Binding Energies of Charged Metal Nanoparticle Configurations

The electrostatic interaction between metal spheres is an influential component in the assembly of many nanoscale materials in chemistry. Here we derive a method to calculate the energy and polarizations of metal spheres in arbitrary configurations to an arbitrary multipole order. This helps provide insight into the preferred configurations of charged metal particles and demonstrates the sensitivity of electrostatic interactions to configuration geometry.Comment: 27 pages, 10 figure

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 $C$, we have $\exists C \subset BP \cdot \oplus C$. 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

PACER: Peripheral Activity Completion Estimation and Recognition

Embedded peripheral devices such as memories, sensors and communications interfaces are used to perform a function external to a host microcontroller. The device manufacturer typically specifies worst-case current consumption and latency estimates for each of these peripheral actions. Peripheral Activity Completion, Estimation and Recognition (PACER) is introduced as a suite of algorithms that can be applied to detect completed peripheral operations in real-time. By detecting activity completion, PACER enables the host to exploit slack between the worst-case estimate and the actual response time. These methods were tested independently and in conjunction with IODVS on multiple common peripheral devices. For the peripheral devices under test, the test fixture confirmed decreases in energy expenditures of up to 80% and latency reductions of up to 67%.Comment: 8 pages, 12 figures, Presented at HIP3ES, 201

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

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

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

For Distinguishing Conjugate Hidden Subgroups, the Pretty Good Measurement is as Good as it Gets

Recently Bacon, Childs and van Dam showed that the ``pretty good measurement'' (PGM) is optimal for the Hidden Subgroup Problem on the dihedral group D_n in the case where the hidden subgroup is chosen uniformly from the n involutions. We show that, for any group and any subgroup H, the PGM is the optimal one-register experiment in the case where the hidden subgroup is a uniformly random conjugate of H. We go on to show that when H forms a Gel'fand pair with its parent group, the PGM is the optimal measurement for any number of registers. In both cases we bound the probability that the optimal measurement succeeds. This generalizes the case of the dihedral group, and includes a number of other examples of interest.Comment: proved additional conditions for optimality due to Yuen, Kennedy and Max; also bounded the probability of success in the one-register and multiregister case