46,246 research outputs found
Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin
In this work we present a determinant expression for the domain-wall boundary
condition partition function of rational (XXX) Richardson-Gaudin models which,
in addition to spins , contains one arbitrarily large spin
.
The proposed determinant representation is written in terms of a set of
variables which, from previous work, are known to define eigenstates of the
quantum integrable models belonging to this class as solutions to quadratic
Bethe equations. Such a determinant can be useful numerically since systems of
quadratic equations are much simpler to solve than the usual highly non-linear
Bethe equations. It can therefore offer significant gains in stability and
computation speed.Comment: 17 pages, 0 figure
A Dichotomy Theorem for Homomorphism Polynomials
In the present paper we show a dichotomy theorem for the complexity of
polynomial evaluation. We associate to each graph H a polynomial that encodes
all graphs of a fixed size homomorphic to H. We show that this family is
computable by arithmetic circuits in constant depth if H has a loop or no edge
and that it is hard otherwise (i.e., complete for VNP, the arithmetic class
related to #P). We also demonstrate the hardness over the rational field of cut
eliminator, a polynomial defined by B\"urgisser which is known to be neither VP
nor VNP-complete in the field of two elements, if VP is not equal to VNP (VP is
the class of polynomials computable by arithmetic circuit of polynomial size)
Circuit complexity, proof complexity, and polynomial identity testing
We introduce a new algebraic proof system, which has tight connections to
(algebraic) circuit complexity. In particular, we show that any
super-polynomial lower bound on any Boolean tautology in our proof system
implies that the permanent does not have polynomial-size algebraic circuits
(VNP is not equal to VP). As a corollary to the proof, we also show that
super-polynomial lower bounds on the number of lines in Polynomial Calculus
proofs (as opposed to the usual measure of number of monomials) imply the
Permanent versus Determinant Conjecture. Note that, prior to our work, there
was no proof system for which lower bounds on an arbitrary tautology implied
any computational lower bound.
Our proof system helps clarify the relationships between previous algebraic
proof systems, and begins to shed light on why proof complexity lower bounds
for various proof systems have been so much harder than lower bounds on the
corresponding circuit classes. In doing so, we highlight the importance of
polynomial identity testing (PIT) for understanding proof complexity.
More specifically, we introduce certain propositional axioms satisfied by any
Boolean circuit computing PIT. We use these PIT axioms to shed light on
AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no
satisfactory explanation as to their apparent difficulty. We show that either:
a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not
have polynomial-size circuits of depth d - a notoriously open question for d at
least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or
b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we
have a lower bound on AC^0[p]-Frege.
Using the algebraic structure of our proof system, we propose a novel way to
extend techniques from algebraic circuit complexity to prove lower bounds in
proof complexity
Monotone Projection Lower Bounds from Extended Formulation Lower Bounds
In this short note, we reduce lower bounds on monotone projections of
polynomials to lower bounds on extended formulations of polytopes. Applying our
reduction to the seminal extended formulation lower bounds of Fiorini, Massar,
Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014;
J. ACM, 2017), we obtain the following interesting consequences.
1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size
projection of the permanent; this both rules out a natural attempt at a
monotone lower bound on the Boolean permanent, and shows that the permanent is
not complete for non-negative polynomials in VNP under monotone
p-projections.
2. The cut polynomials and the perfect matching polynomial (or "unsigned
Pfaffian") are not monotone p-projections of the permanent. The latter, over
the Boolean and-or semi-ring, rules out monotone reductions in one of the
natural approaches to reducing perfect matchings in general graphs to perfect
matchings in bipartite graphs.
As the permanent is universal for monotone formulas, these results also imply
exponential lower bounds on the monotone formula size and monotone circuit size
of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18;
Received: November 10, 2015, Revised: July 27, 2016, Published: December 22,
201
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The use of proofs in diversity arguments
The limits to the reliability that can be claimed for a design-diverse fault-tolerant system are mainly determined by the dependence that must be expected in the failure behaviours of the different versions: claims for independence between version failure processes are not believable. In this note we examine a different approach, in which a simple secondary system is used as a back-up to a more complex primary. The secondary system is sufficiently simple that claims for its perfection (with respect to design faults) are possible, but there is not complete certainty about such perfection. It is shown that assessment of the reliability of the overall fault-tolerant system in this case may take advantage of claims for independence that are more plausible than those involved in design diversity
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
Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials
This paper is our third step towards developing a theory of testing monomials
in multivariate polynomials and concentrates on two problems: (1) How to
compute the coefficients of multilinear monomials; and (2) how to find a
maximum multilinear monomial when the input is a polynomial. We
first prove that the first problem is \#P-hard and then devise a
upper bound for this problem for any polynomial represented by an arithmetic
circuit of size . Later, this upper bound is improved to for
polynomials. We then design fully polynomial-time randomized
approximation schemes for this problem for polynomials. On the
negative side, we prove that, even for polynomials with terms of
degree , the first problem cannot be approximated at all for any
approximation factor , nor {\em "weakly approximated"} in a much relaxed
setting, unless P=NP. For the second problem, we first give a polynomial time
-approximation algorithm for polynomials with terms of
degrees no more a constant . On the inapproximability side, we
give a lower bound, for any on the
approximation factor for polynomials. When terms in these
polynomials are constrained to degrees , we prove a lower
bound, assuming ; and a higher lower bound, assuming the
Unique Games Conjecture
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