Pablo Arrighi

University of Paris-Saclay · Computer Science

The laws of Physics are time-reversible, making no qualitative distinction between the past and the future—yet we can only go towards the future. This apparent contradiction is known as the ‘arrow of time problem’. Its current resolution states that the future is the direction of increasing entropy. But entropy can only increase towards the future...

The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of being connected or not, and be allowed to merge, split and reconnect coherently in a superposition. Second, te...

Discrete-time quantum walks (QWs) are transportation models of single quantum particles over a lattice. Their evolution is driven through causal and local unitary operators. QWs are a powerful tool for quantum simulation of fundamental physics, as some of them have a continuum limit converging to well-known physics partial differential equations, s...

Discrete-time Quantum Walks (QWs) are transportation models of single quantum particles over a lattice. Their evolution is driven through causal and local unitary operators. QWs are a powerful tool for quantum simulation of fundamental physics, as some of them have a continuum limit converging to well-known physics partial differential equations, s...

We draw the current landscape of quantum algorithms, by classifying about 130 quantum algorithms, according to the fundamental mathematical problems they solve, their real-world applications, the main subroutines they employ, and several other relevant criteria. The primary objectives include revealing trends of algorithms, identifying promising fi...

This work provides a relativistic, digital quantum simulation scheme for both 2 + 1 and 3 + 1 dimensional quantum electrodynamics (QED), based on a discrete spacetime formulation of theory. It takes the form of a quantum circuit, infinitely repeating across space and time, parametrised by the discretization step Δ t = Δ x . Strict causality at each...

We propose a definition of graph subshifts of finite type that can be seen as extending both the notions of subshifts of finite type from classical symbolic dynamics and finitely presented groups from combinatorial group theory. These are sets of graphs that are defined by forbidding finitely many local patterns. In this paper, we focus on the ques...

The laws of Physics are time-reversible, making no qualitative distinction between the past and the future -- yet we can only go towards the future. This apparent contradiction is known as the `arrow of time problem'. Its resolution states that the future is the direction of increasing entropy. But entropy can only increase towards the future if it...

The possibility to describe the laws of the Universe in a computational way seems to be correlated to a principle that the density of information is bounded. This principle, that is dual to that of a finite velocity of information, has already been investigated in Physics, and is correlated to the old idea that there is no way to know a magnitude w...

We extend the circuit model of quantum computation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders and making their geometrical layout explicit: we express the quantum switch and the pola...

We propose a definition of graph subshifts of finite type that can be seen as extending both the notions of subshifts of finite type from classical symbolic dynamics and finitely presented groups from combinatorial group theory. These are sets of graphs that are defined by forbidding finitely many local patterns. In this paper, we focus on the ques...

This work provides a relativistic, digital quantum simulation scheme for $3+1$ quantum electrodynamics (QED), based on a discrete spacetime formulation of theory. It takes the form of a quantum circuit, infinitely repeating across space and time, parameterized by the discretization step $\Delta_t=\Delta_x$. Strict causality is ensured as circuit wi...

Tensors and traceouts are generalised, so that systems can be partitioned according to almost arbitrary logical predicates. One might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeo...

We build a quantum cellular automaton (QCA) which coincides with 1+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+1$$\end{document} QED on its known continuum limit...

Gauge-invariance is a fundamental concept in Physics—known to provide mathematical justification for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts directly in terms of cellular automata. More precisely, the notions of gauge-invariance and gauge-equivalence in cellular automata are for...

The formalism of quantum theory over discrete systems is extended in two significant ways. First, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates. Second, quantum evolutions are generalized to act over network configurations, in such a way that nodes be allowed to merge, spl...

We build a quantum cellular automaton (QCA) which coincides with 1 + 1 QED on its known continuum limits. It consists in a circuit of unitary gates driving the evolution of particles on a one dimensional lattice, and having them interact with the gauge field on the links. The particles are massive fermions, and the evolution is exactly U (1) gauge-...

