8,199 research outputs found
A simple constant-probability RP reduction from NP to Parity P
The proof of Toda's celebrated theorem that the polynomial hierarchy is
contained in \P^{# P} relies on the fact that, under mild technical
conditions on the complexity class , we have . More concretely, there is a randomized reduction which transforms
nonempty sets and the empty set, respectively, into sets of odd or even size.
The customary method is to invoke Valiant's and Vazirani's randomized reduction
from NP to UP, followed by amplification of the resulting success probability
from 1/\poly(n) to a constant by combining the parities of \poly(n) trials.
Here we give a direct algebraic reduction which achieves constant success
probability without the need for amplification. Our reduction is very simple,
and its analysis relies on well-known properties of the Legendre symbol in
finite fields
Uncertainty Principles for Compact Groups
We establish an operator-theoretic uncertainty principle over arbitrary
compact groups, generalizing several previous results. As a consequence, we
show that if f is in L^2(G), then the product of the measures of the supports
of f and its Fourier transform ^f is at least 1; here, the dual measure is
given by the sum, over all irreducible representations V, of d_V rank(^f(V)).
For finite groups, our principle implies the following: if P and R are
projection operators on the group algebra C[G] such that P commutes with
projection onto each group element, and R commutes with left multiplication,
then the squared operator norm of PR is at most rank(P)rank(R)/|G|.Comment: 9 pages, to appear in Illinois J. Mat
How close can we come to a parity function when there isn't one?
Consider a group G such that there is no homomorphism f:G to {+1,-1}. In that
case, how close can we come to such a homomorphism? We show that if f has zero
expectation, then the probability that f(xy) = f(x) f(y), where x, y are chosen
uniformly and independently from G, is at most 1/2(1+1/sqrt{d}), where d is the
dimension of G's smallest nontrivial irreducible representation. For the
alternating group A_n, for instance, d=n-1. On the other hand, A_n contains a
subgroup isomorphic to S_{n-2}, whose parity function we can extend to obtain
an f for which this probability is 1/2(1+1/{n \choose 2}). Thus the extent to
which f can be "more homomorphic" than a random function from A_n to {+1,-1}
lies between O(n^{-1/2}) and Omega(n^{-2})
Circuit partitions and #P-complete products of inner products
We present a simple, natural #P-complete problem. Let G be a directed graph,
and let k be a positive integer. We define q(G;k) as follows. At each vertex v,
we place a k-dimensional complex vector x_v. We take the product, over all
edges (u,v), of the inner product . Finally, q(G;k) is the expectation
of this product, where the x_v are chosen uniformly and independently from all
vectors of norm 1 (or, alternately, from the Gaussian distribution). We show
that q(G;k) is proportional to G's cycle partition polynomial, and therefore
that it is #P-complete for any k>1
On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism
It is known that any quantum algorithm for Graph Isomorphism that works
within the framework of the hidden subgroup problem (HSP) must perform highly
entangled measurements across Omega(n log n) coset states. One of the only
known models for how such a measurement could be carried out efficiently is
Kuperberg's algorithm for the HSP in the dihedral group, in which quantum
states are adaptively combined and measured according to the decomposition of
tensor products into irreducible representations. This ``quantum sieve'' starts
with coset states, and works its way down towards representations whose
probabilities differ depending on, for example, whether the hidden subgroup is
trivial or nontrivial.
In this paper we give strong evidence that no such approach can succeed for
Graph Isomorphism. Specifically, we consider the natural reduction of Graph
Isomorphism to the HSP over the the wreath product S_n \wr Z_2. We show, modulo
a group-theoretic conjecture regarding the asymptotic characters of the
symmetric group, that no matter what rule we use to adaptively combine quantum
states, there is a constant b > 0 such that no algorithm in this family can
solve Graph Isomorphism in e^{b sqrt{n}} time. In particular, such algorithms
are essentially no better than the best known classical algorithms, whose
running time is e^{O(sqrt{n \log n})}
Regarding a Representation-Theoretic Conjecture of Wigderson
We show that there exists a family of irreducible representations R_i (of
finite groups G_i) such that, for any constant t, the average of R_i over t
uniformly random elements g_1, ..., g_t of G_i has operator norm 1 with
probability approaching 1 as i limits to infinity. This settles a conjecture of
Wigderson in the negative
Heat and Noise on Cubes and Spheres: The Sensitivity of Randomly Rotated Polynomial Threshold Functions
We establish a precise relationship between spherical harmonics and Fourier
basis functions over a hypercube randomly embedded in the sphere. In
particular, we give a bound on the expected Boolean noise sensitivity of a
randomly rotated function in terms of its "spherical sensitivity," which we
define according to its evolution under the spherical heat equation. As an
application, we prove an average case of the Gotsman-Linial conjecture,
bounding the sensitivity of polynomial threshold functions subjected to a
random rotation
Approximate Representations and Approximate Homomorphisms
Approximate algebraic structures play a defining role in arithmetic
combinatorics and have found remarkable applications to basic questions in
number theory and pseudorandomness. Here we study approximate representations
of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is
large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2$ is small, where x
and y are uniformly random elements of the group G and U_d denotes the unitary
group of degree d. We bound these quantities in terms of the ratio d / d_min
where d_min is the dimension of the smallest nontrivial representation of G. As
an application, we bound the extent to which a function f : G -> H can be an
approximate homomorphism where H is another finite group. We show that if H's
representations are significantly smaller than G's, no such f can be much more
homomorphic than a random function.
We interpret these results as showing that if G is quasirandom, that is, if
d_min is large, then G cannot be embedded in a small number of dimensions, or
in a less-quasirandom group, without significant distortion of G's
multiplicative structure. We also prove that our bounds are tight by showing
that minors of genuine representations and their polar decompositions are
essentially optimal approximate representations
Explicit Multiregister Measurements for Hidden Subgroup Problems
We present an explicit measurement in the Fourier basis that solves an
important case of the Hidden Subgroup Problem, including the case to which
Graph Isomorphism reduces. This entangled measurement uses
registers, and each of the subsets of the registers contributes some
information. While this does not, in general, yield an efficient algorithm, it
generalizes the relationship between Subset Sum and the HSP in the dihedral
group, and sheds some light on how quantum algorithms for Graph Isomorphism
might work
Classical and Quantum Polynomial Reconstruction via Legendre Symbol Evaluation
We consider the problem of recovering a hidden monic polynomial f(X) of
degree d > 0 over the finite field F of p elements given a black box which, for
any x in F, evaluates the quadratic character of f(x). We design a classical
algorithm of complexity O(d^2 p^{d + c}), for any c > 0, and also show that the
quantum query complexity of this problem is O(d). Some of our results extend
those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a
linear polynomial f(X) = X + s (with unknown s); some are new even in this
case.Comment: 14 page
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