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

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

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

Uncertainty Principles for Compact Groups

We establish an operator-theoretic uncertainty principle over arbitrary compact groups, generalizing several previous results. As a consequence, we show that if f is in L^2(G), then the product of the measures of the supports of f and its Fourier transform ^f is at least 1; here, the dual measure is given by the sum, over all irreducible representations V, of d_V rank(^f(V)). For finite groups, our principle implies the following: if P and R are projection operators on the group algebra C[G] such that P commutes with projection onto each group element, and R commutes with left multiplication, then the squared operator norm of PR is at most rank(P)rank(R)/|G|.Comment: 9 pages, to appear in Illinois J. Mat

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

Explicit Multiregister Measurements for Hidden Subgroup Problems

We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses $k=\log_2 |G|$ registers, and each of the $2^k$ subsets of the registers contributes some information. While this does not, in general, yield an efficient algorithm, it generalizes the relationship between Subset Sum and the HSP in the dihedral group, and sheds some light on how quantum algorithms for Graph Isomorphism might work

Classical and Quantum Polynomial Reconstruction via Legendre Symbol Evaluation

We consider the problem of recovering a hidden monic polynomial f(X) of degree d > 0 over the finite field F of p elements given a black box which, for any x in F, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d^2 p^{d + c}), for any c > 0, and also show that the quantum query complexity of this problem is O(d). Some of our results extend those of Wim van Dam, Sean Hallgren and Lawrence Ip obtained in the case of a linear polynomial f(X) = X + s (with unknown s); some are new even in this case.Comment: 14 page