We extend the circuit model of quantum comuptation so that the wiring between gates is soft-coded within registers inside the gates. The addresses in these registers can be manipulated and put into superpositions. This aims at capturing indefinite causal orders, as well as making their geometrical layout explicit. We show how to implement the quant...

We build a quantum cellular automaton (QCA) which coincides with 1+1 QED on its known continuum limits. It consists in a circuit of unitary gates driving the evolution of particles on a one dimensional lattice, and having them interact with the gauge field on the links. The particles are massive fermions, and the evolution is exactly U(1) gauge-inv...

Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justification for the fundamental forces. Usually, one starts from a non-interactive theory which governs `matter', and features a global symmetry. One then extends the theory so as make the global symmetry into a local one (a.k.a gauge-invariance). We formalise a d...

We provide a robust notion of quantum superpositions of graphs. Quantum superpositions of graphs crucially require node names for their correct alignment, as we demonstrate through a non-signalling argument. Nevertheless, node names are a fiducial construct, serving a similar purpose to the labelling of points through a choice of coordinates in con...

We provide first evidence that under certain conditions, 1/2-spin fermions may naturally behave like a Grover search, looking for topological defects in a material. The theoretical framework is that of discrete-time quantum walks (QWs), i.e., local unitary matrices that drive the evolution of a single particle on the lattice. Some QWs are well know...

Two models are first presented, of a one-dimensional discrete-time quantum walk (DTQW) with temporal noise on the internal degree of freedom (i.e., the coin): (i) a model with both a coin-flip and a phase-flip channel, and (ii) a model with random coin unitaries. It is then shown that both these models admit a common limit in the spacetime continuu...

Gauge-invariance is a fundamental concept in Physics---known to provide mathematical justification for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts directly in terms of Cellular Automata. More precisely, the notions of gauge-invariance and gauge-equivalence in Cellular Automata are f...

Causal Graph Dynamics extend Cellular Automata to arbitrary time-varying graphs of bounded degree. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We add a further physi...

We propose a discrete spacetime formulation of quantum electrodynamics in one dimension (a.k.a the Schwinger model) in terms of quantum cellular automata, i.e. translationally invariant circuits of local quantum gates. These have exact gauge covariance and a maximum speed of information propagation. In this picture, the interacting quantum field th...

Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a quantum walks or quantum cellular automata based) enjoying a relativistic continuous-spacetime limit. We provide a...

Quantum cellular automata are arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata provide a...

Two models are first presented, of one-dimensional discrete-time quantum walk (DTQW) with temporal noise on the internal degree of freedom (i.e., the coin): (i) a model with both a coin-flip and a phase-flip channel, and (ii) a model with random coin unitaries. It is then shown that both these models admit a common limit in the spacetime continuum,...

Gauge-invariance is a mathematical concept that has profound implications in Physics—as it provides the justification of the fundamental interactions. It was recently adapted to the Cellular Automaton (CA) framework, in a restricted case. In this paper, this treatment is generalized to non-abelian gauge-invariance, including the notions of gauge-eq...

We provide the first evidence that under certain conditions, electrons may naturally behave like a Grover search, looking for defects in a material. The theoretical framework is that of discrete-time quantum walks (QW), i.e. local unitary matrices that drive the evolution of a single particle on the lattice. Some of these are well-known to recover...

Gauge-invariance is a mathematical concept that has profound implications in Physics -- known to provide justification for the fundamental interactions -- and has recently been applied to the Cellular Automaton (CA) model in a restricted case. In this paper, this invariance applied to CA is generalized to a case known as non-abelian gauge-invarianc...

A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. In a previous paper, we showed that QWs over the honeycomb and triangular lattices can be used to simulate the Dirac equation. We apply a spacetime coordinate transformation upon the lattice of this QW, and show that...

Nowadays, quantum simulation schemes come in two flavours. Either they are continuous-time discrete-space models (a.k.a Hamiltonian-based), pertaining to non-relativistic quantum mechanics. Or they are discrete-spacetime models (a.k.a Quantum Walks or Quantum Cellular Automata-based) enjoying a relativistic continuous spacetime limit. We provide a...

