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World Automation Congress (WAC)

m-Valued Quantum Representations and Operations of Elementary Cellular Automata

Anas N. Al‐Rabadi

Published Jul 1, 2006• 1 citation
Publisher

Abstract

This paper introduces the m-valued Swap-based quantum computations (QC) of ECA. The fundamental m-valued quantum representations of: (1) m-valued orthonormal computational basis states quantum decision trees (QDTs) and (2) m-valued orthonormal composite basis states QDTs are also introduced as quantum modeling for the representations and operations of the QECA dynamics. The new m-valued quantum representations and their associated operations can play an important role in the automated design of future QECA circuits that consume minimal power in the quantum domain.

DOI
10.1109/wac.2006.376009

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Abstract Modeling noisy discrete systems utilizing conservative reversible elementary cellular automata (CRECA) is introduced. Reversibility in the ECA evolution rule and reversibility in the ECA circuit implementation are both achieved using a new Swap-based algorithm called Swap-based CRECA (SCRECA). The new method results in adding variable redundancy to counteract the effect of noise. The problem of obtaining a reversible map from an irreversible map is important because quantum circuits are inherently reversible and thus does not consume power, while irreversible circuits (due to either an irreversible mapping or to noise within a reversible mapping) cannot exist in the quantum domain and its circuits consume power. Since noise is an integral part of any real process and since the reduction of power consumption is a main requirement for the circuit design of future technologies such as in quantum computing (QC), the main features of several future technologies will include reversibility, and thus the...
Quantum cellular automata are alternative quantum-computing paradigms to quantum Turing machines and quantum circuits. Their working mechanisms are inherently automated, therefore measurement free, and they act in a translation invariant manner on all cells or qudits of a register, generating a global rule that updates cell states locally, i.e., based solely on the states of their neighbors. Although desirable features in many applications, it is generally not clear to which extent these fully automated discrete-time local updates can generate and sustain long-range order in the (noisy) systems they act upon. In particular, whether and how quantum cellular automata can perform quantum error correction remain open questions. We close this conceptual gap by proposing quantum cellular automata with quantum-error-correction capabilities. We design and investigate two (quasi)one dimensional quantum cellular automata based on known classical cellular-automata rules with density-classification capabilities, namely the local majority voting and the two-line voting. We investigate the performances of those quantum cellular automata as quantum-memory components by simulating the number of update steps required for the logical information they act upon to be afflicted by a logical bit flip. The proposed designs pave a way to further explore the potential of new types of quantum cellular automata with built-in quantum-error-correction capabilities.
Quantum cellular automata are alternative quantum-computing paradigms to quantum Turing machines and quantum circuits. Their working mechanisms are inherently automated, therefore measurement free, and they act in a translation invariant manner on all cells/qudits of a register, generating a global rule that updates cell states locally, i.e., based solely on the states of their neighbors. Although desirable features in many applications, it is generally not clear to which extent these fully automated discrete-time local updates can generate and sustain long-range order in the (noisy) systems they act upon. In special, whether and how quantum cellular automata can perform quantum error correction remain open questions. We close this conceptual gap by proposing quantum cellular automata with quantum-error-correction capabilities. We design and investigate two (quasi-)one dimensional quantum cellular automata based on known classical cellular-automata rules with density-classification capabilities, namely the local majority voting and the two-line voting. We investigate the performances of those quantum cellular automata as quantum-memory components by simulating the number of update steps required for the logical information they act upon to be afflicted by a logical bit flip. The proposed designs pave a way to further explore the potential of new types of quantum cellular automata with built-in quantum error correction capabilities.
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This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.