arXiv subjects

Robert Beals

Publications and source records attributed to Robert Beals.

3 recordsLinked to original sources

arXiv2025

Finding Blocks of Imprimitivity When There is a Small-Base Action on Blocks

Given a transitive permutation group G of degree n , we seek to determine whether or not G is primitive, and to find a system of blocks of imprimitivity in the case that G is imprimitive. An algorithm of Atkinson solves this problem in time O(n^2) , while a previous algorithm of ours runs in time O(n log^3|G|) , which is advantageous in the small-base case. A simpler algorithm of Schonert and Seress has the same asymptotic O(n log^3|G|) performance. In this paper we extend the small-base algorithms to work with imprimitive groups G which, while not small-base in the action on n points, possess a small-base action on a block system. Using a recent upper bound by Kelsey and Roney-Dougal on the size of a nonredundant base of a primitive group of a given degree, we obtain a time of O(n log^5 n) except in the case that G has a primitive action (either on the n points or on a block system) for which the socle is isomorphic to Alt(m)^d for some m at least 5 and d at least 1. A key component of our improvement is a new variant of sifting, which is a workhorse of permutation group algorithms.

math.GR↗

arXiv2012

Efficient Distributed Quantum Computing

We provide algorithms for efficiently addressing quantum memory in parallel. These imply that the standard circuit model can be simulated with low overhead by the more realistic model of a distributed quantum computer. As a result, the circuit model can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit. In addition, we apply our results to existing memory intensive quantum algorithms. We present a parallel quantum search algorithm and improve the time-space trade-off for the Element Distinctness and Collision problems.

quant-ph↗

arXiv1998

Quantum Lower Bounds by Polynomials

We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

quant-ph↗