2,370 research outputs found
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
Grothendieck inequalities for semidefinite programs with rank constraint
Grothendieck inequalities are fundamental inequalities which are frequently
used in many areas of mathematics and computer science. They can be interpreted
as upper bounds for the integrality gap between two optimization problems: a
difficult semidefinite program with rank-1 constraint and its easy semidefinite
relaxation where the rank constrained is dropped. For instance, the integrality
gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen
as a Grothendieck inequality. In this paper we consider Grothendieck
inequalities for ranks greater than 1 and we give two applications:
approximating ground states in the n-vector model in statistical mechanics and
XOR games in quantum information theory.Comment: 22 page
Simulating Quantum Correlations with Finite Communication
Assume Alice and Bob share some bipartite -dimensional quantum state. A
well-known result in quantum mechanics says that by performing two-outcome
measurements, Alice and Bob can produce correlations that cannot be obtained
locally, i.e., with shared randomness alone. We show that by using only two
bits of communication, Alice and Bob can classically simulate any such
correlations. All previous protocols for exact simulation required the
communication to grow to infinity with the dimension . Our protocol and
analysis are based on a power series method, resembling Krivine's bound on
Grothendieck's constant, and on the computation of volumes of spherical
tetrahedra.Comment: 19 pages, 3 figures, preliminary version in IEEE FOCS 2007; to appear
in SICOM
Adiabatic Quantum State Generation and Statistical Zero Knowledge
The design of new quantum algorithms has proven to be an extremely difficult
task. This paper considers a different approach to the problem, by studying the
problem of 'quantum state generation'. This approach provides intriguing links
between many different areas: quantum computation, adiabatic evolution,
analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing
Markov chains, the complexity class statistical zero knowledge, quantum random
walks, and more.
We first show that many natural candidates for quantum algorithms can be cast
as a state generation problem. We define a paradigm for state generation,
called 'adiabatic state generation' and develop tools for adiabatic state
generation which include methods for implementing very general Hamiltonians and
ways to guarantee non negligible spectral gaps. We use our tools to prove that
adiabatic state generation is equivalent to state generation in the standard
quantum computing model, and finally we show how to apply our techniques to
generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure
Hardy is (almost) everywhere: nonlocality without inequalities for almost all entangled multipartite states
We show that all -qubit entangled states, with the exception of tensor
products of single-qubit and bipartite maximally-entangled states, admit
Hardy-type proofs of non-locality without inequalities or probabilities. More
precisely, we show that for all such states, there are local, one-qubit
observables such that the resulting probability tables are logically contextual
in the sense of Abramsky and Brandenburger, this being the general form of the
Hardy-type property. Moreover, our proof is constructive: given a state, we
show how to produce the witnessing local observables. In fact, we give an
algorithm to do this. Although the algorithm is reasonably straightforward, its
proof of correctness is non-trivial. A further striking feature is that we show
that local observables suffice to witness the logical contextuality of
any -qubit state: two each for two for the parties, and one each for the
remaining parties.Comment: 23 pages. Submitted for publicatio
- ā¦