4,925 research outputs found
Pseudorandomness via the discrete Fourier transform
We present a new approach to constructing unconditional pseudorandom
generators against classes of functions that involve computing a linear
function of the inputs. We give an explicit construction of a pseudorandom
generator that fools the discrete Fourier transforms of linear functions with
seed-length that is nearly logarithmic (up to polyloglog factors) in the input
size and the desired error parameter. Our result gives a single pseudorandom
generator that fools several important classes of tests computable in logspace
that have been considered in the literature, including halfspaces (over general
domains), modular tests and combinatorial shapes. For all these classes, our
generator is the first that achieves near logarithmic seed-length in both the
input length and the error parameter. Getting such a seed-length is a natural
challenge in its own right, which needs to be overcome in order to derandomize
RL - a central question in complexity theory.
Our construction combines ideas from a large body of prior work, ranging from
a classical construction of [NN93] to the recent gradually increasing
independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some
novel analytic machinery which might find other applications
Finding the Minimum-Weight k-Path
Given a weighted -vertex graph with integer edge-weights taken from a
range , we show that the minimum-weight simple path visiting
vertices can be found in time \tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k
M). If the weights are reals in , we provide a
-approximation which has a running time of \tilde{O}(2^k
\poly(k) n^\omega(\log\log M + 1/\varepsilon)). For the more general problem
of -tree, in which we wish to find a minimum-weight copy of a -node tree
in a given weighted graph , under the same restrictions on edge weights
respectively, we give an exact solution of running time \tilde{O}(2^k \poly(k)
M n^3) and a -approximate solution of running time
\tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon)). All of the above
algorithms are randomized with a polynomially-small error probability.Comment: To appear at WADS 201
An Atypical Survey of Typical-Case Heuristic Algorithms
Heuristic approaches often do so well that they seem to pretty much always
give the right answer. How close can heuristic algorithms get to always giving
the right answer, without inducing seismic complexity-theoretic consequences?
This article first discusses how a series of results by Berman, Buhrman,
Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the
early 1970s through the early 1990s, explicitly or implicitly limited how well
heuristic algorithms can do on NP-hard problems. In particular, many desirable
levels of heuristic success cannot be obtained unless severe, highly unlikely
complexity class collapses occur. Second, we survey work initiated by Goldreich
and Wigderson, who showed how under plausible assumptions deterministic
heuristics for randomized computation can achieve a very high frequency of
correctness. Finally, we consider formal ways in which theory can help explain
the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012
issue of SIGACT New
Quantum Algorithms for Fermionic Simulations
We investigate the simulation of fermionic systems on a quantum computer. We
show in detail how quantum computers avoid the dynamical sign problem present
in classical simulations of these systems, therefore reducing a problem
believed to be of exponential complexity into one of polynomial complexity. The
key to our demonstration is the spin-particle connection (or generalized
Jordan-Wigner transformation) that allows exact algebraic invertible mappings
of operators with different statistical properties. We give an explicit
implementation of a simple problem using a quantum computer based on standard
qubits.Comment: 38 pages, 2 psfigur
Efficient Circuits for Quantum Walks
We present an efficient general method for realizing a quantum walk operator
corresponding to an arbitrary sparse classical random walk. Our approach is
based on Grover and Rudolph's method for preparing coherent versions of
efficiently integrable probability distributions. This method is intended for
use in quantum walk algorithms with polynomial speedups, whose complexity is
usually measured in terms of how many times we have to apply a step of a
quantum walk, compared to the number of necessary classical Markov chain steps.
We consider a finer notion of complexity including the number of elementary
gates it takes to implement each step of the quantum walk with some desired
accuracy. The difference in complexity for various implementation approaches is
that our method scales linearly in the sparsity parameter and
poly-logarithmically with the inverse of the desired precision. The best
previously known general methods either scale quadratically in the sparsity
parameter, or polynomially in the inverse precision. Our approach is especially
relevant for implementing quantum walks corresponding to classical random walks
like those used in the classical algorithms for approximating permanents and
sampling from binary contingency tables. In those algorithms, the sparsity
parameter grows with the problem size, while maintaining high precision is
required.Comment: Modified abstract, clarified conclusion, added application section in
appendix and updated reference
Local tests of global entanglement and a counterexample to the generalized area law
We introduce a technique for applying quantum expanders in a distributed
fashion, and use it to solve two basic questions: testing whether a bipartite
quantum state shared by two parties is the maximally entangled state and
disproving a generalized area law. In the process these two questions which
appear completely unrelated turn out to be two sides of the same coin.
Strikingly in both cases a constant amount of resources are used to verify a
global property.Comment: 21 pages, to appear FOCS 201
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