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Complete classification of trapping coins for quantum walks on the two-dimensional square lattice

Kollár, B. and Gilyén, A. and Tkáčová, I. and Kiss, T. and Jex, I. and Štefaňák, M. (2020) Complete classification of trapping coins for quantum walks on the two-dimensional square lattice. Physical Review A, 102 (1). Art. No. 012207. ISSN 2469-9926. https://resolver.caltech.edu/CaltechAUTHORS:20200707-120236607

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Abstract

One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, although the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four-state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four-dimensional coin Hilbert space. The well-known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such an escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the two-dimensional square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or nonexistence of the escaping state and the area covered by the spreading wave packet.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1103/physreva.102.012207DOIArticle
ORCID:
AuthorORCID
Tkáčová, I.0000-0002-4166-8053
Kiss, T.0000-0001-9320-291X
Additional Information:© 2020 American Physical Society. Received 19 February 2020; revised 16 May 2020; accepted 21 May 2020; published 7 July 2020. I.J. and M.Š. received support from the Czech Grant Agency through Grant No. 17-00844S and from MSMT RVO 14000. This publication was funded by the project “Centre for Advanced Applied Sciences,” Registry No. CZ. 02.1.01/0.0/0.0/16_019/0000778, supported by the Operational Programme Research, Development and Education, cofinanced by the European Structural and Investment Funds and the state budget of the Czech Republic. T.K. was supported by the National Research, Development and Innovation Office of Hungary (Projects No. K124351 and No. 2017-1.2.1-NKP-2017-00001). A.G. was supported by ERC Consolidator Grant QPROGRESS and partially supported by QuantERA project QuantAlgo No. 680-91-034, and also by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant No. PHY-1733907). I.T. would like to acknowledge financial support from Technical University of Ostrava under Project No. SP2018/44.
Group:Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
Grantová Agentura České Republiky17-00844S
Ministry of Education, Youth and Sports (Czech Republic)RVO 14000
Operational Programme Research, Development and EducationCZ. 02.1.01/0.0/0.0/16_019/0000778
European Structural and Investment FundsUNSPECIFIED
state of Czech RepublicUNSPECIFIED
National Research, Development and Innovation Office (Hungary)K124351
National Research, Development and Innovation Office (Hungary)2017-1.2.1-NKP-2017-00001
European Research Council (ERC)QPROGRESS
QuantERA680-91-034
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
NSFPHY-1733907
Technical University of OstravaSP2018/44
Issue or Number:1
Record Number:CaltechAUTHORS:20200707-120236607
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200707-120236607
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:104256
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:07 Jul 2020 19:12
Last Modified:07 Jul 2020 19:12

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