Abstract In the same context of lattice gauge theory, a notion of cellular automaton (CA) is considered as a description of the underlying dynamics of fields evaluated on a discrete lattice. The explorative approach of this work is initiated by taking the automata evolution as an elementary description of fields dynamics on one side and pivotal role of locality on other side to connect the notion of states and transformations to density operators in quantum electrodynamics. In order to parameterize the definition of proposed model in the context of lattice field theory the nonlinear interaction is introduced with neighboring cells. In this paper, analytical and explicit expression of atomic density matrix for single trapped ion interacting with laser field based on cellular automata dynamics are discussed. The definition is further explored by introducing some probabilistic theories as the theoretical framework then it is concluded that the graph of cellular automata in continuum time limit of the CA dynamics leads to Maxwell–Bloch equation.
Abstract In the same context of lattice gauge theory, a notion of cellular automaton (CA) is considered as a description of the underlying dynamics of fields evaluated on a discrete lattice. The explorative approach of this work is initiated by taking the automata evolution as an elementary description of fields dynamics on one side and pivotal role of locality on other side to connect the notion of states and transformations to density operators in quantum electrodynamics. In order to parameterize the definition of proposed model in the context of lattice field theory the nonlinear interaction is introduced with neighboring cells. In this paper, analytical and explicit expression of atomic density matrix for single trapped ion interacting with laser field based on cellular automata dynamics are discussed. The definition is further explored by introducing some probabilistic theories as the theoretical framework then it is concluded that the graph of cellular automata in continuum time limit of the CA dynamics leads to Maxwell–Bloch equation.
We consider discrete and integer-valued cellular automata (CA). A particular class of which comprises "Hamiltonian CA" with equations of motion that bear similarities to Hamilton's equations, while they present discrete updating rules. The dynamics is linear , quite similar to unitary evolution described by the Schrödinger equation. This has been essential in our construction of an invertible map between such CA and continuous quantum mechanical models, which incorporate a fundamental discreteness scale. Based on Shannon's sampling theory, it leads, for example, to a one-to-one relation between quantum mechanical and CA conservation laws. The important issue of linearity of the theory is examined here by incorporating higher-order nonlinearities into the underlying action. These produce inconsistent nonlocal (in time) effects when trying to describe continuously such nonlinear CA. Therefore, in the present framework, only linear CA and local quantum mechanical dynamics are compatible.
A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas. KEY WORDS: quantum cellular automaton; quantum lattice gas; quantum computation. to appear in J. Stat. Phys.
We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector.
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 interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.
We construct and analyze an explicit one-dimensional quantum cellular automaton (QCA) that provides a controlled lattice regularization of the free Dirac equation in (1+1) dimensions, and we extend it to a locally U(1) gauge-covariant version. The microscopic model is strictly local, causal in the Lieb–Robinson sense, and unitarily evolves a chain of finite-dimensional quantum systems by a translation invariant, discrete-time rule. At the representation-theoretic level, it realizes a chiral spin-1/2 degree of freedom per site. We prove that, in the joint limit of small lattice spacing and small time step with fixed physical scales, the QCA dynamics converges to the continuum Dirac evolution, with an explicit bound on the error valid for smooth, band-limited initial states. We then couple the QCA to a compact U(1) gauge field living on links, in a way that is exactly gauge-covariant at the microscopic level. A local Gauss constraint defines a gauge-invariant code subspace, and we show that the QCA update preserves this subspace and implements a gauge-covariant discrete Dirac dynamics. This provides a concrete example of a gauge-coded QCA with a controlled continuum limit, which can serve as a microscopic building block for more elaborate SU(3) × SU(2) × U(1) constructions. All results are stated with modest but explicit hypotheses and proofs, so that the framework is mathematically well-defined and amenable to further generalization.
This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It 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. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model 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: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.