An overview of quantum computation models: quantum automata

Quantum automata, as theoretical models of quantum computers, include quantum finite automata (QFA), quantum sequential machines (QSM), quantum pushdown automata (QPDA), quantum Turing machines (QTM), quantum cellular automata (QCA), and the others, for example, automata theory based on quantum logic (orthomodular lattice-valued automata). In this paper, we try to outline a basic progress in the research on these models, focusing on QFA, QSM, QPDA, QTM, and orthomodular lattice-valued automata. Also, other models closely relative to them are mentioned. In particular, based on the existing results in the literature, we finally address a number of problems to be studied in future.     

量子自动机的交换性

定义了量子自动机及广义量子自动机的交换性,并提出了（广义）量子自动机所识别语言的交换性。利用半群及矩阵研究（广义）量子自动机的交换性,得出了（广义）量子自动机交换性的几个等价刻画。研究了（广义）量子自动机的交换性与其所识别语言的交换性的关系,证明了交换的（广义）量子自动机所识别的语言也是交换的。此外,讨论了（广义）量子自动机的广义直积、全直积、限制直积、级联积和圈积等积的交换性,得出了一些积的交换性的充分条件和必要条件。     

Coupling of quantum angular momenta: an insight into analogic/discrete and local/global models of computation

In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli and Rasetti, 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on ‘qubits’ and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occurring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite states-machines which in turn represent discrete versions of topological quantum computation models. We argue that our model embodies a sort of unifying paradigm for computing inspired by Nature and, even more ambitiously, a universal setting in which suitably encoded quantum symbolic manipulations of combinatorial, topological and algebraic problems might find their ‘natural’ computational reference model.     

Qsimulation:一个量子计算模拟器工具

介绍一个可在经典计算机上模拟量子计算的工具Qsimulation。该工具由4个主要部分组成：一个命令式的量子编程语言,一个量子计算解释器,一个用于模拟量子程序执行的图形用户界面以及错误处理模块,它能帮助教师和新手设计并测试简单的量子电路和量子程序。     

辅助量子比特驱动型通用盲量子计算

应用量子隐形传态将Broadbent等人提出的通用盲量子计算(universal blind quantum computation)模型和辅助量子比特驱动型量子计算(ancilla-driven universal quantum computation)模型进行结合, 构造一个新的混合模型来进行计算。此外, 用计算寄存器对量子纠缠的操作来代替量子比特测量操作。因为后者仅限于两个量子比特, 所以代替后的计算优势十分明显。基于上述改进, 设计了实现辅助驱动型通用盲量子计算的协议。协议的实现, 能够使Anders等人的辅助驱动型量子计算增强计算能力, 并保证量子计算的正确性, 从而使得参与计算的任何一方都不能获得另一方的保密信息。     

Mathematical logic and quantum finite state automata

This paper is a review of the connection between formulas of logic and quantum finite-state automata in respect to the language recognition and acceptance probability of quantum finite-state automata. As is well known, logic has had a great impact on classical computation, it is promising to study the relation between quantum finite-state automata and mathematical logic. After a brief introduction to the connection between classical computation and logic, the required background of the logic and quantum finite-state automata is provided and the results of the connection between quantum finite-state automata and logic are presented.     

Notes on automata theory based on quantum logic

The main results are as follows: (1) it deals with a number of basic operations (concatenation, Kleene closure, homomorphism, complement); (2) due to a condition imposed on the implication operator for discussing some basic issues in orthomodular lattice-valued automata, this condition is investigated in detail, and it is discovered that all the relatively reasonable five implication operators in quantum logic do not satisfy this condition, and that one of the five implications satisfies such a condition iff the truth-value lattice is indeed a Boolean algebra; (3) it deals further with orthomodular lattice-valued successor and source operators; (4) an example is provided, implying that some negative results obtained in the literature may still hold in some typical orthomodular lattice-valued automata.     

通用的辅助量子计算

设计了一个通用的辅助量子计算协议。该协议的客户端Alice仅拥有经典计算机或有限的量子技术,这些资源不足以让Alice做通用量子计算,因此Alice需要把她的量子计算任务委派给远程的量子服务器Bob。Bob拥有充分成熟的量子计算机,并会诚实地帮助Alice执行委派的量子计算任务,但他却得不到Alice的任何输入、输出信息。该协议只要求Alice能发送量子态和执行Pauli门操作,协议具有通用性、半盲性、正确性和可验证性。     

An overview of quantum cellular automata

Quantum cellular automata are arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata provide a model/architecture for distributed quantum computation. More generally, they encompass most of discrete-space discrete-time quantum theory. We give an overview of their theory, with particular focus on structure results; computability and universality results; and quantum simulation results.

     

Finite automata based on quantum logic and monadic second-order quantum logic

We introduce monadic second-order quantum logic and prove that the behaviors of finite automata based on quantum logic are precisely the quantum languages definable with sentences of our monadic secondorder quantum logic. This generalizes Büchi’s and Elgot’s fundamental theorems to quantum logic setting. We also consider first-order quantum logic and show that star-free quantum languages and aperiodic quantum languages introduced here coincide with the first-order quantum definable ones. This generalizes Sc...     