Consider a network that evolves reversibly, according to nearest neighbours interactions. Can its dynamics create/destroy nodes? On the one hand, since the nodes are the principal carriers of information, it seems that they cannot be destroyed without jeopardising bijectivity. On the other hand, there are plenty of global functions from graphs to g...

Quantum cellular automata consist in arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata pro...

We propose a discrete spacetime formulation of quantum electrodynamics in one-dimension (a.k.a the Schwinger model) in terms of quantum cellular automata, i.e. translationally invariant circuits of local quantum gates. These have exact gauge covariance and a maximum speed of information propagation. In this picture, the interacting quantum field th...

It is shown that discrete-time quantum walks can be used to digitize, i.e., to time discretize fermionic models of continuous-time lattice gauge theory. The resulting discrete-time dynamics is thus not only manifestly unitary, but also ultralocal, i.e., the particle's speed is upper bounded, as in standard relativistic quantum field theories. The l...

A discrete-time Quantum Walk (QW) is an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QW admit, as their continuum limit, a well-known equation of Physics. In arXiv:1803.01015 the QW is over the honeycomb and triangular lattices, and simulates the Dirac equation. We apply a spacetime coordinate tr...

It is shown that discrete-time quantum walks can be used to digitize, i.e., to time discretize fermionic models of continuous-time lattice gauge theory. The resulting discrete-time dynamics is thus not only manifestly unitary, but also ultralocal, i.e. the particle's speed is upper bounded, as in standard relativistic quantum field theories. The la...

Gauge-invariance is a fundamental concept in physics—known to provide mathematical justifications for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts, directly in terms of Cellular Automata. More precisely, we describe a step-by-step gauging procedure to enforce local symmetries upon a...

We extend Cellular Automata to time-varying discrete geometries. In other words we formalize, and prove theorems about, the intuitive idea of a discrete manifold which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same. For this purpose we develop a corresponde...

Consider a network that evolves reversibly, according to nearest neighbours interactions. Can its dynamics create/destroy nodes?. On the one hand, since the nodes are the principal carriers of information, it seems that they cannot be destroyed without jeopardising bijectivity. On the other hand, there are plenty of global functions from graphs to...

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 equati...

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 equati...

Gauge-invariance is a fundamental concept in physics---known to provide the mathematical justification for all four fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts, directly in terms of Cellular Automata. More precisely, we describe a step-by-step gauging procedure to enforce local symmetri...

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of QWs (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit...

A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limi...

We describe a type system for the linear-algebraic λ-calculus. The type system accounts for the linear-algebraic aspects of this extension of λ-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as term...

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 familiar PDEs (e.g. the Dirac equation). Recently it was discovered that prior grouping and encoding allows for more general continuum limit equations (e.g. the...

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 familiar PDEs (e.g. the Dirac equation). Recently it was discovered that prior grouping and encoding allows for more general continuum limit equations (e.g. the...

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 H...

Consider a graph having quantum systems lying at each node. Suppose that the whole thing evolves in discrete time steps, according to a global, unitary causal operator. By causal we mean that information can only propagate at a bounded speed, with respect to the distance given by the graph. Suppose, moreover, that the graph itself is subject to the...

Consider a graph having quantum systems lying at each node. Suppose that the whole thing evolves in discrete time steps, according to a global, unitary causal operator. By causal we mean that information can only propagate at a bounded speed, with respect to the distance given by the graph. Suppose, moreover, that the graph itself is subject to the...

Causal Graph Dynamics extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We study a...

Three reasonable hypotheses lead to the thesis that physical phenomena can be described and simulated with cellular automata. In this work, we attempt to describe the motion of a particle upon which a constant force is applied, with a cellular automaton, in Newtonian physics, in Special Relativity, and in General Relativity. The results are very di...

Three reasonable hypotheses lead to the thesis that physical phenomena can be described and simulated with cellular automata. In this work, we attempt to describe the motion of a particle upon which a constant force is applied, with a cellular automaton, in Newtonian physics, in Special Relativity, and in General Relativity. The results are very di...

This paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computi...

Si l’ordinateur quantique universel est encore lointain, les physiciens espèrent fabriquer bientôt un ordinateur quantique imparfait, capable de simuler des systèmes quantiques. En jeu, la compréhension de systèmes inaccessibles par les moyens habituels.

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), and thus provide us with discrete toy models of familiar particles (e.g. the electron). In this paper, we study the continuum limit of a...

Causal Graph Dynamics extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We study a...

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. translation-invariant and time-independent), and causal (i.e. information propagates at a bounded speed, in a strict sense). This quantum walk mod...

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 introd...

The GOOL system is a lightweight translator between OOP languages (Java, C++, C#, Objective C, ...). It relies upon a minimal, abstract OOP language called GOOL (General Object-Oriented Language) in order to represent the common features between these languages.

We formalize the intuitive idea of a labelled discrete surface which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same.

Decoupling a vectorial PDE consists in solving the system for each component, thereby obtaining scalar PDEs that prescribe the evolution of each component independently. We present a general approach to decoupling of Quantum Walks, again defined as a procedure to obtain an evolution law for each scalar component of the QW, in such a way that it doe...

This paper introduces a simple formalism for dealing with deterministic, non- deterministic and stochastic cellular automata in an unified and composable manner. This formalism allows for local probabilistic correlations, a feature which is not present in usual definitions. We show that this feature allows for strictly more behaviors (for instance,...

The possibility to describe the laws of the Universe in a compu-tational way seems to be correlated to a principle that the density of information is bounded. This principle, that is dual to that of a finite velocity of information, has already been investigated in Physics, and is correlated to the old idea that there is no way to know a magnitude...

Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their equality; to name all vertices relative to a point; the fact that they have a well-defined notion of translation, and that they can be endowed with a compact metric. We propose a notion of graph associated to a language,...

The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. We then provide explicit tool...

We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as...

The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend the lambda-calculus with the possibility of making arbitrary linear combinations of terms. In this paper we provide a fine-grained, System F-like type system for the linear-algebraic lambda-calculus. We show that this "scalar" type system enjoys both the subject-reduction...

We extend the theory of Cellular Automata to arbitrary, time-varying graphs. In other words we formalize, and prove theorems about, the intuitive idea of a labelled graph which evolves in time - but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph. The not...

Reversible Cellular Automata (RCA) are a physics-like model of computation consisting of an array of identical cells, evolving in discrete time steps by iterating a global evolution G. Further, G is required to be shift-invariant (it acts the same everywhere), causal (information cannot be transmitted faster than some fixed number of cells per time...

We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as...

This paper demonstrates how to add a measurement operator to quantum λ-calculi. A proof of the consistency of the semantics is given through a proof of confluence presented in a sufficiently general way to allow this technique to be used for other languages. The method described here may be applied to probabilistic rewrite systems in general, and t...

Notoriously, quantum computation shatters complexity theory, but is innocuous to computability theory. Yet several works have shown how quantum theory as it stands could breach the physical Church-Turing thesis. We draw a clear line as to when this is the case, in a way that is inspired by Gandy. Gandy formulates postulates about physics, such as h...

Cellular automata (CA) consist of an array of identical cells, each of which may take one of a finite number of possible states. The entire array evolves in discrete time steps by iterating a global evolution G. Further, this global evolution G is required to be shift-invariant (it acts the same everywhere) and causal (information cannot be transmi...

The Linear-Algebraic λ-Calculus [Arrighi, P. and G. Dowek, Linear-algebraic λ-calculus: higher-order, encodings and confluence, Lecture Notes in Computer Science (RTA'08) 5117 (2008), pp. 17–31] extends the λ-calculus with the possibility of making arbitrary linear combinations of terms α.t+β.u. Since one can express fixed points over sums in this...

We define and study quantum cellular automata (QCA).We show that they are reversible and that the neighborhood of the inverse is the opposite of the neighborhood. We also show that QCA always admit, modulo shifts, a two-layered block representation. Note that the same two-layered block representation result applies also over infinite configurations...