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Quantum Walks: Applications and Fundamentals

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Quantum Information".

Deadline for manuscript submissions: closed (31 March 2022) | Viewed by 7550

Special Issue Editor


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Guest Editor
CNRS, LIS, Aix-Marseille Université, 13000 Marseille, France
Interests: natural computing; discrete dynamical system; quantum simulation; quantum algorithms

Special Issue Information

Dear Colleagues,

Quantum walks (QWs), and their generalizations, the multiparticle quantum lattice gas automata (QLGA), and the quantum cellular automata (QCA), underlie simulations of physics, many a search algorithm, and are participants in aspects of quantum machine learning (QML). The emergence of quantum devices presents opportunities to extend the reach of classical computing through quantum algorithms. Often, techniques developed to simulate fundamental physics, among them QW and QCA, find applications in quantum/hybrid algorithms and accelerate optimizations. In the current noisy intermediate-scale quantum (NISQ) regime of quantum devices, competing criteria of noise and decoherence and the availability of limited resources must be considered in simulations, and QWs are no exception. Circuits for QWs will avail quantum error correction (QEC)-based schemes for fault tolerance in the future, when a higher number of qubits, error-corrected, are available.

Considering the recent advances achieved in the field of QW, this Special Issue will collect new ideas and describe promising methods arising from the field of QW-based quantum computation and quantum simulation.

This Special Issue will accept unpublished original papers and comprehensive reviews focused on (but not restricted to) the following research areas, where QW and QCA play a main role:

  • QW- or (QCA-) based quantum simulation;
  • QW-based quantum algorithmics with and without QEC;
  • Fault-tolerant quantum cellular automata;
  • QW and their application to quantum machine learning;
  • QW and their application to quantum communication and cryptography.

Dr. Giuseppe Di Molfetta
Guest Editor

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Keywords

  • quantum walks
  • quantum cellular automata
  • quantum simulation
  • quantum algorithmics

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Published Papers (3 papers)

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12 pages, 278 KiB  
Article
Return Probability of Quantum and Correlated Random Walks
by Chusei Kiumi, Norio Konno and Shunya Tamura
Entropy 2022, 24(5), 584; https://doi.org/10.3390/e24050584 - 21 Apr 2022
Cited by 3 | Viewed by 1698
Abstract
The analysis of the return probability is one of the most essential and fundamental topics in the study of classical random walks. In this paper, we study the return probability of quantum and correlated random walks in the one-dimensional integer lattice by the [...] Read more.
The analysis of the return probability is one of the most essential and fundamental topics in the study of classical random walks. In this paper, we study the return probability of quantum and correlated random walks in the one-dimensional integer lattice by the path counting method. We show that the return probability of both quantum and correlated random walks can be expressed in terms of the Legendre polynomial. Moreover, the generating function of the return probability can be written in terms of elliptic integrals of the first and second kinds for the quantum walk. Full article
(This article belongs to the Special Issue Quantum Walks: Applications and Fundamentals)
11 pages, 3216 KiB  
Article
Dirac Spatial Search with Electric Fields
by Julien Zylberman and Fabrice Debbasch
Entropy 2021, 23(11), 1441; https://doi.org/10.3390/e23111441 - 31 Oct 2021
Cited by 3 | Viewed by 2197
Abstract
Electric Dirac quantum walks, which are a discretisation of the Dirac equation for a spinor coupled to an electric field, are revisited in order to perform spatial searches. The Coulomb electric field of a point charge is used as a non local oracle [...] Read more.
Electric Dirac quantum walks, which are a discretisation of the Dirac equation for a spinor coupled to an electric field, are revisited in order to perform spatial searches. The Coulomb electric field of a point charge is used as a non local oracle to perform a spatial search on a 2D grid of N points. As other quantum walks proposed for spatial search, these walks localise partially on the charge after a finite period of time. However, contrary to other walks, this localisation time scales as N for small values of N and tends asymptotically to a constant for larger Ns, thus offering a speed-up over conventional methods. Full article
(This article belongs to the Special Issue Quantum Walks: Applications and Fundamentals)
Show Figures

