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Evolutionary Logic Synthesis of Quantum Finite State Machines for Sequence Detection

Martin Lukáč, Marek A. Perkowski

Published Feb 1, 2010• 2 citations
Publisher

Abstract

In this paper we presented a methodology and we showed some experimental results confirming that our approach is possible in simulated environment. Also because all simulated elements of the presented experiments are based on existing Quantum operations, the simulated detectors are Quantum-realizable. It is well known that the state assignment problem is a NP-complete problem [Esc93] and the finding a minimal State Assignment has been solved only for particular subsets of FSM's [LPD95] or using Quantum computing [ANdMM08]. This problem is here naturally solved (without addressing it). The setup of this experimental approach automatically generates a state assignment such that when the detection is successful the state assignment is as well.

DOI
10.5772/8049

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In this paper we presented a methodology and we showed some experimental results confirming that our approach is possible in simulated environment. Also because all simulated elements of the presented experiments are based on existing Quantum operations, the simulated detectors are Quantum-realizable. It is well known that the state assignment problem is a NP-complete problem [Esc93] and the finding a minimal State Assignment has been solved only for particular subsets of FSM's [LPD95] or using Quantum computing [ANdMM08]. This problem is here naturally solved (without addressing it). The setup of this experimental approach automatically generates a state assignment such that when the detection is successful the state assignment is as well.
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.
In this paper we present the synthesis process of quantum finite state machines for sequence detection based on the synthesis of quantum sequential circuits. Such machines have many applications for instance in robotics. We show the relation between the quantum circuit and the natural ability of quantum systems to retain the quantum state. We consider cases where a single qubit in a circuit represents a quantum state and another is used as the input qubit. We analyze some of the well known circuits as sequence detectors and describe their behaviors. We show that the standard quantum logic synthesis can be used for both the synthesis of quantum circuits as well as for quantum finite state machines used as sequence detectors.
This paper demonstrates an application of an evolutionary approach for solving a class of non-trivial, hardware-design problems. Array processing features of the computer language APL simplify the implementation of an evolutionary solution in which simulation is performed by a genetic algorithm on a population of candidate solutions until one or more are satisfactory quantum algorithms. The objective of the simulation model is the automatic discovery of quantum computer algorithms. The algorithms are expressed in a circuit model that specifies the sequences in which quantum operators are to be applied. The automatically configured circuits operate as quantum computers that for the present have their domains of application limited to evaluation of only a few specific functions. The simulations use a small collection of basic and relatively low-level operators to obtain perfect results for five different target functions. These functions were chosen to demonstrate how, on a personal computer, an evolutionary method is able to discover novel designs of computing hardware. The results establish that it is feasible to use genetic algorithms in an evolutionary method to invent correct hardware designs. The progress of simulated evolution is directed by input/output constraints and a fitness function. Alternative configurations of circuit models can obtain algorithms with promise for future quantum computers. Current simulations yield primitive quantum algorithms with a total of twelve or fewer inputs and outputs. The discovered algorithms produce the correct results for evaluating the five selected logical and arithmetic functions.
This paper demonstrates an application of an evolutionary approach for solving a class of non-trivial, hardware-design problems. Array processing features of the computer language APL simplify the implementation of an evolutionary solution in which simulation is performed by a genetic algorithm on a population of candidate solutions until one or more are satisfactory quantum algorithms. The objective of the simulation model is the automatic discovery of quantum computer algorithms. The algorithms are expressed in a circuit model that specifies the sequences in which quantum operators are to be applied. The automatically configured circuits operate as quantum computers that for the present have their domains of application limited to evaluation of only a few specific functions. The simulations use a small collection of basic and relatively low-level operators to obtain perfect results for five different target functions. These functions were chosen to demonstrate how, on a personal computer, an evolutionary method is able to discover novel designs of computing hardware. The results establish that it is feasible to use genetic algorithms in an evolutionary method to invent correct hardware designs. The progress of simulated evolution is directed by input/output constraints and a fitness function. Alternative configurations of circuit models can obtain algorithms with promise for future quantum computers. Current simulations yield primitive quantum algorithms with a total of twelve or fewer inputs and outputs. The discovered algorithms produce the correct results for evaluating the five selected logical and arithmetic functions.
Quantum computation established itself as a promising emerging technology and, hence, attracted considerable attention in the domain of computer-aided-design (CAD). However, quantum mechanical phenomena such as superposition, phase shifts, or entanglement lead to a logic model which poses serious challenges to the development of a proper design flow for quantum circuits. Consequently, researchers addressed synthesis of quantum circuits not as a single design step, but considered sub-tasks such as synthesis of Boolean components or synthesis of restricted subsets of quantum functionality. Generating a particularly desired quantum state is another of these sub-tasks. However, logic synthesis of quantum circuits accomplishing that has hardly been considered yet. In this work, we propose a generic method which automatically synthesizes a quantum circuit generating any desired quantum state from an initially given basis state. The proposed method allows for both, a theoretical determination of upper bounds as well as an experimental evaluation of the number of quantum gates needed for this important design step.
We present a model of Quantum Finite State Machines (QFSM) based on the quantum circuit representation. In this model, properties of quantum bits are exploited in order to implement Quantum Finite State Machines without an explicit external memory. We show that because of the the natural ability of quantum systems to retain the quantum state, quantum circuits can be directly used to implement some classes of the quantum finite state machines. In particular, the circuit model allows to remove the logic-memory distinction as it is used in the classically established model of finite state machines. Rather, the quantum register implementing the QFSM as a circuit is alternatively used as memory and as logic. In order to allow such behavior the quantum circuit (processor) is controlled by a classical circuit that provides the necessary timing of the quantum operations as well as the interface between the quantum and the classical world. Moreover, additional mechanism for preserving unknown quantum qubit state is used and a novel computational protocol allowing the implementation of the QFSM is proposed. This framework is is called the Classically Controlled Quantum Computer (CCQC). The CCQC thus combines a quantum processor with a classical controller in such manner that computation is carried from the classical inputs, through quantum processing to classical outputs. We discuss a high level synthesis approach to quantum sequence detection based on the synthesis of quantum sequential circuits. Such machines can find many applications for instance in robotics; for example a robot is required to recognize certain sequences of input sensor values.