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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Quantum Finite State Machines as Sequential Quantum Circuits

Martin Lukáč, Marek A. Perkowski

Published Jan 1, 2009• 16 citations
Publisher

Abstract

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.

DOI
10.1109/ismvl.2009.46

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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.
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.
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.
We present an analysis of the Reversible and Quantum Finite State Machines (QFSM) realized as Quantum Circuits using the three well known sequences applied in the analysis of the classical Finite State Machines (FSM). The synchronizing, the homing and the distinguishing sequences are applied to both strictly Reversible FSM (RFSM) and QFSM in order to determine the power of these new techniques in the above mentioned new models of sequential devices. In particular, care is taken to demonstrate these classical techniques on the RFSM and the one-way QFSMs. We show certain properties of the RFSM/QFSM with respect to these sequences and we show what are the restrictions and advantages.
We define a formal framework for equivalence checking of sequential quantum circuits. The model we adopt is a quantum state machine, which is a natural quantum generalization of Mealy machines. A major difficulty in checking quantum circuits (but not present in checking classical circuits) is that the state spaces of quantum circuits are continuums. This difficulty is resolved by our main theorem showing that equivalence checking of two quantum Mealy machines can be done with input sequences that are taken from some chosen basis (which are finite) and have a length quadratic in the dimensions of the state Hilbert spaces of the machines. Based on this theoretical result, we develop an (and to the best of our knowledge, the first) algorithm for checking equivalence of sequential quantum circuits with running timeO(23m+5l(23m + 23l))\mathcal {O}(2^{3m+5l}(2^{3m}{\,+\,}2^{3l}))O(23m+5l(23m+23l)), wheremmmandllldenote the numbers of input and internal qubits, respectively. The complexity of our algorithm is comparable with that of the known algorithms for checking classical sequential circuits in the sense that both are exponential in the number of (qu)bits. Several case studies and experiments are presented.
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.
Future quantum information processing devices will require the use of exotic quantum states, such as specially crafted entangled states, to achieve certain desired computations on demand. Thus far, synthesis schemes for such states have been devised on a case-by-case basis using ad hoc techniques. In this paper we present a systematic method for finding a quantum circuit that can synthesize any pure or mixed n-qubit state. We then give examples of the use of our algorithm for finding synthesis pathways for especially exotic quantum states such as maximal mixed states. It is not known how to prepare general instances of such states by other means. Thus our quantum state synthesis algorithm should be of use not only in quantum information processing, but also in experimental quantum physics.