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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum state transition diagram: a bridge from classical computing to quantum computing

Loyd R. Hook, Sam Lee

Published Mar 25, 2010• 3 citations
Publisher

Abstract

Very few papers have been written on the topic of a quantum version of the finite state machine, (or finite state automata). Furthermore, these papers only serve to define what a quantum finite state machine might be in the mathematical sense using the early languages of Turing machines. This paper seeks to further develop the notion of a quantum finite state machine (FSM) using constructs developed for the classical FSM and utilized for classical FSM design. In particular the quantum state transition diagram (QSTD) is constructed to further the understanding and realization of quantum finite state machines and quantum computers.

DOI
10.1117/12.847450

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Very few papers have been written on the topic of a quantum version of the finite state machine, (or finite state automata). Furthermore, these papers only serve to define what a quantum finite state machine might be in the mathematical sense using the early languages of Turing machines. This paper seeks to further develop the notion of a quantum finite state machine (FSM) using constructs developed for the classical FSM and utilized for classical FSM design. In particular the quantum state transition diagram (QSTD) is constructed to further the understanding and realization of quantum finite state machines and quantum computers.
Quantum computing is concerned with computer technology based on the principles of quantum mechanics, with operations performed at the quantum level. Quantum computational models make it possible to analyze the resources required for computations. Quantum automata can be classified thusly: quantum finite automata, quantum sequential machine, quantum pushdown automata, quantum Turing machine and orthomodular lattice-valued automata. These models are useful for determining the expressive power and boundaries of various computational features. In light of the current state of quantum computation theory research, a systematic review of the literature seems timely. This article seeks to provide a comprehensive and systematic analysis of quantum finite automata models, quantum finite automata models with density operators and quantum finite automata models with classical states, interactive proof systems, quantum communication complexity and query complexity as described in the literature. The statistics of quantum finite automata related papers are shown and open problems are identified for more advanced research. The current status of quantum automata theory is distributed into various categories. This research work also highlights the previous research, current status and future directions of quantum automata models.
There are several papers that define quantum finite automaton analogous to classical finite automaton. Some papers describe the power of quantum automaton. In general, in quantum automata unitary matrices are associated to input symbols. Even though quantum circuits (or unitary matrices) are shown in specific cases, there is no general method given to construct a circuit or unitary matrix is found in the literature. This research focuses on construction of unitary matrices for each input symbol from the classical definition of a Deterministic Finite State Machine (DFSM). An algorithm is developed to associate a quantum circuit to an input symbol which considered as a mapping of classical DFSM states. IBM quantum composer is used to design circuits and build examples.
Quantum finite automata were introduced by C.Moore and J.P.Crutchfield in [MC 97] and by A.Kondacs and J.Watrous in [KW 97]. This notion is not a generalization of the deterministic finite automata, but rather a generalization of deterministic reversible (permutation) automata. In [AF 98] A.Ambainis and R.Freivalds raised the question what kind of probabilistic automata can be viewed as a special case of quantum finite automata. To answer that question and study relationship between quantum finite automata and probabilistic finite automata, we introduce a notion of probabilistic reversible automata (PRA, or doubly stochastic automata). We give the necessary condition for a language to be recognized by PRA. We find that there is a strong relationship between different possible models of PRA and corresponding models of quantum finite automata. In these thesis we regard quantum automata, probabilistic reversible and deterministic reversible automata as reversible automata. At least two non-equivalent definitions of language recognition are used for one-way reversible finite automata in various papers. Both of these definitions have their own advantages and disadvantages, summarized in our classification of one-way reversible automata. Further, we introduce a notion of quantum finite multitape automata and prove that there is a language recognized by a quantum finite multitape automaton but not by deterministic finite multitape automata. Additionally we discover unexpected probabilistic automata recognizing complicated languages. Finally, we revise the concept of quantum pushdown automata (QPA), first introduced by C.Moore and J.P.Cruthchfield in [MC 97]. We give the definition of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of [KW 97]. It is established that the unitary criteria of QPA are not equivalent to the corresponding unitary criteria of quantum Turing machines [BV 97]. We show that QPA can recognize every regular language. We also present some simple non-context-free languages recognized by QPA (hence not recognizable by deterministic and nondeterministic pushdown automata).
Abstract There are many papers presenting quantum computing models. The definitions parallel the classical definitions of finite state automata, push down automata, context-free grammars etc. Classical computing model definitions define languages precisely. We can state that a string belongs to a language or does not belong to it with no room for error. Quantum definitions do not possess this certainty. Sacrificing the certainty and adopting a quantum definition of a computing model does not appear to provide any concrete power to the model. Therefore, the path of this paper is to begin from the classical definition and end in a quantum circuit. In this paper we start from a Deterministic Push-Down Automaton (DPDA). We present circuit for state transition and stack operations. The circuits presented can be viewed as independent algorithms. As an example, the approach used to construct the circuit for state transition can be utilized to build the circuit for a function presented as a Boolean matrix.
The design of state machines for quantum algorithms remains a largely unexplored topic. The abstract models for quantum computation proposed by Feynman [1], Deutsch [2] etc have been related to the theory of Quantum Turing Machines (QTM) and the work was later expanded by Bernstein and Vazirani [3] to lay the foundations of quantum complexity theory. However, no papers deal with the following problem: “how to realize the quantum state machine in hardware as an operational quantum circuit that the internal states of which can be measured and used as inputs in a system of such machines”. This is an equivalent to the fundamental problem of “structural synthesis of FSM”. This paper creates models of quantum state machines that are useful from an engineering perspective. The objective of the paper is to propose models that can be used for circuit-level implementation. We restrict our study to binary logic realized in quantum, the so-called binary permutative quantum circuits; the “permutative machines”. The presented synthesis procedure is analogous to classical logic circuits synthesis and is a necessary step in the entire design procedure of quantum automata. Thus state minimization, state assignment and excitation function realization aspects are also of interest and some preliminary experimental results are presented.