Minor-embedding in adiabatic quantum computation: I. The parameter setting problem

We show that the NP-hard quadratic unconstrained binary optimization (QUBO) problem on a graph G can be solved using an adiabatic quantum computer that implements an Ising spin-1/2 Hamiltonian, by reduction through minor-embedding of G in the quantum hardware graph U. There are two components to this reduction: embedding and parameter setting. The embedding problem is to find a minor-embedding G _emb of a graph G in U, which is a subgraph of U such that G can be obtained from G _emb by contracting edges. The parameter setting problem is to determine the corresponding parameters, qubit biases and coupler strengths, of the embedded Ising Hamiltonian. In this paper, we focus on the parameter setting problem. As an example, we demonstrate the embedded Ising Hamiltonian for solving the maximum independent set (MIS) problem via adiabatic quantum computation (AQC) using an Ising spin-1/2 system. We close by discussing several related algorithmic problems that need to be investigated in order to facilitate the design of adiabatic algorithms and AQC architectures.     

An analysis of the effect of the stochastic component of urban cellular automata models

Urban cellular automata models have proved useful tools in urban growth prediction because of their simplicity and their ability to reproduce complex emergent dynamics. Complex emergent dynamic systems involve processes that are difficult to predict, in which randomness plays a key role. In view of the fact that randomness is particularly relevant to complex processes, the aim of this paper is to analyze the sensitivity of the results of urban cellular automata models to the different methods used to incorporate the stochastic component in the models. The urban growth patterns obtained using different stochastic components are analyzed and compared using a number of spatial metrics. The results show that the differences observed in the simulated patterns are sufficiently relevant to justify the need for this type of analysis, which allows for the selection of the stochastic component that best suits the dynamics of the area.     

Improved constructions of mixed state quantum automata

Quantum finite automata with mixed states are proved to be super-exponentially more concise rather than quantum finite automata with pure states. It was proved earlier by A. Ambainis and R. Freivalds that quantum finite automata with pure states can have an exponentially smaller number of states than deterministic finite automata recognizing the same language. There was an unpublished “folk theorem” proving that quantum finite automata with mixed states are no more super-exponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable.     

Interference automata

We propose a computing model, the Two-Way Optical Interference Automata (2OIA), that makes use of the phenomenon of optical interference. We introduce this model to investigate the increase in power, in terms of language recognition, of a classical Deterministic Finite Automaton (DFA) when endowed with the facility of interference. The question is in the spirit of Two-Way Finite Automata With Quantum and Classical States (2QCFA) [A. Ambainis, J. Watrous, Two-way finite automata with quantum and classical states, Theoret. Comput. Sci. 287 (1) (2002) 299–311] wherein the classical DFA is augmented with a quantum component of constant size. We test the power of 2OIA against the languages mentioned in the above paper. We give efficient 2OIA algorithms to recognize languages for which 2QCFA machines have been shown to exist, as well as languages whose status vis-a-vis 2QCFA has been posed as open questions. Having a DFA as a component, it trivially recognizes regular languages. We show that our model can recognize all languages recognized by 1-way deterministic blind counter automata. Finally we show the existence of a language that cannot be recognized by a 2OIA but which can be recognized by an O(n³)$O\left(n^3\right)$ space Turing machine.     

Simulation of one-way cellular automata by boolean circuits

We present a relationship between two major models of parallel computation: the one-way cellular automata and the boolean circuits. The starting point is the boolean circuit of small depth designed by Ladner and Fischer to simulate any rational transducer. We extend this construction to simulate one-way cellular automata by boolean circuits.     

Spreadable cellular automata: modelling and simulations

The spreadable phenomena which describes the expansion in time of a given spatial property has been studied using models based on partial differential equations. These spreadable dynamics are generally non-linear and then difficult to simulate particularly in two dimensions. In this article, we propose cellular automata (CA) models as an alternative modelling tool that can easily simulate spreadable systems. CA are capable of describing complex systems based on simple evolution rules, which provide numerical schemes directly implemented on computers without approximation or rounding errors. We design local CA dynamics which allow us to maintain a spatial property on non-decreasing subdomains. Several numerical results are performed to illustrate spreadable phenomena. The simulation results corroborate the general shape theory that exhibits the convergence to a specific domain independently on initial conditions.     

Compositional and holistic quantum computational semantics

In quantum computational logic meanings of sentences are identified with quantum information quantities: systems of qubits or, more generally, mixtures of systems of qubits. We consider two kinds of quantum computational semantics: (1) a compositional semantics, where the meaning of a compound sentence is determined by the meanings of its parts; (2) a holistic semantics, which makes essential use of the characteristic “holistic” features of the quantum-theoretic formalism. We prove that the compositional and the holistic semantics characterize the same logic.     

On the complexity of simulating space-bounded quantum computations

This paper studies the space-complexity of predicting the long-term behavior of a class of stochastic processes based on evolutions and measurements of quantum mechanical systems. These processes generalize a wide range of both quantum and classical space-bounded computations, including unbounded error computations given by machines having algebraic number transition amplitudes or probabilities. It is proved that any space s quantum stochastic process from this class can be simulated probabilistically with unbounded error in space O(s), and therefore deterministically in space O(s²).