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arXiv:2112.11828

Quantum walks, limits and transport equations

Giuseppe Di Molfetta

Published Dec 22, 2021• 1 citation

Abstract

This manuscript gathers and subsumes a long series of works on using QW to simulate transport phenomena. Quantum Walks (QWs) consist of single and isolated quantum systems, evolving in discrete or continuous time steps according to a causal, shift-invariant unitary evolution in discrete space. We start reminding some necessary fundamentals of linear algebra, including the definitions of Hilbert space, tensor state, the definition of linear operator and then we briefly present the principles of quantum mechanics on which this thesis is grounded. After having reviewed the literature of QWs and the main historical approaches to their study, we then move on to consider a new property of QWs, the plasticity. Plastic QWs are those ones admitting both continuous time-discrete space and continuous spacetime time limit. We show that such QWs can be used to quantum simulate a large class of physical phenomena described by transport equations. We investigate this new family of QWs in one and two spatial dimensions, showing that in two dimensions, the PDEs we can simulate are more general and include dispersive terms. We show that the above results do not need to rely on the grid and we prove that such QW-based quantum simulators can be defined on 2-complex simplicia, i.e. triangular lattices. Finally, we extend the above result to any arbitrary triangulation, proving that such QWs coincide in the continuous limit to a transport equation on a general curved surface, including the curved Dirac equation in 2+1 spacetime dimensions.

DOI: 10.48550/arxiv.2112.11828

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This manuscript gathers and subsumes a long series of works on using QW to simulate transport phenomena. Quantum Walks (QWs) consist of single and isolated quantum systems, evolving in discrete or continuous time steps according to a causal, shift-invariant unitary evolution in discrete space. We start reminding some necessary fundamentals of linear algebra, including the definitions of Hilbert space, tensor state, the definition of linear operator and then we briefly present the principles of quantum mechanics on which this thesis is grounded. After having reviewed the literature of QWs and the main historical approaches to their study, we then move on to consider a new property of QWs, the plasticity. Plastic QWs are those ones admitting both continuous time-discrete space and continuous spacetime time limit. We show that such QWs can be used to quantum simulate a large class of physical phenomena described by transport equations. We investigate this new family of QWs in one and two spatial dimensions, showing that in two dimensions, the PDEs we can simulate are more general and include dispersive terms. We show that the above results do not need to rely on the grid and we prove that such QW-based quantum simulators can be defined on 2-complex simplicia, i.e. triangular lattices. Finally, we extend the above result to any arbitrary triangulation, proving that such QWs coincide in the continuous limit to a transport equation on a general curved surface, including the curved Dirac equation in 2+1 spacetime dimensions.

Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and non relativistic regimes (Di Molfetta and Arrighi, 2020). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in (Di Molfetta and Arrighi, 2020) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coins") admit nontrivial continuum limits in 1D+1. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. We also demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk (CTQW) when the coin is allowed to transition through the continuum limit process. Finally, we introduce the Plastic Quantum Walk, or a quantum walk which admits both continuous time and continuous spacetime limits and, as a novel result, we use our 1D+1 results to obtain necessary and sufficient conditions concerning which DTQWs admit plasticity in 2D+1, showing the resulting Hamiltonians. We consider coin operators as general 4 parameter unitary matrices, with parameters which are functions of the lattice step size 𝜖. This dependence on 𝜖 encapsulates all functions of 𝜖 for which a Taylor series expansion in 𝜖 is well defined, making our results very general.

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, such as the Dirac or the Schr\"odinger equation. In this paper, we show how to recover the Dirac equation in

(3+1)

dimensions with a QW evolving in a tetrahedral space. This paves the way to simulate the Dirac equation on a curved space-time. This also suggests an ordered scheme for propagating matter over a spin network, of interest in loop quantum gravity, where matter propagation has remained an open problem.

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 Dirac equation in (1+ 1) curved spacetime). In this paper, we extend these results to arbitrary space dimension and internal degree of freedom. We recover an entire class of PDEs encompassing the massive Dirac equation in (3 + 1) curved spacetime. This means that the metric field can be represented by a field of local unitaries over a lattice.

A Plastic Quantum Walk admits both continuous time and continuous spacetime. The model has been recently proposed by one of the authors in \cite{molfetta2019quantum}, leading to a general quantum simulation scheme for simulating fermions in the relativistic and non relativistic regimes. The extension to two physical dimensions is still missing and here, as a novel result, we demonstrate necessary and sufficient conditions concerning which discrete time quantum walks can admit plasticity, showing the resulting Hamiltonians. We consider coin operators as general

4

parameter unitary matrices, with parameters which are function of the lattice step size

\varepsilon

. This dependence on

\varepsilon

encapsulates all functions of

\varepsilon

for which a Taylor series expansion in

\varepsilon

is well defined, making our results very general.

There are many problems that cannot be solved using current classical comput- ers. One manner to approach a solution of these systems is by using quantum computers. However, building a quantum computer is really challenging from the experimental side. Quantum simulators have been capable to solve some of these problems, as they are realizable experimentally.Discrete Time Quantum Walks (DTQWs) have been proved to be an useful tool to quantum simulate physical systems. In the continuous limit, a family of differential equations can be achieved, in particular, the Dirac equation can be recovered. In this thesis we study QWs as possible schemes for quantum simula- tion. Specifically, we can summarize our results in: i) We introduce a QW-based model in which a brane theory can be simulated in the continuum, opening the possibility to study more general theories with extra dimensions; ii) Elec- tromagnetic gauge invariance in QWs is discussed, presenting some similarities and differences to previous models. This QW model also makes a connection to gauge invariance in lattice gauge theories (LGT); iii) We introduce QWs over non- rectangular lattices, such a triangular or honeycomb structures, for the purpose of simulating the Dirac equation in the continuum. We also extent these models, by introducing local coin operators, that allow us to reproduce the dynamics of quantum particles under a curved space time.

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 wide class of QWs, and show that it leads to all those PDEs corresponding to the Hamiltonian form of the massive curved Dirac equation in (1 + 1) dimensions. Therefore a certain QW, which we make explicit, provides us with a unitary discrete toy model of the electron as 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.