Simulations of one quantum system by an other has an implication in realization of quantum machine that can imitate any quantum system and solve problems that are not accessible to classical computers. One of the approach to engineer quantum simulations is to discretize the space-time degree of freedom in quantum dynamics and define the quantum cellular automata (QCA), a local unitary update rule on a lattice. Different models of QCA are constructed using set of conditions which are not unique and are not always in implementable configuration on any other system. Dirac Cellular Automata (DCA) is one such model constructed for Dirac Hamiltonian (DH) in free quantum field theory. Here, starting from a split-step discrete-time quantum walk (QW) which is uniquely defined for experimental implementation, we recover the DCA along with all the fine oscillations in position space and bridge the missing connection between DH-DCA-QW. We will present the contribution of the parameters resulting in the fine oscillations on the Zitterbewegung frequency and entanglement. The tuneability of the evolution parameters demonstrated in experimental implementation of QW will establish it as an efficient tool to design quantum simulator and approach quantum field theory from principles of quantum information theory.
Simulations of one quantum system by an other has an implication in realization of quantum machine that can imitate any quantum system and solve problems that are not accessible to classical computers. One of the approach to engineer quantum simulations is to discretize the space-time degree of freedom in quantum dynamics and define the quantum cellular automata (QCA), a local unitary update rule on a lattice. Different models of QCA are constructed using set of conditions which are not unique and are not always in implementable configuration on any other system. Dirac Cellular Automata (DCA) is one such model constructed for Dirac Hamiltonian (DH) in free quantum field theory. Here, starting from a split-step discrete-time quantum walk (QW) which is uniquely defined for experimental implementation, we recover the DCA along with all the fine oscillations in position space and bridge the missing connection between DH-DCA-QW. We will present the contribution of the parameters resulting in the fine oscillations on the Zitterbewegung frequency and entanglement. The tuneability of the evolution parameters demonstrated in experimental implementation of QW will establish it as an efficient tool to design quantum simulator and approach quantum field theory from principles of quantum information theory.
Dirac Cellular Automata (DCA) provide a framework for simulating Dirac dynamics, yet the rigid coupling between spatial and temporal resolutions can introduce artificial phase-matching symmetries on finite grids that suppress interference phenomena such as Zitterbewegung. In this work, we propose a Split-Step Dirac Cellular Automaton (SDCA) that enables continuous-time Dirac evolution at fixed spatial discretization. By employing a Trotterized fractional-step scheme in the momentum representation, SDCA breaks the phase-matching cancellation present in the standard DCA and recovers the interference dynamics of the continuous-time limit. We benchmark the SDCA through analytical and numerical studies and demonstrate its implementation on IBM Quantum processors. Despite the increased circuit depth required for finer temporal resolution, the NISQ implementation reproduces the characteristic velocity oscillations and entanglement-entropy dynamics of the continuous-time model. We further investigate hardware-topology trade-offs and dynamic circuit implementations of the Quantum Fourier Transform (QFT), highlighting the competing effects of gate errors, measurement, and feed-forward latency. These results demonstrate that SDCA provides a practical framework for improving temporal resolution while maintaining a fixed spatial quantum register, enabling the exploration of relativistic quantum dynamics on near-term quantum devices.
Quantum simulation is an important way to study the Dirac particles in a general situation. Discrete quantum walk (DQW), is a powerful quantum simulation scheme, and implementable in well controllable table-top set-ups. We first identify that the conventional DQW can't exactly simulate Dirac Cellular Automaton (DCA), a discretized theory of free Dirac Hamiltonian (DH). We found some choice of coin parameters of the split-step (SS) DQW, a generalization of DQW can fully simulate single-particle DCA. Next we question whether the same SS-DQW can simulate dynamics of free Dirac particle with extra degrees of freedom like colors, flavors besides the spin or chirality. One such example is Neutrino oscillation. By moving from the U(2) coined SS-DQW to the U(6) coined SS-DQW we have simulated the exact probability profile of Neutrino flavor transitions. We further probe towards simulating single particle massive DH in presence of background potentials and space-time curvature. By using a SS-DQW with position-time dependent coin parameters, and we realize that it will give us an unbounded effective Hamiltonian, at the continuum limit of position-time. So we have introduced a modified version of SS-DQW which will produce a bounded effective Hamiltonian. This modified SS-DQW with U(2) coin operations produces single-particle massive DH in presence of abelian gauge potentials and space-time curvature. Introducing higher dimensional---U(N) coin operations in the modified SS-DQW we can include non-abelian potentials in the same DH. In order to simulate two-particle DH in presence of curved space-time and external potentials, we have used two particle modified SS-DQW, where the shift operations act separately on each particle, the coin operations which act simultaneously on both particles contain all kinds of interactions.
We show that the Dirac quantum cellular automaton [A. Bisio, G. M. D'Ariano, and A. Tosini, Ann. Phys. (N.Y.) 354, 244 (2015)] shares many properties in common with the discrete-time quantum walk. These similarities can be exploited to study the automaton as a unitary process that takes place at regular time steps on a one-dimensional lattice, in the spirit of general quantum cellular automata. In this way, it becomes an alternative to the quantum walk, with a dispersion relation that can be controlled by a parameter that plays a similar role to the coin angle in the quantum walk. The Dirac Hamiltonian is recovered under a suitable limit. We provide two independent analytical approximations to the long-term probability distribution. It is shown that, starting from localized conditions, the asymptotic value of the entropy of entanglement between internal and motional degrees of freedom overcomes the known limit that is approached by the quantum walk for the same initial conditions and is similar to the ones achieved by highly localized states of the Dirac equation.
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.
We propose a Continuous-Time Quantum Walks (CTQW) model for one-dimensional Dirac dynamics simulation with higher-order approximation. Our model bridges CTQW with a discrete-time model called Dirac Cellular Automata (DCA) via Quantum Fourier Transformation (QFT). From our continuous-time model, we demonstrate how varying time intervals and position space sizes affect both quantum entanglement between the internal space and external (position) space of the quantum state and the relativistic effect called Zitterbewegung. We find that the time interval changes the transition range for each site, and the position space sizes affect the value of transition amplitude. Therefore, it shows that the size of spacetime plays a crucial role in the observed quantum entanglement and relativistic phenomena in quantum computers. These results enhance the understanding of the interplay between internal and external spaces in Dirac dynamics through the insights of quantum information theory and enrich the application of the quantum walks-based algorithm.
The thesis focuses on the analytical and the numerical study of the Weyl and Dirac Quantum Cellular Automata (QCAs). A QCA describes the unitary local evolution of a denumerable collection of quantum systems in mutual interaction. The Weyl QCA can be derived under the assumptions of linearity, unitarity, locality, homogeneity and isotropy and one can show that it recovers the relativistic evolution of the Weyl equation in the limit of small wave-vectors. Under these assumptions a QCA can be described in terms of a set of so-called transition matrices. In the specific case of the Weyl automaton the transition matrices generate a semigroup allowing for the explicit computation of the propagator in position space as a sum over paths. The computation of the propagator is simplified by encoding the lattice paths as binary strings, so that the problems reduces to a combinatorial problem for the binary strings. The thesis also addresses the Thirring QCA, featuring an on-site interaction. We provide a starting point for the study of the perturbation theory in this case.