The Dirac equation can be modelled as a quantum walk, with the quantum walk being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translationinvariant and time-independent), and causal (i.e. information propagates at a bounded speed, in a strict sense). This quantum walk model was proposed independently by Succi and Benzi, Bialynicki-Birula and Meyer: we rederive it in a simple way in all dimensions and for hyperbolic symmetric systems in general. We then prove that for any time t, the model converges to the continuous solution of the Dirac equation at time t, i.e. the probability of observing a discrepancy between the model and the solution is an O(ε 2 ), with ε the discretization step. At the practical level, this result is of interest for the quantum simulation of relativistic particles. At the theoretical level, it reinforces the status of this quantum walk model as a simple, discrete toy model of relativistic particles.

A discrete-time Quantum Walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we study the continuum limit of a wide class of QWs, and show that it leads to an entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac equation in (1 + 1) curved spacetime. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of a test particle in curved spacetime, in spite of the fixed background lattice. Mathematically we have introduced two novel ingredients for taking the continuum limit of a QW, but which apply to any quantum cellular automata: encoding and grouping.

Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in the industrial setting. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach set computations such that set operations are performed in low dimensions, while matrix operations like exponentiation are carried out in the full dimension. Our method is applicable both in dense-and discrete-time settings. For a set of standard benchmarks, it shows a speed-up of up to two orders of magnitude compared to the respective state-of-the art tools, with only modest losses in accuracy. For the dense-time case, we show an experiment with more than 10.000 variables, roughly two orders of magnitude higher than possible with previous approaches.

We formalize a notion of discrete Lorentz transforms for quantum walks (QW) and quantum cellular automata (QCA), in +(1 1)-dimensional discrete spacetime. The theory admits a diagrammatic representation in terms of a few local, circuit equivalence rules. Within this framework, we show the first-order-only covariance of the Dirac QW. We then introduce the clock QW and the clock QCA, and prove that they are exactly discrete Lorentz covariant. The theory also allows for non-homogeneous Lorentz transforms, between non-inertial frames.