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Simple quantum coins enable pretty good state transfer on every hypercube.

Hanmeng Zhan1

  • 1Computer Science, Worcester Polytechnic Institute, Worcester, MA, USA.

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|October 9, 2025
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Summary
This summary is machine-generated.

Researchers developed weighted Grover coins for pretty good state transfer in quantum walks on hypercubes, applicable to all dimensions. This advances quantum algorithms and state transfer capabilities on various graphs.

Keywords:
coined quantum walkdiscrete quantum walkgraph spectrahypercubepretty good state transfer

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Area of Science:

  • Quantum Information Science
  • Graph Theory
  • Quantum Algorithms

Background:

  • Pretty good state transfer is crucial for quantum information processing.
  • Previous work established state transfer for prime dimensions on hypercubes using Grover coins.
  • Generalizing state transfer to all dimensions on hypercubes remained an open challenge.

Purpose of the Study:

  • To develop novel weighted Grover coins for achieving pretty good state transfer on hypercubes of any dimension.
  • To generalize the methodology for enabling state transfer on other graph structures.
  • To establish a sufficient condition for pretty good state transfer in quantum walks.

Main Methods:

  • Construction of real-valued weighted Grover coins.
  • Analysis of coined quantum walks on hypercube graphs.
  • Generalization of the coin construction and analysis to arbitrary graph structures.

Main Results:

  • Successfully designed weighted Grover coins enabling pretty good state transfer on hypercubes for all dimensions.
  • Demonstrated that only a single arc weight modification per vertex is necessary.
  • Established a general condition for achieving pretty good state transfer on other graphs.

Conclusions:

  • The developed weighted Grover coins provide a universal method for pretty good state transfer on hypercubes.
  • The findings offer a significant advancement in quantum walk algorithms and state transfer protocols.
  • The generalized approach opens new avenues for quantum information processing on diverse graph architectures.