Quantum Field as a quantum cellular automaton: the Dirac free evolution in one dimension

We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of homogeneity, parity and time-reversal invariance. The comparison between the automaton and the Dirac evolutions is rigorously set as a discrimination problem between unitary channels. We derive an exact lower bound for the probability of error in the discrimination as an explicit function of the mass, the number and the momentum of the particles, and the duration of the evolution. Computing this bound with experimentally achievable values, we see that in that regime the QCA model cannot be discriminated from the usual Dirac evolution. Finally, we show that the evolution of one-particle states with narrow-band in momentum can be effi- ciently simulated by a dispersive differential equation for any regime. This analysis allows for a comparison with the dynamics of wave-packets as it is described by the usual Dirac equation. This paper is a first step in exploring the idea that quantum field theory could be grounded on a more fundamental quantum cellular automaton model and that physical dynamics could emerge from quantum information processing. In this framework, the discretization is a central ingredient and not only a tool for performing non-perturbative calculation as in lattice gauge theory. The automaton model, endowed with a precise notion of local observables and a full probabilistic interpretation, could lead to a coherent unification of an hypothetical discrete Planck scale with the usual Fermi scale of high-energy physics.Comment: 21 pages, 4 figure

The Dirac Quantum Cellular Automaton in one dimension: Zitterbewegung and scattering from potential

We study the dynamical behaviour of the quantum cellular automaton of Refs. [1, 2], which reproduces the Dirac dynamics in the limit of small wavevectors and masses. We present analytical evaluations along with computer simulations, showing how the automaton exhibits typical Dirac dynamical features, as the Zitterbewegung and the scattering behaviour from potential that gives rise to the so-called Klein paradox. The motivation is to show concretely how pure processing of quantum information can lead to particle mechanics as an emergent feature, an issue that has been the focus of solid-state, optical and atomic-physics quantum simulator.Comment: 8 pages, 7 figure

Data-Driven Inference, Reconstruction, and Observational Completeness of Quantum Devices

The range of a quantum measurement is the set of output probability distributions that can be produced by varying the input state. We introduce data-driven inference as a protocol that, given a set of experimental data as a collection of output distributions, infers the quantum measurement which is, i) consistent with the data, in the sense that its range contains all the distributions observed, and, ii) maximally noncommittal, in the sense that its range is of minimum volume in the space of output distributions. We show that data-driven inference is able to return a measurement up to symmetries of the state space (as it is solely based on observed distributions) and that such limit accuracy is achieved for any data set if and only if the inference adopts a (hyper)-spherical state space (for example, the classical or the quantum bit). When using data-driven inference as a protocol to reconstruct an unknown quantum measurement, we show that a crucial property to consider is that of observational completeness, which is defined, in analogy to the property of informational completeness in quantum tomography, as the property of any set of states that, when fed into any given measurement, produces a set of output distributions allowing for the correct reconstruction of the measurement via data-driven inference. We show that observational completeness is strictly stronger than informational completeness, in the sense that not all informationally complete sets are also observationally complete. Moreover, we show that for systems with a (hyper)-spherical state space, the only observationally complete simplex is the regular one, namely, the symmetric informationally complete set.Comment: 15 pages, 12 figures, minor update

Solutions of a two-particle interacting quantum walk

We study the solutions of the interacting Fermionic cellular automaton introduced in Ref. [Phys Rev A 97, 032132 (2018)]. The automaton is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particles solutions of the automaton, which exploits the symmetries of the evolution operator. In the two-particles sector, the evolution operator is given by the sequence of two steps, the first one corresponding to a unitary interaction activated by two-particle excitation at the same site, and the second one to two independent one-dimensional Dirac quantum walks. The interaction step can be regarded as the discrete-time version of the interacting term of some Hamiltonian integrable system, such as the Hubbard or the Thirring model. The present automaton exhibits scattering solutions with nontrivial momentum transfer, jumping between different regions of the Brillouin zone that can be interpreted as Fermion-doubled particles, in stark contrast with the customary momentum-exchange of the one dimensional Hamiltonian systems. A further difference compared to the Hamiltonian model is that there exist bound states for every value of the total momentum, and even for vanishing coupling constant. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.Comment: 16 pages, 6 figure

Path-integral solution of the one-dimensional Dirac quantum cellular automaton

Quantum cellular automata have been recently considered as a fundamental approach to quantum field theory, resorting to a precise automaton, linear in the field, for the Dirac equation in one dimension. In such linear case a quantum automaton is isomorphic to a quantum walk, and a convenient formulation can be given in terms of transition matrices, leading to a new kind of discrete path integral that we solve analytically in terms of Jacobi polynomials versus the arbitrary mass parameter.Comment: 5 page

The Feynman problem and Fermionic entanglement: Fermionic theory versus qubit theory

The present paper is both a review on the Feynman problem, and an original research presentation on the relations between Fermionic theories and qubits theories, both regarded in the novel framework of operational probabilistic theories. The most relevant results about the Feynman problem of simulating Fermions with qubits are reviewed, and in the light of the new original results the problem is solved. The answer is twofold. On the computational side the two theories are equivalent, as shown by Bravyi and Kitaev (Ann. Phys. 298.1 (2002): 210-226). On the operational side the quantum theory of qubits and the quantum theory of Fermions are different, mostly in the notion of locality, with striking consequences on entanglement. Thus the emulation does not respect locality, as it was suspected by Feynman (Int. J. Theor. Phys. 21.6 (1982): 467-488).Comment: 46 pages, review about the "Feynman problem". Fixed many typo

Path-sum solution of the Weyl Quantum Walk in 3+1 dimensions

We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group $\mathbb Z^3$, that in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of $\mathbb Z^3$ that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.Comment: 13 page

Informational features of Fermionic systems

n/