Local unitary quantum cellular automata

Abstract

In this paper we present a quantization of Cellular Automata. Our formalism is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. It is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: universality, strong connections to the circuit model, simple implementation on quantum hardware, and a wealth of applications.

Unitarity plus causality implies localizability

“…Quantum cellular automata (QCA). As regards QCA more specifically, a corollary of our results is that the axiomatic definition of n-dimensional QCA proposed in [26] does admit an operational, Block-structured QCA description after all -and conversely that these seemingly specific Block-structured QCA [24] are in fact general instances axiomatic definition. As regards Quantum Walks this answers a long-standing open problem formulated in [1].…”

“…Several works have followed this route for instance [28,7,22,25], but amongst them [24] stand out at this stage as it just directly posits, after some interesting arguments, that their evolutions take a form akin to the one described in Theorem 2.…”

“…2 (Notice that even the shift QCA can be implemented as a quantum circuit according to this theorem, but of course this is at the price of introducing ancilla. This question of the special role of the shift and its implementability is emphasized in [24] and carefully analyzed in [13]. )…”

Quantum Cellular Automata Controlled Self-Organizing Networks

“…The updating transformations operate probabilistically on the cells, and for a well-formed QCA, all the probabilities of the unitary transformations have to preserve the squared sum of these properties, which is equal to one. As has also been shown, this 'well-formed' property can be verified algorithmically (Dam, 1996), (Perez-Delgado and Cheung, 2005).…”

Performance of 1D quantum cellular automata in the presence of error

“…First, they are of interest because, as a quantum generalization of cellular automata, they have been shown to be capable of universal quantum computation. [1][2][3][4][5][6][7][8] Second, QCA allow implementation of specific quantum information processing tasks with applications to quantum cryptography, 9 entanglement generation, 10,11 quantum state transfer, [11][12][13] and phase covariant cloning. 14 Finally there is a body of recent work focusing on the emergence of quantum field theory from an underlying QCA model.…”