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Spatial search on Johnson graphs by continuous-time quantum walk


Spatial search on graphs is one of the most important algorithmic applications of quantum walks. To show that a quantum-walk-based search is more efficient than a random-walk-based search is a difficult problem, which has been addressed in several ways. Usually, graph symmetries aid in the calculation of the algorithm’s computational complexity, and Johnson graphs are an interesting class regarding symmetries because they are regular, Hamilton-connected, vertex- and distance-transitive. In this work, we show that spatial search on Johnson graphs by continuous-time quantum walk achieves the Grover lower bound \(\pi \sqrt{N}/2\) with success probability 1 asymptotically for every fixed diameter, where N is the number of vertices. The proof is mathematically rigorous and can be used for other graph classes.

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The authors thank the anonymous referees for providing valuable comments. The work of H. Tanaka was supported by JSPS KAKENHI Grant Number JP20K03551. The work of R. Portugal was supported by FAPERJ Grant Number CNE E-26/202.872/2018 and CNPq Grant Number 308923/2019-7.

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Correspondence to Renato Portugal.

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Tanaka, H., Sabri, M. & Portugal, R. Spatial search on Johnson graphs by continuous-time quantum walk. Quantum Inf Process 21, 74 (2022).

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  • Continuous-time quantum walk
  • Spatial quantum search
  • Johnson graph