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Quantum Walk on the Generalized Birkhoff Polytope Graph.

Rafael Cação1, Lucas Cortez1, Ismael de Farias1

  • 1Department of Industrial, Manufacturing & Systems Engineering, Texas Tech University, Lubbock, TX 79430, USA.

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|October 23, 2021
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Summary
This summary is machine-generated.

Quantum walks show significant speedups on specific transportation problems. On generalized Birkhoff polytope graphs, quantum walks achieve a greater than quadratic speedup in mixing time, outperforming classical methods.

Keywords:
countinggeneralized Birkhoff polytopequantum walksamplingtransportation problem

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Area of Science:

  • Quantum computation and information theory
  • Discrete mathematics and graph theory
  • Operations research and optimization

Background:

  • Quantum walks offer potential speedups over classical random walks in various graph structures.
  • Generalized Birkhoff polytope graphs (GBPGs) are relevant to transportation linear programming (TLP) problems.
  • Previous research indicates at most quadratic speedups for quantum walks on many common graph types.

Purpose of the Study:

  • To investigate discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs).
  • To explore quantum speedups for mixing times on a specific subclass of GBPGs arising from TLP.
  • To analyze the impact of initial states on quantum walk mixing times in these graphs.

Main Methods:

  • Numerical simulations of discrete-time quantum walks on GBPGs.
  • Analysis of mixing times for different initial states (single node vs. maximal clique).
  • Comparison of quantum mixing times with classical mixing times for the studied TLP problems.

Main Results:

  • A greater than quadratic quantum speedup in mixing time was observed on a subclass of GBPGs (TLP with two consumers and m suppliers).
  • For a single-node initial state, quantum mixing time is independent of 'm', despite increasing graph diameter.
  • For a maximal clique initial state, quantum mixing time is O(m/ϵ), outperforming the classical O(m^1.5/ϵ).

Conclusions:

  • GBPGs, particularly those from TLP, can exhibit super-quadratic quantum speedups for mixing time.
  • The independence of quantum mixing time from 'm' for a single-node start is a novel finding.
  • Quantum walks provide a provably faster mixing approach for certain transportation optimization problems.