A Simple and Fast Algorithm for Computing the NN-th Term of a Linearly Recurrent Sequence

We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and $\mathsf{M}(d)$ denotes the number of arithmetic operations for computing the product of two polynomials of degree $d$. The state-of-the-art algorithm, due to Charles Fiduccia (1985), has the same arithmetic complexity up to a constant factor. Our algorithm is simpler, faster and obtained by a totally different method. We also discuss several algorithmic applications, notably to polynomial modular exponentiation, powering of matrices and high-order lifting.Comment: 34 page

Towards an Adaptive Skeleton Framework for Performance Portability

The proliferation of widely available, but very different, parallel architectures makes the ability to deliver good parallel performance on a range of architectures, or performance portability, highly desirable. Irregularly-parallel problems, where the number and size of tasks is unpredictable, are particularly challenging and require dynamic coordination. The paper outlines a novel approach to delivering portable parallel performance for irregularly parallel programs. The approach combines declarative parallelism with JIT technology, dynamic scheduling, and dynamic transformation. We present the design of an adaptive skeleton library, with a task graph implementation, JIT trace costing, and adaptive transformations. We outline the architecture of the protoype adaptive skeleton execution framework in Pycket, describing tasks, serialisation, and the current scheduler.We report a preliminary evaluation of the prototype framework using 4 micro-benchmarks and a small case study on two NUMA servers (24 and 96 cores) and a small cluster (17 hosts, 272 cores). Key results include Pycket delivering good sequential performance e.g. almost as fast as C for some benchmarks; good absolute speedups on all architectures (up to 120 on 128 cores for sumEuler); and that the adaptive transformations do improve performance

Introduction to topological quantum computation with non-Abelian anyons

Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.Comment: 51 pages, 51 figure

Partition Information and its Transmission over Boolean Multi-Access Channels

In this paper, we propose a novel partition reservation system to study the partition information and its transmission over a noise-free Boolean multi-access channel. The objective of transmission is not message restoration, but to partition active users into distinct groups so that they can, subsequently, transmit their messages without collision. We first calculate (by mutual information) the amount of information needed for the partitioning without channel effects, and then propose two different coding schemes to obtain achievable transmission rates over the channel. The first one is the brute force method, where the codebook design is based on centralized source coding; the second method uses random coding where the codebook is generated randomly and optimal Bayesian decoding is employed to reconstruct the partition. Both methods shed light on the internal structure of the partition problem. A novel hypergraph formulation is proposed for the random coding scheme, which intuitively describes the information in terms of a strong coloring of a hypergraph induced by a sequence of channel operations and interactions between active users. An extended Fibonacci structure is found for a simple, but non-trivial, case with two active users. A comparison between these methods and group testing is conducted to demonstrate the uniqueness of our problem.Comment: Submitted to IEEE Transactions on Information Theory, major revisio

Studies of braided non-Abelian anyons using anyonic tensor networks

The content of this thesis can be broadly summarised into two categories: first, I constructed modified numerical algorithms based on tensor networks to simulate systems of anyons in low dimensions, and second, I used those methods to study the topological phases the anyons form when they braid around one another. In the first phase of my thesis, I extended the anyonic tensor network algorithms, by incorporating U(1) symmetry to give a modified ansatz, Anyon-U(1) tensor networks, which are capable of simulating anyonic systems at any rational filling fraction. In the second phase, I used the numerical methods to study some models of non-Abelian anyons that naturally allows for exchange of anyons. I proposed a lattice model of anyons, which I dubbed anyonic Hubbard model, which is a pair of coupled chains of anyons (or simply called anyonic ladder). Each site of the ladder can either host a single anyonic charge, or it can be empty. The anyons are able to move around, interact with one another, and exchange positions with other anyons, when vacancies exist. Exchange of anyons is a non-trivial process which may influence the formation of different kinds of new phases of matter. I studied this model using the two prominent species of anyons: Fibonacci and Ising anyons, and made a number of interesting discoveries about their phase diagrams. I identified new phases of matter arising from both the interaction between these anyons and their exchange braid statistics.Comment: 150 pages, PhD thesis, Macquarie University, Sydney. Chapter 6 of this thesis titled "Phase transitions in braided non-Abelian anyonic system" contains results which are yet to be finalised and publishe

Algorithms for computing Fibonacci numbers quickly

A study of the running time of several known algorithms and several new algorithms to compute the n[superscript th] element of the Fibonacci sequence is presented. Since the size of the n[superscript th] Fibonacci number grows exponentially with n, the number of bit operations, instead of the number of integer operations, was used as the unit of time. The number of bit operations used to compute f[subscript n] is reduced to less than 1/2 of the number of bit operations used to multiply two n bit numbers. The algorithms were programmed in Ibuki Common Lisp and timing runs were made on a Sequent Balance 21000. Multiplication was implemented using the standard n² algorithm. Times for the various algorithms are reported as various constants times n². An algorithm based on generating factors of Fibonacci numbers had the smallest constant. The Fibonacci sequence, arranged in various ways, is searched for redundant information that could be eliminated to reduce the number of operations. Cycles in the b[superscript th] bit of f[subscript n] were discovered but are not yet completely understood