Dynamical Triangulation Induced by Quantum Walk

Quentin, Aristote; Eon, Nathanaël; Molfetta, Giuseppe Di

doi:10.3390/sym12010128

Cited by 6 publications

(4 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Other possible extensions are currently being developed: an extension to larger dimensions and the generalization to dynamic triangulations. An example of the latter is a very recent result where the triangulation is changed through Pachner moves, induced by the quantum walker density itself, allowing the surface to transform into any topologically equivalent one [4]. This model extends the quantum walk over triangular lattice, introduced here.…”

Section: Discussion and Perspectivesmentioning

confidence: 83%

Quantum walks, limits and transport equations

Molfetta¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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. Collaborations and publications• Chapter 4 introduce the notion of Plasticity for QW on the infinite line and is only partially resulting from the paper A quantum walk with both a continuous-time limit and a continuous-spacetime limit, co-authored with Pablo Arrighi (AMU), see [25].• Chapter 5 extends the Plastic QW to the grid and refers to the paper Continuous Time Limit of the DTQW in 2D+ 1 and Plasticity, coauthored with Michael Manighalam (University of Boston) and recently accepted in QINP, see [58]. • Sections in Chapter 6 are results from the paper Dirac equation as a quantum walk over the honeycomb and triangular lattices, co-authored with Ivan Marquez, Armando Perez (Universidad de Valencia) and Pablo Arrighi, see [13]. • Section 7.2 generalises results obtained in the paper From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular Quantum Walks, co-authored with Ivan Marquez, Armando Perez and Pablo Arrighi, see [14]. • Section 7.3 refers to a recent result obtained in the paper Dynamical Triangulation Induced by Quantum Walk, co-authored with Quentin Aristote (ENS Paris) and Nathanael Eon, see [4]. • Section 7.3 refers to the paper Grover Search as a Naturally Occurring Phenomenon, co-authored with Mathieu Roget (ENS Lyon), Stephan Guillet (ENS Lyon) and Pablo Arrighi, see [70].

show abstract

Section: Discussion and Perspectivesmentioning

confidence: 83%

Quantum walks, limits and transport equations

Molfetta¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Applications include search algorithms [9][10][11][12] and graph isomorphism algorithms 13 to modeling and simulating quantum [14][15][16][17][18] and classical dynamics 19,20 . These models have sparked various theoretical investigations covering areas in mathematics, computer science, quantum information and statistical mechanics and have been defined in any physical dimensions 21,22 and over several topologies [23][24][25] . QW appear in multiple variants and can be defined on arbitrary graphs.…”

Section: Quantum Control Using Quantum Memorymentioning

confidence: 99%

Quantum control using quantum memory

Roget¹,

Herzog²,

Molfetta³

2020

Sci Rep

Self Cite

View full text Add to dashboard Cite

We propose a new quantum numerical scheme to control the dynamics of a quantum walker in a two dimensional space–time grid. More specifically, we show how, introducing a quantum memory for each of the spatial grid, this result can be achieved simply by acting on the initial state of the whole system, and therefore can be exactly controlled once for all. As example we prove analytically how to encode in the initial state any arbitrary walker’s mean trajectory and variance. This brings significantly closer the possibility of implementing dynamically interesting physics models on medium term quantum devices, and introduces a new direction in simulating aspects of quantum field theories (QFTs), notably on curved manifold.

show abstract

“…On the other hand, the intrinsic randomness that emerges from measuring a quantum system in superposition of states can be used to give guarantees on the randomness of a graph. Quantum walks on graphs are well-known [1] and already lend themselves to many applications, including search algorithms [8], element distinctness [2] and isomorphism tests [6], discretization of differentiable surfaces [3,4] and cryptography [20].…”

Section: Introductionmentioning

confidence: 99%

Growing Random Graphs with Quantum Rules

Jnane

Molfetta²,

Miatto

2020

Electron. Proc. Theor. Comput. Sci.

Self Cite

View full text Add to dashboard Cite

Random graphs are a central element of the study of complex dynamical networks such as the internet, the brain, or socioeconomic phenomena. New methods to generate random graphs can spawn new applications and give insights into more established techniques. We propose two variations of a model to grow random graphs and trees, based on continuous-time quantum walks on the graphs. After a random characteristic time, the position of the walker(s) is measured and new nodes are attached to the nodes where the walkers collapsed. Such dynamical systems are reminiscent of the class of spontaneous collapse theories in quantum mechanics. We investigate several rates of this spontaneous collapse for an individual quantum walker and for multiple non-interacting walkers. We conjecture (and report some numerical evidence) that the models are scale-free.

show abstract

Dynamical Triangulation Induced by Quantum Walk

Cited by 6 publications

References 27 publications

Quantum walks, limits and transport equations

Quantum walks, limits and transport equations

Quantum control using quantum memory

Growing Random Graphs with Quantum Rules

Contact Info

Product

Resources

About