A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions

The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov recently characterized the minimal degree deg_{\eps}(f) among all polynomials (over the reals) that approximate a symmetric function f:{0,1}^n-->{0,1} up to worst-case error \eps: deg_{\eps}(f) = ~\Theta(deg_{1/3}(f) + \sqrt{n\log(1/\eps)}). In this note we show how a tighter version (without the log-factors hidden in the ~\Theta-notation), can be derived quite easily using the close connection between polynomials and quantum algorithms.Comment: 7 pages LaTeX. 2nd version: corrected a few small inaccuracie

Understanding the Complexity of Lifted Inference and Asymmetric Weighted Model Counting

In this paper we study lifted inference for the Weighted First-Order Model Counting problem (WFOMC), which counts the assignments that satisfy a given sentence in first-order logic (FOL); it has applications in Statistical Relational Learning (SRL) and Probabilistic Databases (PDB). We present several results. First, we describe a lifted inference algorithm that generalizes prior approaches in SRL and PDB. Second, we provide a novel dichotomy result for a non-trivial fragment of FO CNF sentences, showing that for each sentence the WFOMC problem is either in PTIME or #P-hard in the size of the input domain; we prove that, in the first case our algorithm solves the WFOMC problem in PTIME, and in the second case it fails. Third, we present several properties of the algorithm. Finally, we discuss limitations of lifted inference for symmetric probabilistic databases (where the weights of ground literals depend only on the relation name, and not on the constants of the domain), and prove the impossibility of a dichotomy result for the complexity of probabilistic inference for the entire language FOL

Weakly Non-Equilibrium Properties of Symmetric Inclusion Process with Open Boundaries

We study close to equilibrium properties of the one-dimensional Symmetric Inclusion Process (SIP) by coupling it to two particle-reservoirs at the two boundaries with slightly different chemical potentials. The boundaries introduce irreversibility and induce a weak particle current in the system. We calculate the McLennan ensemble for SIP, which corresponds to the entropy production and the first order non-equilibrium correction for the stationary state. We find that the first order correction is a product measure, and is consistent with the local equilibrium measure corresponding to the steady state density profile.Comment: 17 pages, revise

Stanley character polynomials

Stanley considered suitably normalized characters of the symmetric groups on Young diagrams having a special geometric form, namely multirectangular Young diagrams. He proved that the character is a polynomial in the lengths of the sides of the rectangles forming the Young diagram and he conjectured an explicit form of this polynomial. This Stanley character polynomial and this way of parametrizing the set of Young diagrams turned out to be a powerful tool for several problems of the dual combinatorics of the characters of the symmetric groups and asymptotic representation theory, in particular to Kerov polynomials.Comment: Dedicated to Richard P. Stanley on the occasion of his seventieth birthda

You can hide but you have to run: direct detection with vector mediators

We study direct detection in simplified models of Dark Matter (DM) in which interactions with Standard Model (SM) fermions are mediated by a heavy vector boson. We consider fully general, gauge-invariant couplings between the SM, the mediator and both scalar and fermion DM. We account for the evolution of the couplings between the energy scale of the mediator mass and the nuclear energy scale. This running arises from virtual effects of SM particles and its inclusion is not optional. We compare bounds on the mediator mass from direct detection experiments with and without accounting for the running. In some cases the inclusion of these effects changes the bounds by several orders of magnitude, as a consequence of operator mixing which generates new interactions at low energy. We also highlight the importance of these effects when translating LHC limits on the mediator mass into bounds on the direct detection cross section. For an axial-vector mediator, the running can alter the derived bounds on the spin-dependent DM-nucleon cross section by a factor of two or more. Finally, we provide tools to facilitate the inclusion of these effects in future studies: general approximate expressions for the low energy couplings and a public code runDM to evolve the couplings between arbitrary energy scales.Comment: 26 pages + appendices, 9 + 2 figures. The runDM code is available at https://github.com/bradkav/runDM/. v2: references added, version published in JHE

Spectral Norm of Symmetric Functions

The spectral norm of a Boolean function $f:\{0,1\}^n \to \{-1,1\}$ is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as $r(f)\log(n/r(f))$ where $r(f) = \max\{r_0,r_1\}$, and $r_0$ and $r_1$ are the smallest integers less than $n/2$ such that $f(x)$ or $f(x) \cdot parity(x)$ is constant for all $x$ with $\sum x_i \in [r_0, n-r_1]$. We mention some applications to the decision tree and communication complexity of symmetric functions

Diffusion-limited annihilation in inhomogeneous environments

We study diffusion-limited (on-site) pair annihilation $A+A\to 0$ and (on-site) fusion $A+A\to A$ which we show to be equivalent for arbitrary space-dependent diffusion and reaction rates. For one-dimensional lattices with nearest neighbour hopping we find that in the limit of infinite reaction rate the time-dependent $n$-point density correlations for many-particle initial states are determined by the correlation functions of a dual diffusion-limited annihilation process with at most $2n$ particles initially. By reformulating general properties of annihilating random walks in one dimension in terms of fermionic anticommutation relations we derive an exact representation for these correlation functions in terms of conditional probabilities for a single particle performing a random walk with dual hopping rates. This allows for the exact and explicit calculation of a wide range of universal and non-universal types of behaviour for the decay of the density and density correlations.Comment: 27 pages, Latex, to appear in Z. Phys.

Efficient Quantum Tensor Product Expanders and k-designs

Quantum expanders are a quantum analogue of expanders, and k-tensor product expanders are a generalisation to graphs that randomise k correlated walkers. Here we give an efficient construction of constant-degree, constant-gap quantum k-tensor product expanders. The key ingredients are an efficient classical tensor product expander and the quantum Fourier transform. Our construction works whenever k=O(n/log n), where n is the number of qubits. An immediate corollary of this result is an efficient construction of an approximate unitary k-design, which is a quantum analogue of an approximate k-wise independent function, on n qubits for any k=O(n/log n). Previously, no efficient constructions were known for k>2, while state designs, of which unitary designs are a generalisation, were constructed efficiently in [Ambainis, Emerson 2007].Comment: 16 pages, typo in references fixe