Figure 1

Figure 1
<p>Evolution of <math display="inline"><semantics> <msubsup> <mi>P</mi> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> with time <span class="html-italic">j</span> for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>120</mn> <mn>2</mn> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>right</b>).</p>
Full article ">Figure 2
<p>Evolutions of <math display="inline"><semantics> <msubsup> <mi>P</mi> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> with time <span class="html-italic">j</span> for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>120</mn> <mn>2</mn> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and mass <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.08</mn> </mrow> </semantics></math> (<b>left</b>) or <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math> (<b>right</b>).</p>
Full article ">Figure 3
<p>Density profiles at time <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>61</mn> </mrow> </semantics></math> corresponding to the first maximum of <math display="inline"><semantics> <msubsup> <mi>P</mi> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> (<b>up</b>) and at time <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>111</mn> </mrow> </semantics></math> corresponding to the first minimum of <math display="inline"><semantics> <msubsup> <mi>P</mi> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> (<b>down</b>) for <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>120</mn> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Density profiles at time <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>47</mn> </mrow> </semantics></math> corresponding to the first maximum of <math display="inline"><semantics> <msubsup> <mi>P</mi> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> (<b>up</b>) and time <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>137</mn> </mrow> </semantics></math> corresponding to the first minimum of <math display="inline"><semantics> <msubsup> <mi>P</mi> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> (<b>down</b>) for <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>120</mn> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>Density profiles at times <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (<b>up</b>) and <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> (<b>down</b>) for <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>120</mn> <mn>2</mn> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Density contours at time <math display="inline"><semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>61</mn> </mrow> </semantics></math> corresponding to the first maximum of <math display="inline"><semantics> <msubsup> <mi>P</mi> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>120</mn> <mn>2</mn> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math> and two different initial conditions: <math display="inline"><semantics> <mrow> <msup> <mi>ψ</mi> <mi>L</mi> </msup> <mo>=</mo> <msup> <mi>ψ</mi> <mi>R</mi> </msup> </mrow> </semantics></math> (<b>up</b>) and <math display="inline"><semantics> <mrow> <msup> <mi>ψ</mi> <mi>R</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>down</b>).</p>
Full article ">Figure 7
<p>Time <span class="html-italic">T</span> for the first maximum of <math display="inline"><semantics> <msubsup> <mrow> <mover accent="true"> <mi>P</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> in function of <math display="inline"><semantics> <msqrt> <mi>N</mi> </msqrt> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The function fitting <span class="html-italic">T</span> for small <span class="html-italic">N</span> is approximately <math display="inline"><semantics> <mrow> <mn>0.96</mn> <msqrt> <mi>N</mi> </msqrt> <mo>−</mo> <mn>1.66</mn> </mrow> </semantics></math> and appears in yellow.</p>
Full article ">Figure 8
<p>Renormalised probability <math display="inline"><semantics> <msubsup> <mrow> <mover accent="true"> <mi>P</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mi>j</mi> <mo>*</mo> </msubsup> </semantics></math> against time <span class="html-italic">j</span> for <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>Q</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <msub> <mo>Ω</mo> <mn>0</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>30</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (blue), <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>46</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (orange), <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>60</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (green) and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>76</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (red), <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>90</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (navy blue), <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>106</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (brown), <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>120</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (cyan), <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>180</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (yellow) and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mn>240</mn> <mn>2</mn> </msup> </mrow> </semantics></math> (purple).</p>
Full article ">
13 pages, 761 KiB  
Article
Quantum Walk on the Generalized Birkhoff Polytope Graph
by Rafael Cação, Lucas Cortez, Ismael de Farias, Ernee Kozyreff, Jalil Khatibi Moqadam and Renato Portugal
Entropy 2021, 23(10), 1239; https://doi.org/10.3390/e23101239 - 23 Sep 2021
Viewed by 2328
Abstract
We study discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs), which arise in the solution-set to certain transportation linear programming problems (TLPs). It is known that quantum walks mix at most quadratically faster than random walks on cycles, two-dimensional lattices, hypercubes, and [...] Read more.
We study discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs), which arise in the solution-set to certain transportation linear programming problems (TLPs). It is known that quantum walks mix at most quadratically faster than random walks on cycles, two-dimensional lattices, hypercubes, and bounded-degree graphs. In contrast, our numerical results show that it is possible to achieve a greater than quadratic quantum speedup for the mixing time on a subclass of GBPG (TLP with two consumers and m suppliers). We analyze two types of initial states. If the walker starts on a single node, the quantum mixing time does not depend on m, even though the graph diameter increases with it. To the best of our knowledge, this is the first example of its kind. If the walker is initially spread over a maximal clique, the quantum mixing time is O(m/ϵ), where ϵ is the threshold used in the mixing times. This result is better than the classical mixing time, which is O(m1.5/ϵ). Full article
(This article belongs to the Special Issue Quantum Walks: Applications and Fundamentals)
Show Figures

Figure 1

Figure 1
<p>Classical mixing time as a function of the number of sources <span class="html-italic">m</span> for <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> from <math display="inline"><semantics> <mrow> <mn>0.01</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 2
<p>Limiting distribution <math display="inline"><semantics> <msub> <mi>π</mi> <mi>v</mi> </msub> </semantics></math> of a quantum walk on GBPG for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> (left panel) and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> (right panel), and initial state (15) localized on node 1.</p>
Full article ">Figure 3
<p>Limiting distribution <math display="inline"><semantics> <msub> <mi>π</mi> <mi>v</mi> </msub> </semantics></math> of a quantum walk on GBPG for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> (left panel) and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> (right panel), using an initial state distributed on the clique <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Log-log plot of the total variation distance between the instantaneous and limiting distributions <math display="inline"><semantics> <mrow> <mo>∥</mo> <mi>p</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>−</mo> <mi>π</mi> <mo>∥</mo> </mrow> </semantics></math> (upper black curve), and between the average and limiting distributions <math display="inline"><semantics> <mrow> <mo>∥</mo> <mover accent="true"> <mi>p</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>π</mi> <mo>∥</mo> </mrow> </semantics></math> (lower blue curve) as a function of time steps for GBPG for <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>. The equation of the straight line is <math display="inline"><semantics> <mrow> <mn>0.7761</mn> <mo>/</mo> <msup> <mi>t</mi> <mrow> <mn>1.0007</mn> </mrow> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>Quantum mixing time as a function of the number of sources <span class="html-italic">m</span> for <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> from <math display="inline"><semantics> <mrow> <mn>0.01</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Quantum mixing time as a function of the number of sources <span class="html-italic">m</span> for <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> from <math display="inline"><semantics> <mrow> <mn>0.01</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>0.1</mn> </mrow> </semantics></math> using the alternative initial state (16).</p>
Full article ">Figure 7
<p>Instantaneous mixing time to the limiting distribution as a function of <span class="html-italic">m</span> for <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> from <math display="inline"><semantics> <mrow> <mn>0.06</mn> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mn>0.4</mn> </mrow> </semantics></math> using the initial state (16).</p>
Full article ">
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