Quantum Multiplexers, Parrondo Games, and Proper Quantization

A quantum logic gate of particular interest to both electrical engineers and game theorists is the quantum multiplexer. This shared interest is due to the facts that an arbitrary quantum logic gate may be expressed, up to arbitrary accuracy, via a circuit consisting entirely of variations of the quantum multiplexer, and that certain one player games, the history dependent Parrondo games, can be quantized as games via a particular variation of the quantum multiplexer. However, to date all such quantizations have lacked a certain fundamental game theoretic property. The main result in this dissertation is the development of quantizations of history dependent quantum Parrondo games that satisfy this fundamental game theoretic property. Our approach also yields fresh insight as to what should be considered as the proper quantum analogue of a classical Markov process and gives the first game theoretic measures of multiplexer behavior.Comment: Doctoral dissertation, Portland State University, 138 pages, 22 figure

Asymptotically Improved Grover's Algorithm in any Dimensional Quantum System with Novel Decomposed nn-qudit Toffoli Gate

As the development of Quantum computers becomes reality, the implementation of quantum algorithms is accelerating in a great pace. Grover's algorithm in a binary quantum system is one such quantum algorithm which solves search problems with numeric speed-ups than the conventional classical computers. Further, Grover's algorithm is extended to a $d$-ary quantum system for utilizing the advantage of larger state space. In qudit or $d$-ary quantum system n-qudit Toffoli gate plays a significant role in the accurate implementation of Grover's algorithm. In this paper, a generalized $n$-qudit Toffoli gate has been realized using qudits to attain a logarithmic depth decomposition without ancilla qudit. Further, the circuit for Grover's algorithm has been designed for any d-ary quantum system, where d >= 2, with the proposed $n$-qudit Toffoli gate so as to get optimized depth as compared to state-of-the-art approaches. This technique for decomposing an n-qudit Toffoli gate requires access to higher energy levels, making the design susceptible to leakage error. Therefore, the performance of this decomposition for the unitary and erasure models of leakage noise has been studied as well

Constructing all qutrit controlled Clifford+T gates in Clifford+T

For a number of useful quantum circuits, qudit constructions have been found which reduce resource requirements compared to the best known or best possible qubit construction. However, many of the necessary qutrit gates in these constructions have never been explicitly and efficiently constructed in a fault-tolerant manner. We show how to exactly and unitarily construct any qutrit multiple-controlled Clifford+T unitary using just Clifford+T gates and without using ancillae. The T-count to do so is polynomial in the number of controls $k$, scaling as $O(k^{3.585})$. With our results we can construct ancilla-free Clifford+T implementations of multiple-controlled T gates as well as all versions of the qutrit multiple-controlled Toffoli, while the analogous results for qubits are impossible. As an application of our results, we provide a procedure to implement any ternary classical reversible function on $n$ trits as an ancilla-free qutrit unitary using $O(3^n n^{3.585})$ T gates.Comment: 14 page