General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit...

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Bibliographic Details
Published in:Quantum information processing 2020-10, Vol.19 (10), Article 379
Main Authors: Manighalam, Michael, Kon, Mark
Format: Article
Language:English
Subjects:
Coins
Data Structures and Information Theory
Dirac equation
Fermions
Mathematical Physics
Physics
Physics and Astronomy
Quantum Computing
Quantum Information Technology
Quantum Physics
Spacetime
Spintronics
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Summary:Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.
ISSN:1570-0755
1573-1332
DOI:10.1007/s11128-020-02880-6