Abstract
Discretizing spacetime is often a natural step towards modelling physical systems. For quantum systems, if we also demand a strict bound on the speed of information propagation, we get quantum cellular automata (QCAs). These originally arose as an alternative paradigm for quantum computation, though more recently they have found application in understanding topological phases of matter and have} been proposed as models of periodically driven (Floquet) quantum systems, where QCA methods were used to classify their phases. QCAs have also been used as a natural discretization of quantum field theory, and some interesting examples of QCAs have been introduced that become interacting quantum field theories in the continuum limit. This review discusses all of these applications, as well as some other interesting results on the structure of quantum cellular automata, including the tensor-network unitary approach, the index theory and higher dimensional classifications of QCAs.
Highlights
It is a little confusing that the name quantum cellular automata or other similar sounding names have been used for several unrelated concepts, so it will help to go through what we do not mean by QCAs in this review
The input and readout of data occurs only on the end qubits of the line. Later it was shown in [88] that two species in an ...ABAB... pattern suffices for universal quantum computation. These kinds of models are what some authors mean by the name quantum cellular automata, whereas, from our point of view, a better name would be classically controlled QCAs, as they typically involve different unitaries applied at each timestep
There is, e.g., no welldeveloped theory of error correction in quantum cellular automata, in contrast to the quantum circuit model. (It may be straightforward to develop such a theory, and it would be interesting to see if it could be done without requiring few-qubit operations.) the current trend in building quantum computers is oriented towards quantum computers with a focus on few-qubit addressability in order to use the quantum circuit model
Summary
Cellular automata (CAs) are fascinating systems: despite having extremely simple dynamics, we often see the emergence of highly complex behaviour. Quantum cellular automata (QCAs) are the quantum version of CAs. The initial rough idea can be traced back at least as far as [12], where it was proposed that to simulate quantum physics it makes sense to consider quantum computers as opposed to classical. It was only later that an axiomatic definition of QCAs was given, which captured the spirit of CAs, while ensuring that the dynamics were quantum [19] It was not initially clear how useful this definition was in dimensions greater than one, at least until it was shown in [20] that any QCA defined in this axiomatic way can be written as a local finite-depth circuit by appending local ancillas.
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