Three routes to the exact asymptotics for the one-dimensional quantum walk
Open accessAbstract
We demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path-integral representation. We calculate the asymptotics using a method that is significantly easier to use than the Darboux method. It also provides a single integral representation for the wavefunction that works over the full range of positions, n , including throughout the transitional range where the behaviour changes from oscillatory to exponential. Previous analyses of this system have run into difficulties in the transitional range, because the approximations on which they were based break down here. The fact that there are two different kinds of approach to this problem (path integral versus Schrödinger wave mechanics) is ultimately a manifestation of the equivalence between the path-integral formulation of quantum mechanics and the original formulation developed in the 1920s. We also discuss how and why our approach is related to the two methods that have already been used to analyse these systems.
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Positive-energy states of a relativistic particle localized beneath its Compton wavelength exist.
“These examples demonstrate the existence of positive-energy states of a relativistic particle localized beneath its Compton wavelength [17], despite well-known claims to the contrary.”
From Relativistic quantum walks (2006) · cites this paper as [16]
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Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., & Watrous, J. (2001) · Proceedings of the thirty-third annual ACM symposium on Theory of computing · doi:10.1145/380752.380757 · Cited by 562 · see citing works
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QUANTUM WALKS AND THEIR ALGORITHMIC APPLICATIONS
AMBAINIS, A. (2003) · International Journal of Quantum Information · doi:10.1142/s0219749903000383 · Cited by 632 · see citing works
Decoherence in quantum walks – a review
KENDON, V. (2007) · Mathematical Structures in Computer Science · doi:10.1017/s0960129507006354 · Cited by 252 · see citing works
Strauch, F. W. (2006) · Physical Review A · doi:10.1103/physreva.73.054302 · Cited by 115 · see citing works
Relativistic effects and rigorous limits for discrete- and continuous-time quantum walks
Strauch, F. W. (2007) · Journal of Mathematical Physics · doi:10.1063/1.2759837 · Cited by 54 · see citing works
The meeting problem in the quantum walk
Štefanák, M., Kiss, T., Jex, I., & Mohring, B. (2006) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/39/48/009 · Cited by 24 · see citing works
Position-dependent and cooperative quantum Parrondo walks
Bulger, D., Freckleton, J., & Twamley, J. (2008) · New Journal of Physics · doi:10.1088/1367-2630/10/9/093014 · Cited by 20 · see citing works
Evanescence in coined quantum walks
Carteret, H. A., Richmond, B., & Temme, N. M. (2005) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/38/40/011 · Cited by 14 · see citing works
Self-citation
- Authors
- Carteret, H. A., Ismail, M. E. H., & Richmond, B.
- Affiliations
- University of Waterloo (CA)
- University of Central Florida (US)
- ORCID
- Ismail, Mourad E H
- Venue
- Journal of Physics A: Mathematical and General
- Publisher
- IOP Publishing
- Published
- 2003-08-05
- Volume
- 36
- Issue
- 33
- Pages
- 8775-8795
- ISSN
- 0305-4470 (ISSN-L)
- 1361-6447
- OpenAlex
- W1986225399
- Cited by
- 52
- References
- 2
- Citations by year
- 2025: 2 · 2023: 1 · 2022: 2 · 2021: 1 · 2020: 2 · 2018: 2 · 2017: 1 · 2014: 3 · 2013: 1 · 2012: 2
- FWCI
- 3.55
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of Physics A Mathematical and General (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Journal of International Crisis and Risk Communication Research (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9970865)
- Mitacs
- Natural Sciences and Engineering Research Council of Canada
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Keywords
- asymptotic behavior, Jacobi polynomials, path integral representation, Darboux method, wave functions, wave mechanics, infinite line
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