Evanescence in coined quantum walks
Open accessAbstract
In this paper, we complete the analysis begun by two of the authors in a previous work on the discrete quantum walk on the infinite line (Carteret et al 2003 J. Phys. A: Math. Gen. 36 8775–95 ). We obtain uniformly convergent asymptotics for the 'exponential decay' regions at the leading edges of the main peaks in the Schrödinger (or wave mechanics) picture. This calculation required us to generalize the method of stationary phase and we describe this extension in some detail, including self-contained proofs of all the technical lemmas required. We also rigorously establish the exact Feynman equivalence between the path-integral and wave-mechanics representations for this system using some techniques from the theory of special functions. Taken together with the previous work, we can now prove every theorem by both routes.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
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 [17]
Section 4 discusses evanescent behavior in a coined quantum walk, and the Conclusions describe an analogy with quantum optics. Searches for Compton wavelength, positive-energy relativistic states, and sub-Compton localization found no passage addressing the claimed existence of such states. (LLM, conf=0.77)
Experimental — accuracy not yet measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
Asymptotics and Special Functions
Olver, F. W. J. (1997) · A K Peters/CRC Press eBooks · doi:10.1201/9781439864548 · Cited by 3,368 · see citing works
Andrews, G. E., Askey, R., & Roy, R. (1999) · Cambridge University Press eBooks · doi:10.1017/cbo9781107325937 · Cited by 2,324 · see citing works
Aharonov, Y., Davidovich, L., & Zagury, N. (1993) · Physical Review A · doi:10.1103/physreva.48.1687 · Cited by 1,622 · see citing works
Quantum random walks: An introductory overview
Kempe, J. (2003) · Contemporary Physics · doi:10.1080/00107151031000110776 · Cited by 1,413 · see citing works
Quantum computation and decision trees
Farhi, E., & Gutmann, S. (1998) · Physical Review A · doi:10.1103/physreva.58.915 · Cited by 1,215 · see citing works
From quantum cellular automata to quantum lattice gases
Meyer, D. A. (1996) · Journal of Statistical Physics · doi:10.1007/bf02199356 · Cited by 556 · see citing works
Τemme, N. Μ. (1996) · Wiley eBooks · doi:10.1002/9781118032572 · Cited by 541 · see citing works
An Example of the Difference Between Quantum and Classical Random Walks
Childs, A. M., Farhi, E., & Gutmann, S. (2002) · Quantum Information Processing · doi:10.1023/a:1019609420309 · Cited by 481 · see citing works
Superpositions of time evolutions of a quantum system and a quantum time-translation machine
Aharonov, Y., Anandan, J., Popescu, S., et al. (1990) · Physical Review Letters · doi:10.1103/physrevlett.64.2965 · Cited by 241 · see citing works
Evanescent and real waves in quantum billiards and Gaussian beams
Berry, M. (1994) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/27/11/008 · Cited by 227 · see citing works
Quantum Walk on the LinePreprint
Ashwin, N., & Vishwanath, A. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0010117 · Cited by 200 · see citing works
Quantum walk on the line as an interference phenomenon
Knight, P. L., Roldán, E., & Sipe, J. E. (2003) · Physical Review A · doi:10.1103/physreva.68.020301 · Cited by 176 · see citing works
Optical cavity implementations of the quantum walk
Knight, P. L., Roldán, E., & Sipe, J. E. (2003) · Optics Communications · doi:10.1016/j.optcom.2003.09.024 · Cited by 74 · see citing works
On Asymptotics of Jacobi Polynomials
Chen, L.-C., & Ismail, M. E. H. (1991) · SIAM Journal on Mathematical Analysis · doi:10.1137/0522092 · Cited by 69 · see citing works
Complementarity and quantum walks
Kendon, V. M., & Sanders, B. C. (2005) · Physical Review A · doi:10.1103/physreva.71.022307 · Cited by 65 · see citing works
Three routes to the exact asymptotics for the one-dimensional quantum walk
Carteret, H. A., Ismail, M. E. H., & Richmond, B. (2003) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/36/33/305 · Cited by 52 · see citing works
Self-citationNew generating functions for Jacobi and related polynomials
Srivastava, H. M., & Singhal, J. P. (1973) · Journal of Mathematical Analysis and Applications · doi:10.1016/0022-247x(73)90244-8 · Cited by 42 · see citing works
Spectra of Quantized Walks and a $\sqrt{δε}$ rulePreprint
Szegedy, M. (2004) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0401053 · Cited by 18 · see citing works
Quantum walks and their algorithmic applicationsPreprint
Ambainis, A. (2004) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0403120 · Cited by 16 · see citing works
Quantum Walks On GraphsPreprint
Aharonov, D., Ambainis, A., Kempe, J., & Vazirani, U. V. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0012090 · Cited by 10 · see citing works
Propagating quantum walks: the origin of interference structures
Knight, P. L., Roldán, E., & Sipe, J. E. (2004) · Journal of Modern Optics · doi:10.1080/09500340410001669390 · Cited by 8 · see citing works
Justification of the stationary phase approximation in time–domain asymptotics
McClure, J. P., & Wong, R. (1997) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.1997.0057 · Cited by 5 · see citing works
Flagged first, then by citations
Research works printed but not linked: 4
- [5]5. Ambainis A, Bach E, Nayak A, Vishwanath A and Watrous J 2001 One-Dimensional Quantum Walks: Proc. 33rd ACM Symposium on Theory of Computation (STOC'01) (Association for Computing Machinery, New York, July 2001) pp 37-49
- [10]10. Carteret H A, Ismail M E H and Richmond B 2003 Three routes to the exact asymptotics of the one-dimensional quantum walk J. Phys. A: Math. Gen. 36 8775-95 (Preprint quant-ph/0303105) · arXiv
- [15]15. Feynman R P and Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York: McGraw-Hill)
- [31]31. Saff E B and Varga R S 1979 The sharpness of Lorentz's theorem on incomplete polynomials Trans. Am. Math. Soc. 249 163-86
Cited by
Sorted by citations
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
- Authors
- Carteret, H. A., Richmond, B., & Temme, N. M.
- Affiliations
- Université de Montréal (CA)
- University of Waterloo (CA)
- Centrum Wiskunde & Informatica (NL)
- Venue
- Journal of Physics A: Mathematical and General
- Publisher
- IOP Publishing
- Published
- 2005-09-21
- Volume
- 38
- Issue
- 40
- Pages
- 8641-8665
- ISSN
- 0305-4470 (ISSN-L)
- 1361-6447
- OpenAlex
- W2050628998
- Cited by
- 14
- References
- 22
- Citations by year
- 2025: 1 · 2023: 1 · 2012: 3
- FWCI
- 2.47
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of Physics A Mathematical and General (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- Funders
- Mitacs
- University of Waterloo
- Institut Périmètre de physique théorique
- 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 › Mathematical Physics › Spectral Theory in Mathematical Physics
- Keywords
- discrete-time quantum walks, asymptotic analysis, exponential decay, stationary phase method, path integral, wave mechanics, special functions
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
