Limiting properties of stochastic quantum walks on directed graphs

Glos, Adam; Miszczak, Jarosław Adam; Ostaszewski, Mateusz

doi:10.1088/1751-8121/aa9a4a

Cited by 5 publications

(9 citation statements)

References 35 publications

(88 reference statements)

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“…We note that it is not true that the number of TSCCs of G L lower bounds dim ker L in general, even with consistent H; for example, see the example of section 2 of aforementioned [9] for which G L has two TSCCs yet the QDS has a single invariant state.…”

Section: Consistent Generatorsmentioning

confidence: 93%

GKSL generators and digraphs: computing invariant states

Androulakis¹,

Wiedemann²

2019

J. Phys. A: Math. Theor.

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In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. In this article we define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup.In order to define this class we provide a characterization of when the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be traceless as demanded by the GKSL Theorem). Moreover, we consider the converse construction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corresponding semigroup. 2

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Section: Consistent Generatorsmentioning

confidence: 93%

GKSL generators and digraphs: computing invariant states

Androulakis¹,

Wiedemann²

2019

J. Phys. A: Math. Theor.

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“…In both models we choose the Hamiltonian H to be the adjacency matrix of the underlying graphḠ. Both models were analyzed in context of propagation [3,7] and convergence [4,5,6].…”

Section: Quantum Stochastic Walksmentioning

confidence: 99%

“…In particular, we demonstrate the examples with all described evolution regimes for quantum stochastic walk, namely local environment interaction, global environment interaction, and nonmoralizing global environment interaction. The examples are based on the results presented in [3,4,5,6,7].…”

Section: Usage Examplesmentioning

confidence: 99%

“…The model is used to study quantum-to-classical transition and to probe the influence of quantum evolution on the efficiency of information processing [3]. Quantum stochastic walks have been widely analysed in context of propagation [3,4,5,6,7], application in computer science [8,9,10] and physics [11,12,13]. In particular, new model of fast-propagating quantum walk on arbitrary directed graph was developed [4,6].…”

Section: Introductionmentioning

confidence: 99%

“…Quantum stochastic walks have been widely analysed in context of propagation [3,4,5,6,7], application in computer science [8,9,10] and physics [11,12,13]. In particular, new model of fast-propagating quantum walk on arbitrary directed graph was developed [4,6]. To this end, we describe a package for Julia programming language developed for the purpose of simulating and analyzing quantum stochastic walks [14] and we compare its efficiency with the existing software.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

QSWalk.jl: Julia package for quantum stochastic walks analysis

Glos

Miszczak

Ostaszewski

2019

Computer Physics Communications

Self Cite

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The paper describes QSWalk.jl package for Julia programming language, developed for the purpose of simulating the evolution of open quantum systems. The package enables the study of quantum procedures developed using stochastic quantum walks on arbitrary directed graphs. We provide a detailed description of the implemented functions, along with a number of usage examples. The package is compared with the existing software offering a similar functionality.Keywords: directed graph, moral graph, quantum walk, open quantum system. PROGRAM SUMMARYProgram Title: QSWalk.jl Licensing provisions: MIT Distribution format: Source code available at https://github.com/QuantumWalks/QSWalk. jl Programming language: Julia Nature of problem: The package implements functions for simulating quantum stochastic walks, including local regime, global regime, and nonmoralizing global regime [1]. It can be used for arbitrary quantum continuous evolution based on GKSL master equation on arbitrary graphs. Solution method: We utilize Expokit routines for fast sparse matrix exponentials on vectors. For dense matrices exponentiation is computed separately, which is faster for small matrices. Restrictions: Currently package requires Julia v0.6.

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Quantum-Mechanical Lossy Transmission Lines-Analysis based on Damped Harmonic Oscillator GKSL Theory

Kumar

Parthasarathy

2021

Few-Body Syst

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Limiting properties of stochastic quantum walks on directed graphs

Cited by 5 publications

References 35 publications

GKSL generators and digraphs: computing invariant states

GKSL generators and digraphs: computing invariant states

QSWalk.jl: Julia package for quantum stochastic walks analysis

Quantum-Mechanical Lossy Transmission Lines-Analysis based on Damped Harmonic Oscillator GKSL Theory

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