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A simple coin for a $2d$ entangled walk

by Ahmadullah Zahed, Kallol Sen

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Submission summary

Authors (as registered SciPost users): Kallol Sen
Submission information
Preprint Link: scipost_202302_00030v1  (pdf)
Date accepted: 2023-05-30
Date submitted: 2023-02-17 12:21
Submitted by: Sen, Kallol
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics
Approaches: Theoretical, Computational


We analyze the effect of a simple coin operator, built out of Bell pairs, in a $2d$ Discrete Quantum Random Walk (DQRW) problem. The specific form of the coin enables us to find analytical and closed form solutions to the recursion relations of the DQRW. The coin induces entanglement between the spin and position degrees of freedom, which oscillates with time and reaches a constant value asymptotically. We probe the entangling properties of the coin operator further, by two different measures. First, by integrating over the space of initial tensor product states, we determine the {\it Entangling Power} of the coin operator. Secondly, we compute the {\it Generalized Relative R\'{e}nyi Entropy} between the corresponding density matrices for the entangled state and the initial pure unentangled state. Both the {\it Entangling Power} and {\it Generalized Relative R\'{e}nyi Entropy} behaves similar to the entanglement with time. Finally, in the continuum limit, the specific coin operator reduces the $2d$ DQRW into two $1d$ massive fermions coupled to synthetic gauge fields, where both the mass term and the gauge fields are built out of the coin parameters.

Published as SciPost Phys. Core 6, 048 (2023)

Reports on this Submission

Anonymous Report 1 on 2023-5-24 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202302_00030v1, delivered 2023-05-24, doi: 10.21468/SciPost.Report.7248


Closed-form solution for a problem of wide applicability, which may well simplify future calculations.


The scope of the paper is somewhat limited, it remains to be seen if indeed this closed form solution is practical.


The manuscript presents a closed-form solution, in terms of a special function, of a problem of wide applicability: The quantum walk in discrete time on a two-dimensional lattice. The closed-form solution of the 1D problem is known, I have not found the 2D case in the literature. So I do think that this is a valuable contribution to the literature. The topic is timely (in particular, with applications in the context of quantum search algorithms). I find this work suitable for publication in SciPost Physics Core.

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: good

Author:  Kallol Sen  on 2023-05-25  [id 3683]

(in reply to Report 1 on 2023-05-24)

The authors thank the reviewer for insightful comments on the work. The practical relevance of the exact solutions in 2d lies in the context of the algorithm. We have focussed on the question of finding a coin operator that induces entanglement in the 2d walk. A coin built from minimal set of bell pairs admits analytic solution while preserving the versatile dynamical aspects of the walk. This is precisely the context in which the analytical solutions become appealing. This serves as a building block for constructing more generalized walk algorithms in higher dimensions and for quantum searches. Also, with analytical solutions, essential dynamical features of the walk e.g. entanglement, entropy and entangling power are extrapolated for asymptotic expansions leading to interesting continuum limits connecting quantum field theories with real time simulations of their discrete versions.

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