Theory Of Automata Quantum Download

In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They provide a mathematical abstraction of real-world quantum computers. Several types of automata may be defined, including measure-once and measure-many automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata.

The languages accepted by QFAs are not the regular languages of deterministic finite automata, nor are they the stochastic languages of probabilistic finite automata. Study of these quantum languages remains an active area of research.

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There is a simple, intuitive way of understanding quantum finite automata. One begins with a graph-theoretic interpretation of deterministic finite automata (DFA). A DFA can be represented as a directed graph, with states as nodes in the graph, and arrows representing state transitions. Each arrow is labelled with a possible input symbol, so that, given a specific state and an input symbol, the arrow points at the next state. One way of representing such a graph is by means of a set of adjacency matrices, with one matrix for each input symbol. In this case, the list of possible DFA states is written as a column vector. For a given input symbol, the adjacency matrix indicates how any given state (row in the state vector) will transition to the next state; a state transition is given by matrix multiplication.

A well-known theorem states that, for each DFA, there is an equivalent NFA, and vice versa. This implies that the set of languages that can be recognized by DFA's and NFA's are the same; these are the regular languages. In the generalization to QFAs, the set of recognized languages will be different. Describing that set is one of the outstanding research problems in QFA theory.

By contrast, in a QFA, the manifold is complex projective space C P N {\displaystyle \mathbb {C} P^{N}} , and the transition matrices are unitary matrices. Each point in C P N {\displaystyle \mathbb {C} P^{N}} corresponds to a quantum-mechanical probability amplitude or pure state; the unitary matrices can be thought of as governing the time evolution of the system (viz in the Schrdinger picture). The generalization from pure states to mixed states should be straightforward: A mixed state is simply a measure-theoretic probability distribution on C P N {\displaystyle \mathbb {C} P^{N}} .

Measure-many automata were introduced by Kondacs and Watrous in 1997.[1] The general framework resembles that of the measure-once automaton, except that instead of there being one projection, at the end, there is a projection, or quantum measurement, performed after each letter is read. A formal definition follows.

The primary difference between real-world quantum computers and the theoretical framework presented above is that the initial state preparation cannot ever result in a point-like pure state, nor can the unitary operators be precisely applied. Thus, the initial state must be taken as a mixed state

There is no quantum analog to the push-down automaton or stack machine. This is due to the no-cloning theorem: there is no way to make a copy of the current state of the machine, push it onto a stack for later reference, and then return to it.

The above constructions indicate how the concept of a quantum finite automaton can be generalized to arbitrary topological spaces. For example, one may take some (N-dimensional) Riemann symmetric space to take the place of C P N {\displaystyle \mathbb {C} P^{N}} . In place of the unitary matrices, one uses the isometries of the Riemannian manifold, or, more generally, some set of open functions appropriate for the given topological space. The initial state may be taken to be a point in the space. The set of accept states can be taken to be some arbitrary subset of the topological space. One then says that a formal language is accepted by this topological automaton if the point, after iteration by the homeomorphisms, intersects the accept set. But, of course, this is nothing more than the standard definition of an M-automaton. The behaviour of topological automata is studied in the field of topological dynamics.

In the past decade or so, there has been a growing number of papers postulating a correspondence relating quantum cellular automata (QCAs) to quantum field theories (QFTs). If valid, this would relate two important but seemingly unconnected physical systems and offer the potential for a deeper understanding of both. For example, it could lead to algorithms for simulating QFTs on a quantum computer. Moreover, it is possible that nature actually obeys a QCA model and it is the corresponding QFT that is a mere approximation. However, verifying the QCA/QFT correspondence beyond the single-particle case has proven challenging, especially in more than one spatial dimension. Here we construct quantum cellular automata that, in the long-wavelength limit, yield a multiparticle theory for free fermions and bosons in three spatial dimensions: the Dirac and Maxwell field theories, i.e., free QED.

A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two-dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.

In quantum information science, a major task is to find the quantum models that can outperform their classical counterparts. Automaton is a fundamental computing model that has wide applications in many fields. It has been shown that the quantum version of automaton can solve certain problem using a much smaller state space compared to the classical automaton. Here we report an experimental demonstration of an optical quantum automaton, which is used to solve the promise problems of determining whether the length of an input string can be divided by a prime number P with no remainder or with a remainder of R. Our quantum automaton can solve such problem using a state space with only three orthonormal states, whereas the classical automaton needs no less than P states. Our results demonstrate the quantum benefits of a quantum automaton over its classical counterpart and paves the way for implementing quantum automaton for more complicated and practical applications.

Computer is not the only device whose capability can be boosted by introducing quantum elements. Finite automaton (FA) is another good example. As one of the most fundamental models in computer science, FA can be used to model problems in many fields, including mathematics, artificial intelligence, games or linguistics.17,18 The concept of quantum finite automaton (QFA) was invented as the quantum version of FA by Kondacs and Watrous19 and also by Moore and Crutchfield.20 It has been theoretically predicted that QFA can solve certain problems more efficiently than its classical counterpart.21,22 However, to the best of our knowledge, no experiment has been performed to prove the quantum benefits of QFA yet.

In this letter, we have built a proof-of-principle optical QFA that can solve some certain problems more space-efficiently than a classical FA. In our experiment, we have proven that a QFA composed with a three-dimensional quantum state suffice to solve a promise problem whereas a classical FA needs much larger state space to solve the same problem.

A deterministic finite automaton (DFA) is a finite-state machine that accepts or rejects input strings of symbols. Here deterministic refers to the fact that a DFA only produces a unique computation for each input string. We only consider the case of DFAs as it has been proven that DFAs have equivalent computing power to non-deterministic finite automata.17,23,24

Here we highlight the key differences between DFA and QFA. By definition, in both DFA and QFA, the state transition will happen n times, where n is the length of the input string. As a result, there is no difference between DFA and QFA in time complexity. The key difference between DFA and QFA is in space complexity. For a DFA, the state inside is a classical system, which means all of its possible states are always orthogonal to each other. As a result, to distinguish between strings with the length of k * P and k * P + R, the classical system inside the DFA must have a state space with at least P orthonormal states.22,26 In contrast, for a QFA, the state inside is a quantum system, which means it can be in a superposition state and its possible states do not need to be orthogonal to each other. By utilizing this feature, the QFA with a state space of just three orthonormal states can be used to solve the same promise problem. Our proof-of-principle optical experiment have demonstrated this quantum advantage of QFA in space efficiency. By using a photonic quantum system with just three orthonormal states, our QFA solves a promise problem which would needs a DFA with at least P orthonormal states. 006ab0faaa