Some formal results for the valence bond basis

In a system with an even number of SU(2) spins, there is an overcomplete set of states--consisting of all possible pairings of the spins into valence bonds--that spans the S=0 Hilbert subspace. Operator expectation values in this basis are related to the properties of the closed loops that are formed by the overlap of valence bond states. We construct a generating function for spin correlation functions of arbitrary order and show that all nonvanishing contributions arise from configurations that are topologically irreducible. We derive explicit formulas for the correlation functions at second, fourth, and sixth order. We then extend the valence bond basis to include triplet bonds and discuss how to compute properties that are related to operators acting outside the singlet sector. These results are relevant to analytical calculations and to numerical valence bond simulations using quantum Monte Carlo, variational wavefunctions, or exact diagonalization.Comment: 22 pages, 14 figure

New Results on Quantum Property Testing

We present several new examples of speed-ups obtainable by quantum algorithms in the context of property testing. First, motivated by sampling algorithms, we consider probability distributions given in the form of an oracle $f:[n]\to[m]$. Here the probability \PP_f(j) of an outcome $j\in[m]$ is the fraction of its domain that $f$ maps to $j$. We give quantum algorithms for testing whether two such distributions are identical or $\epsilon$-far in $L_1$-norm. Recently, Bravyi, Hassidim, and Harrow \cite{BHH10} showed that if \PP_f and \PP_g are both unknown (i.e., given by oracles $f$ and $g$), then this testing can be done in roughly $\sqrt{m}$ quantum queries to the functions. We consider the case where the second distribution is known, and show that testing can be done with roughly $m^{1/3}$ quantum queries, which we prove to be essentially optimal. In contrast, it is known that classical testing algorithms need about $m^{2/3}$ queries in the unknown-unknown case and about $\sqrt{m}$ queries in the known-unknown case. Based on this result, we also reduce the query complexity of graph isomorphism testers with quantum oracle access. While those examples provide polynomial quantum speed-ups, our third example gives a much larger improvement (constant quantum queries vs polynomial classical queries) for the problem of testing periodicity, based on Shor's algorithm and a modification of a classical lower bound by Lachish and Newman \cite{lachish&newman:periodicity}. This provides an alternative to a recent constant-vs-polynomial speed-up due to Aaronson \cite{aaronson:bqpph}.Comment: 2nd version: updated some references, in particular to Aaronson's Fourier checking proble

Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of $\beta=2$ symmetry class.Comment: 34page