, which requires time . The
algorithm is based on the use of a quantum subroutine for finding hidden shifts
in dimensions, which is a variant of algorithms proposed by Kuperberg.Comment: 22 pages, 2 figures; v3: further minor changes, essentially published
versio
Quantum algorithm for the Boolean hidden shift problem
The hidden shift problem is a natural place to look for new separations
between classical and quantum models of computation. One advantage of this
problem is its flexibility, since it can be defined for a whole range of
functions and a whole range of underlying groups. In a way, this distinguishes
it from the hidden subgroup problem where more stringent requirements about the
existence of a periodic subgroup have to be made. And yet, the hidden shift
problem proves to be rich enough to capture interesting features of problems of
algebraic, geometric, and combinatorial flavor. We present a quantum algorithm
to identify the hidden shift for any Boolean function. Using Fourier analysis
for Boolean functions we relate the time and query complexity of the algorithm
to an intrinsic property of the function, namely its minimum influence. We show
that for randomly chosen functions the time complexity of the algorithm is
polynomial. Based on this we show an average case exponential separation
between classical and quantum time complexity. A perhaps interesting aspect of
this work is that, while the extremal case of the Boolean hidden shift problem
over so-called bent functions can be reduced to a hidden subgroup problem over
an abelian group, the more general case studied here does not seem to allow
such a reduction.Comment: 10 pages, 1 figur
Quantum Algorithms for Some Hidden Shift Problems
Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure
Constituent Parsing as Sequence Labeling
We introduce a method to reduce constituent parsing to sequence labeling. For
each word w_t, it generates a label that encodes: (1) the number of ancestors
in the tree that the words w_t and w_{t+1} have in common, and (2) the
nonterminal symbol at the lowest common ancestor. We first prove that the
proposed encoding function is injective for any tree without unary branches. In
practice, the approach is made extensible to all constituency trees by
collapsing unary branches. We then use the PTB and CTB treebanks as testbeds
and propose a set of fast baselines. We achieve 90.7% F-score on the PTB test
set, outperforming the Vinyals et al. (2015) sequence-to-sequence parser. In
addition, sacrificing some accuracy, our approach achieves the fastest
constituent parsing speeds reported to date on PTB by a wide margin.Comment: EMNLP 2018 (Long Papers). Revised version with improved results after
fixing evaluation bu
Quantum algorithm for a generalized hidden shift problem
Consider the following generalized hidden shift problem:
given a function f on {0,...,M â 1} Ă ZN promised to be
injective for fixed b and satisfying f(b, x) = f(b + 1, x + s)
for b = 0, 1,...,M â 2, find the unknown shift s â ZN.
For M = N, this problem is an instance of the abelian
hidden subgroup problem, which can be solved efficiently on
a quantum computer, whereas for M = 2, it is equivalent
to the dihedral hidden subgroup problem, for which no
efficient algorithm is known. For any fixed positive ïżœ, we give
an efficient (i.e., poly(logN)) quantum algorithm for this
problem provided M â„ N^â. The algorithm is based on the
âpretty good measurementâ and uses H. Lenstraâs (classical)
algorithm for integer programming as a subroutine
Quantum Algorithms for Abelian Difference Sets and Applications to Dihedral Hidden Subgroups
Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference set and present a general algorithm that can be used to tackle any hidden shift problem for any difference set in any abelian group. We discuss special cases of this framework which include a) Paley difference sets based on quadratic residues in finite fields which allow to recover the shifted Legendre function quantum algorithm, b) Hadamard difference sets which allow to recover the shifted bent function quantum algorithm, and c) Singer difference sets which allow us to define instances of the dihedral hidden subgroup problem which can be efficiently solved on a quantum computer
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