36 research outputs found
The Power of Quantum Fourier Sampling
A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and
Shepherd, shows that quantum computers can efficiently sample from probability
distributions that cannot be exactly sampled efficiently on a classical
computer, unless the PH collapses. Aaronson and Arkhipov take this further by
considering a distribution that can be sampled efficiently by linear optical
quantum computation, that under two feasible conjectures, cannot even be
approximately sampled classically within bounded total variation distance,
unless the PH collapses.
In this work we use Quantum Fourier Sampling to construct a class of
distributions that can be sampled by a quantum computer. We then argue that
these distributions cannot be approximately sampled classically, unless the PH
collapses, under variants of the Aaronson and Arkhipov conjectures.
In particular, we show a general class of quantumly sampleable distributions
each of which is based on an "Efficiently Specifiable" polynomial, for which a
classical approximate sampler implies an average-case approximation. This class
of polynomials contains the Permanent but also includes, for example, the
Hamiltonian Cycle polynomial, and many other familiar #P-hard polynomials.
Although our construction, unlike that proposed by Aaronson and Arkhipov,
likely requires a universal quantum computer, we are able to use this
additional power to weaken the conjectures needed to prove approximate sampling
hardness results
Fast generalized DFTs for all finite groups
For any finite group , we give an arithmetic algorithm to compute
generalized Discrete Fourier Transforms (DFTs) with respect to , using
operations, for any . Here,
is the exponent of matrix multiplication
A new algorithm for fast generalized DFTs
We give an new arithmetic algorithm to compute the generalized Discrete
Fourier Transform (DFT) over finite groups . The new algorithm uses
operations to compute the generalized DFT over
finite groups of Lie type, including the linear, orthogonal, and symplectic
families and their variants, as well as all finite simple groups of Lie type.
Here is the exponent of matrix multiplication, so the exponent
is optimal if . Previously, "exponent one" algorithms
were known for supersolvable groups and the symmetric and alternating groups.
No exponent one algorithms were known (even under the assumption )
for families of linear groups of fixed dimension, and indeed the previous
best-known algorithm for had exponent despite being the focus
of significant effort. We unconditionally achieve exponent at most for
this group, and exponent one if . Our algorithm also yields an
improved exponent for computing the generalized DFT over general finite groups
, which beats the longstanding previous best upper bound, for any .
In particular, assuming , we achieve exponent , while the
previous best was
Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields
The fastest known algorithm for factoring univariate polynomials over finite
fields is the Kedlaya-Umans (fast modular composition) implementation of the
Kaltofen-Shoup algorithm. It is randomized and takes time to factor polynomials of degree over the finite field
with elements. A significant open problem is if the
exponent can be improved. We study a collection of algebraic problems and
establish a web of reductions between them. A consequence is that an algorithm
for any one of these problems with exponent better than would yield an
algorithm for polynomial factorization with exponent better than
Pseudorandomness of the Sticky Random Walk
We extend the pseudorandomness of random walks on expander graphs using the
sticky random walk. Building on prior works, it was recently shown that
expander random walks can fool all symmetric functions in total variation
distance (TVD) upto an error, where
is the second largest eigenvalue of the expander, is the size of
the arbitrary alphabet used to label the vertices, and , where is the fraction of vertices labeled in the graph.
Golowich and Vadhan conjecture that the dependency on the term is not tight. In this paper, we resolve the conjecture in the
affirmative for a family of expanders. We present a generalization of the
sticky random walk for which Golowich and Vadhan predict a TVD upper bound of
using a Fourier-analytic approach. For this family of
graphs, we use a combinatorial approach involving the Krawtchouk functions to
derive a strengthened TVD of . Furthermore, we present
equivalencies between the generalized sticky random walk, and, using
linear-algebraic techniques, show that the generalized sticky random walk
parameterizes an infinite family of expander graphs.Comment: 21 pages, 2 figure
Special Section On Foundations of Computer Science
This special section comprises eight fully refereed papers whose extended abstracts were presented at the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007) in Providence, Rhode Island, October 21–23, 2007. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2007 proceedings
Which groups are amenable to proving exponent two for matrix multiplication?
The Cohn-Umans group-theoretic approach to matrix multiplication suggests
embedding matrix multiplication into group algebra multiplication, and bounding
in terms of the representation theory of the host group. This
framework is general enough to capture the best known upper bounds on
and is conjectured to be powerful enough to prove , although
finding a suitable group and constructing such an embedding has remained
elusive. Recently it was shown, by a generalization of the proof of the Cap Set
Conjecture, that abelian groups of bounded exponent cannot prove
in this framework, which ruled out a family of potential constructions in the
literature.
In this paper we study nonabelian groups as potential hosts for an embedding.
We prove two main results:
(1) We show that a large class of nonabelian groups---nilpotent groups of
bounded exponent satisfying a mild additional condition---cannot prove in this framework. We do this by showing that the shrinkage rate of powers
of the augmentation ideal is similar to the shrinkage rate of the number of
functions over that are degree polynomials;
our proof technique can be seen as a generalization of the polynomial method
used to resolve the Cap Set Conjecture.
(2) We show that symmetric groups cannot prove nontrivial bounds on
when the embedding is via three Young subgroups---subgroups of the
form ---which is a
natural strategy that includes all known constructions in .
By developing techniques for negative results in this paper, we hope to
catalyze a fruitful interplay between the search for constructions proving
bounds on and methods for ruling them out.Comment: 23 pages, 1 figur
On Multidimensional and Monotone k-SUM
The well-known k-SUM conjecture is that integer k-SUM requires time Omega(n^{ceil{k/2}-o(1)}). Recent work has studied multidimensional k-SUM in F_p^d, where the best known algorithm takes time tilde O(n^{ceil{k/2}}). Bhattacharyya et al. [ICS 2011] proved a min(2^{Omega(d)},n^{Omega(k)}) lower bound for k-SUM in F_p^d under the Exponential Time Hypothesis. We give a more refined lower bound under the standard k-SUM conjecture: for sufficiently large p, k-SUM in F_p^d requires time Omega(n^{k/2-o(1)}) if k is even, and Omega(n^{ceil{k/2}-2k(log k)/(log p)-o(1)}) if k is odd.
For a special case of the multidimensional problem, bounded monotone d-dimensional 3SUM, Chan and Lewenstein [STOC 2015] gave a surprising tilde O(n^{2-2/(d+13)}) algorithm using additive combinatorics. We show this algorithm is essentially optimal. To be more precise, bounded monotone d-dimensional 3SUM requires time Omega(n^{2-frac{4}{d}-o(1)}) under the standard 3SUM conjecture, and time Omega(n^{2-frac{2}{d}-o(1)}) under the so-called strong 3SUM conjecture. Thus, even though one might hope to further exploit the structural advantage of monotonicity, no substantial improvements beyond those obtained by Chan and Lewenstein are possible for bounded monotone d-dimensional 3SUM