Gatis Midrijanis
We present a language $L_n$ which is recognizable by a probabilistic finite automaton (PFA) with probability $1 - ε$ for all $ε> 0$ with $O(log^2n)$ states, with a deterministic finite automaton (DFA) with O(n) states, but a quantum finite automaton (QFA) needs at least $2^{Ω(n/ \log n)}$ states.
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We compare the two approaches to the empirical logic of automata. The first, called partition logic (logic of microstatements), refers to experiments on individual automata. The second one, the logic of simulation (logic of macrostatements), deals with ensembles of automata.
To study relationship between quantum finite automata and probabilistic finite automata, we introduce a notion of probabilistic reversible automata (PRA, or doubly stochastic automata). We find that there is a strong relationship between different possible models of PRA and corresponding models of quantum finite automata. We also propose a classification of reversible finite 1-way automata.
We study the recognition of R-trivial idempotent (R1) languages by various models of "decide-and-halt" quantum finite automata (QFA) and probabilistic reversible automata (DH-PRA). We introduce bistochastic QFA (MM-BQFA), a model which generalizes both Nayak's enhanced QFA and DH-PRA. We apply tools from algebraic automata theory and systems of linear inequalities to give a complete characterization of R1 languages recognized by all these models. We also find that "forbidden constructions" known so far do not include all of the languages that cannot be recognized by measure-many QFA.
Maximum Independent Set via Probabilistic and Quantum Cellular Automata
本文提出利用量子细胞自动机求解最大独立集问题，通过结合耗散与幺正演化，提高收敛效率，为量子优化提供新方法。
We prove that endowing a real-time probabilistic or quantum computer with the ability of postselection increases its computational power. For this purpose, we provide a new model of finite automata with postselection, and compare it with the model of Lāce et al. We examine the related language classes, and also establish separations between the classical and quantum versions, and between the zero-error vs. bounded-error modes of recognition in this model.
Quantum finite automata were introduced by C.Moore, J.P. Crutchfield, and by A.Kondacs and J.Watrous. This notion is not a generalization of the deterministic finite automata. Moreover, it was proved that not all regular languages can be recognized by quantum finite automata. A.Ambainis and R.Freivalds proved that for some languages quantum finite automata may be exponentially more concise rather than both deterministic and probabilistic finite automata. In this paper we introduce the notion of quantum finite multitape automata and prove that there is a language recognized by a quantum finite …
The theory of cellular automata in operational probabilistic theories is developed. We start introducing the composition of infinitely many elementary systems, and then use this notion to define update rules for such infinite composite systems. The notion of causal influence is introduced, and its relation with the usual property of signalling is discussed. We then introduce homogeneity, namely the property of an update rule to evolve every system in the same way, and prove that systems evolving by a homogeneous rule always correspond to vertices of a Cayley graph. Next, we define the notion o…
Quantum Systems from Random Probabilistic Automata
该研究用量子形式体系分析随机概率元胞自动机，通过比特散射模拟验证其量子系统特征，引入统计可观测量，指出能量与动量守恒是其演化核心，其演化类似二维带随机势的单个狄拉克费米子。
Protecting information via probabilistic cellular automata
本文研究概率元胞自动机 sweep rule，在温度-磁场平面中标定双稳定相共存区域，并证明其属于弱二维 Ising 普适类，为理解几何局部纠错策略保护复杂噪声系统中的信息提供基础。
Discretization of phase space usually nullifies chaos in dynamical systems. We show that if randomness is associated with discretization dynamical chaos may survive and be indistinguishable from that of the original chaotic system, when an entropic, coarse-grained analysis is performed. Relevance of this phenomenon to the problem of quantum chaos is discussed.
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 intera…
The minimal probabilistic and quantum finite automata recognizing uncountably many languages with fixed cutpoints
本文针对固定截断点，证明了2状态二元和3状态一元概率有限自动机及2状态一元量子有限自动机可识别不可数多种语言，并给出了新构造。
In this work we study a non-linear generalization based on affine transformations of probabilistic and quantum automata proposed recently by Díaz-Caro and Yakaryılmaz \cite{DCY16A} referred as affine automata. First, we present efficient simulations of probabilistic and quantum automata by means of affine automata which allows us to characterize the class of exclusive stochastic languages. Then, we initiate a study on the succintness of affine automata. In particular, we show that an infinite family of unary regular languages can be recognized by 2-state affine automata but the state numbers o…
A classical local cellular automaton can describe an interacting quantum field theory for fermions. We construct a simple classical automaton for a particular version of the Thirring model with imaginary coupling. This interacting fermionic quantum field theory obeys a unitary time evolution and shows all properties of quantum mechanics. Classical cellular automata with probabilistic initial conditions admit a description in the formalism of quantum mechanics. Our model exhibits interesting features as spontaneous symmetry breaking or solitons. The same model can be formulated as a generalized…
Tensor-Networks-based Learning of Probabilistic Cellular Automata Dynamics
本研究开发了基于矩阵乘积算子的算法，用于学习概率序列到序列过程并应用于概率元胞自动机，可准确学习不同条件下的概率元胞自动机过程，且学习能力与规则间按位差异及规则可能性相关。
We introduce 2-way finite automata with quantum and classical states (2qcfa's). This is a variant on the 2-way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa's; the internal state of a 2qcfa may include a quantum part that may be in a (mixed) quantum state, but the tape head position is required to be classical. We show two languages for which 2qcfa's are better than classical 2-way automata. First, 2qcfa's can recognize palindromes, a language that cannot be recognized by 2-way deterministic or probabilistic finite automata. Second, in polynomial…
Cellular automata (CA) consist of an array of identical cells, each of which may take one of a finite number of possible states. The entire array evolves in discrete time steps by iterating a global evolution G. Further, this global evolution G is required to be shift-invariant (it acts the same everywhere) and causal (information cannot be transmitted faster than some fixed number of cells per time step). At least in the classical, reversible and quantum cases, these two top-down axiomatic conditions are sufficient to entail more bottom-up, operational descriptions of G. We investigate whethe…
New results on classical and quantum counter automata
本文证明零误差单向量子单计数器自动机在承诺问题上强于概率对应物，并推翻了关于盲计数器自动机识别Kleene闭包的猜想，获得多个分离结果。
How do cellular automata behave in the limit of a very large number of cells? Is there a continuum limit with simple properties? We attack this problem by mapping certain classes of automata to quantum field theories for which powerful methods exist for this type of problem. Indeed, many cellular automata admit an interpretation in terms of fermionic particles. Reversible automata on space-lattices with a local updating rule can be described by a partition function or Grassmann functional integral for interacting fermions moving in this space. We discuss large classes of automata that are equi…
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