The continuous limit of quantum walks (QWs) on the line is revisited through a recently developed method. In all cases but one, the limit coincides with the dynamics of a Dirac fermion coupled to an artificial electric and/or relativistic gravitational field. All results are carefully discussed and illustrated by numerical simulations.

The continuous limit of one-dimensional discrete-time quantum walks with timeand space-dependent coefficients is investigated. A given quantum walk does not generally admit a continuous limit but some families (1-jets) of quantum walks do. All families (1-jets) admitting a continuous limit are identified. The continuous limit is described by a Dirac-like equation or, alternately, a couple of Klein-Gordon equations. Variational principles leading to these equations are also discussed, together with local invariance properties. C 2012 American Institute of Physics.

A new family of discrete-time quantum walks (DTQWs) on the line with an exact discrete $U(N)$ gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual $U(N)$ gauge fields in $2D$ spacetime. A discrete generalization of the usual $U(N)$ curvature is also constructed. An alternate interpretation of these results in terms of superimposed $U(1)$ Maxwell fields and $SU(N)$ gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e. non-quantum) motions in classical $SU(2)$ fields. The results presented in this article constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.Comment: 7 pages, 2 figure

We analyze the simulation of Dirac neutrino oscillations using quantum walks, both in a vacuum and in matter. We show that this simulation, in the continuum limit, reproduces a set of coupled Dirac equations that describe neutrino flavor oscillations, and we make use of this to establish a connection with neutrino phenomenology, thus allowing one to fix the parameters of the simulation for a given neutrino experiment. We also analyze how matter effects for neutrino propagation can be simulated in the quantum walk. In this way, important features, such as the MSW effect, can be incorporated. Thus, the simulation of neutrino oscillations with the help of quantum walks might be useful to illustrate these effects in extreme conditions, such as the solar interior or supernovae.

A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to well-known physics partial differential equations, such as the Dirac equation. We show that these simulation results need not rely on the grid: the Dirac equation in (2+1)-dimensions can also be simulated, through local unitaries, on the honeycomb or the triangular lattice, both of interest in the study of quantum propagation on the non-rectangular grids, as in graphene-like materials. The latter, in particular, we argue, opens the door for a generalization of the Dirac equation to arbitrary discrete surfaces. * pablo.arrighi@univ-amu.fr † giuseppe.dimolfetta@lis-lab.fr ‡ ivan.marquez@uv.es § armando.perez@uv.es

We introduce an analytically treatable discrete time quantum walk in a one-dimensional lattice which combines non-Markovianity and hyperballistic diffusion associated with a Gaussian whose variance, σ 2 t , growing cubicly with time, σ ∝ t 3 . These properties have have been numerically found in several systems, namely tight-binding lattice models. For its rules, our model can be understood as the quantum version of the classical non-Markovian 'elephant random walk' process, for which the quantum coin operator only changes the value of the diffusion constant though, contrarily to the classical coin.a. Introduction. The random walk problem has been a cornerstone in the classical description of systems for which a deterministic approach is either impossible or too complex to be carried out in an efficient way. Equilibrium and non-equilibrium problems like Hamiltonian Monte Carlo, belief propagation, genetic and search algorithms or pricing financial derivatives [1][2][3][4][5] are systematically understood as a random walk in phase-space of the respective system. The first fundamental property of a random walk process, X = {X t }, concerns the time dependence of the variance, σ 2 t ∝ t. Second, because it derives from a Bernoulli process, the random walk, abides by the ubiquitous Markovian property [6], according to which a memoryless random process is defined as a orderly succession of events where the conditional probability distribution of the future state X t (discrete time t > t 0 ) does only depend on its present state, P (X t | X t−1 , . . . , X t0 ) = P (X t | X t−1 ).While in the classical treatment of a Physical system probability is above all a tool for getting quantitative answers, in quantum theory, probability is intrinsic [7] and thus quantum walks emerged as the formal quantum equivalent to random walks [8,9]. Physically, quantum walks describe situations where a quantum particle is moving on a discrete grid, which allows simulating a wide range of transport phenomena [10][11][12][13][14] including the description of some types of topological insulators and yields an important approach in quantum computing processes [15][16][17] . In other words, the particle dynamically explores a large Hilbert space, H P , spanned by its positions on a lattice corresponding to basis states {|l }, (l ∈ Z), that is augmented by a Hilbert space, H C , spanned by the particle internal statese.g. a two-dimensional basis {|↑ , |↓ }. The evolution of a quantum walk on the full Hilbert space, H ≡ H C ⊗ H P , is ruled by the combined application of two unitary operatorsÛwhereÎ is the identity operator on the H P subspace. Bear- * giuseppe.dimolfetta@lis-lab.fr ing in mind the analogy of quantum walks with the classical random walk, the operatorĈ acts on subspace H C and plays the same role as the coin. For that reason, it is named quantum coin and the internal states related to the subspace H C the coin states. On the other hand, the shift operator, S, is state-dependent and following Ref.[8] readŝAssuming the quantum coin...

We analyze the properties of a two and three dimensional quantum walk that are inspired by the idea of a brane-world model put forward by Rubakov and Shaposhnikov [1]. In that model, particles are dynamically confined on the brane due to the interaction with a scalar field. We translated this model into an alternate quantum walk with a coin that depends on the external field, with a dependence which mimics a domain wall solution. As in the original model, fermions (in our case, the walker), become localized in one of the dimensions, not from the action of a random noise on the lattice (as in the case of Anderson localization), but from a regular dependence in space. On the other hand, the resulting quantum walk can move freely along the "ordinary" dimensions.

Discrete-time quantum walks (DTQWs) in random artificial electric and gravitational fields are studied analytically and numerically. The analytical computations are carried by a new method which allows a direct exact analytical determination of the equations of motion obeyed by the average density operator. It is proven that randomness induces decoherence and that the quantum walks behave asymptotically like classical random walks. Asymptotic diffusion coefficients are computed exactly. The continuous limit is also obtained and discussed.