Quantum geometry and quantum algorithms

Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the colored Jones polynomial. The construction is based on the complete solution of Chern-Simons topological quantum field theory and its connection to Wess-Zumino-Witten conformal field theory. The colored Jones polynomial is expressed as the expectation value of the evolution of the q-deformed spin-network quantum automaton. A quantum circuit is constructed capable of simulating the automaton and hence of computing such expectation value. The latter is efficiently approximated using a standard sampling procedure in quantum computation.Comment: Submitted to J. Phys. A: Math-Gen, for the special issue ``The Quantum Universe'' in honor of G. C. Ghirard

Quantum Computing and Quantum Algorithms

The field of quantum computing and quantum algorithms is studied from the ground up. Qubits and their quantum-mechanical properties are discussed, followed by how they are transformed by quantum gates. From there, quantum algorithms are explored as well as the use of high-level quantum programming languages to implement them. One quantum algorithm is selected to be implemented in the Qiskit quantum programming language. The validity and success of the resulting computation is proven with matrix multiplication of the qubits and quantum gates involved

Quantum games and quantum algorithms

A quantum algorithm for an oracle problem can be understood as a quantum strategy for a player in a two-player zero-sum game in which the other player is constrained to play classically. I formalize this correspondence and give examples of games (and hence oracle problems) for which the quantum player can do better than would be possible classically. The most remarkable example is the Bernstein-Vazirani quantum search algorithm which I show creates no entanglement at any timestep.Comment: 10 pages, plain TeX; to appear in the AMS Contemporary Mathematics volume: Quantum Computation and Quantum Information Science; revised remarks about other quantum games formalisms; for related work see http://math.ucsd.edu/~dmeyer/research.htm

Quantum Algorithms Revisited

Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multi-particle interference. We use this approach to review (and improve) some of the existing quantum algorithms and to show how they are related to different instances of quantum phase estimation. We provide an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision.Comment: 18 pages, LaTeX, 7 figures. Submitted to Proc. Roy. Soc. Lond.

On Quantum Algorithms

Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. In effect, they follow the same logical paradigm as (multi-particle) interferometers. We show how most known quantum algorithms, including quantum algorithms for factorising and counting, may be cast in this manner. Quantum searching is described as inducing a desired relative phase between two eigenvectors to yield constructive interference on the sought elements and destructive interference on the remaining terms.Comment: 15 pages, 8 figure

Quantum Analog-Digital Conversion

Many quantum algorithms, such as Harrow-Hassidim-Lloyd (HHL) algorithm, depend on oracles that efficiently encode classical data into a quantum state. The encoding of the data can be categorized into two types; analog-encoding where the data are stored as amplitudes of a state, and digital-encoding where they are stored as qubit-strings. The former has been utilized to process classical data in an exponentially large space of a quantum system, where as the latter is required to perform arithmetics on a quantum computer. Quantum algorithms like HHL achieve quantum speedups with a sophisticated use of these two encodings. In this work, we present algorithms that converts these two encodings to one another. While quantum digital-to-analog conversions have implicitly been used in existing quantum algorithms, we reformulate it and give a generalized protocol that works probabilistically. On the other hand, we propose an deterministic algorithm that performs a quantum analog-to-digital conversion. These algorithms can be utilized to realize high-level quantum algorithms such as a nonlinear transformation of amplitude of a quantum state. As an example, we construct a "quantum amplitude perceptron", a quantum version of neural network, and hence has a possible application in the area of quantum machine learning.Comment: 7 page