Quantum walks on lattices can give rise to one-particle relativistic wave equations in the long-wavelength limit. In going to multiple particles, quantum cellular automata (QCA) are natural generalizations of quantum walks. In one spatial dimension, the quantum walk can be promoted'' to a QCA that, in the long-wavelength limit, gives rise to the Dirac quantum field theory (QFT) for noninteracting fermions. This QCA/QFT correspondence has both theoretical and practical applications, but there are obstacles to similar constructions in two or more spatial dimensions. Here we show that a method of construction employing distinguishable particles confined to the completely antisymmetric subspace yields a QCA in two spatial dimensions that gives rise to the two-dimensional (2D) Dirac QFT. Generalizing to 3D will entail some additional complications, but no conceptual barriers. We examine how this construction evades the no-go'' results in earlier work.
It has been shown that certain quantum walks give rise to relativistic wave equations, such as the Dirac and Weyl equations, in their long-wavelength limits. This intriguing result raises the question of whether something similar can happen in the multiparticle case. We construct a one-dimensional quantum cellular automaton (QCA) model, which matches the quantum walk in the single particle case and which approaches the quantum field theory of free fermions in the long-wavelength limit. However, we show that this class of constructions does not generalize to higher spatial dimensions in any straightforward way and that no construction with similar properties is possible in two or more spatial dimensions. This rules out the most common approaches based on QCAs. We suggest possible methods to overcome this barrier while retaining locality.
Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the
Z8×Z2 subgroup of the Witt group, including the
U(1)2 and
U(1)4 QCAs. The same algebraic structure yields new infinite families of generalized
U(1)2 and
U(1)4 non-Clifford QCAs in spatial dimensions
d=4k−1. We also reformulate the 4-dimensional
w2w3 QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families:
w2nw3m QCAs in dimension
d=2n+3m−1 and
w2w4k−1 QCAs in dimension
d=4k. For the 5-dimensional
w32 and
w23 QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.
Quantum cellular automata (QCA) constitute space and time homogeneous discrete models for quantum field theories (QFTs). Although QFTs are defined without reference to particles, computations are done in terms of Feynman diagrams, which are explicitly interpreted in terms of interacting particles. Similarly, the easiest QCA to construct are quantum lattice gas automata (QLGA). A natural question then is, which QCA are not QLGA? Here we construct a nontrivial example of such a QCA; it provides a simple model in
1+1 dimensions with no particle interpretation at the scale where the QCA dynamics are homogeneous.
The quantum walk is a subset of classical random walks.Considerable research has lately been conducted on quantum walks in relation to quantum computing.The walk may be thought of as a type of quantum cellular automata.In a broader sense, there is a one-to-one link between them.Furthermore, we look at another type of quantum cellular automata that may be thought of as a quantum version of stochastic cellular automata.The quantum walk formalism is a widely used and quite practical paradigm for modelling quantum systems such as Dirac equation simulations, discrete dynamics in both the low and high energy regimes, and the development of a multitude of quantum algorithms.The walker in a QW moves in a quantum superposition of pathways, and the ensuing interference serves as the foundation for a broad range of quantum algorithms, including quantum search, graph isomorphism issues, and rating nodes in a network.To generate a block representation of n-dimensional quantum cellular automata, we apply the following general result to them.As a result, we have demonstrated that their generic, axiomatic formulation nevertheless produces a unifying, operational representation of them.Some non-trivial ramifications are investigated, such as the fact that bijective non-reversible CA are not physical as closed systems, or that quantum information can move faster than classical informationwithin some specified dynamics.Finally, we present n-dimensional quantum cellular automata that can simulate anything else.This means that the initial configuration and local transition rule of any one-dimensional QCA may be stored within the universal QCA's initial configuration.The simulated QCA will then correspond to many phases of the universal QCA.In the sense that each cell in the simulated QCA is represented as a group of nearby cells in the universal QCA, the simulation preserves the topology.Because the encoding is linear, it bears no cost of the calculation.
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.