Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms
Quantum lattice algorithms originated with the Feynman checkerboard model for the one-dimensional Dirac equation. They offer discrete models of quantum mechanics in which the complex numbers representing wavefunction values on a discrete spatial lattice evolve through d...
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| Format: | Article |
| Language: | English |
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EDP Sciences
2015-12-01
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| Series: | ESAIM: Proceedings and Surveys |
| Online Access: | http://dx.doi.org/10.1051/proc/201552005 |
| _version_ | 1797971199006867456 |
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| author | Dellar Paul J. |
| author_facet | Dellar Paul J. |
| author_sort | Dellar Paul J. |
| collection | DOAJ |
| description | Quantum lattice algorithms originated with the Feynman checkerboard model for the
one-dimensional Dirac equation. They offer discrete models of quantum mechanics in which
the complex numbers representing wavefunction values on a discrete spatial lattice evolve
through discrete unitary operations. This paper draws together some of the identical, or
at least unitarily equivalent, algorithms that have appeared in three largely disconnected
strands of research. Treated as conventional numerical algorithms, they are all only first
order accurate under refinement of the discrete space/time grid, but may be raised to
second order by a unitary change of variables. Much more efficient implementations arise
from replacing the evolution through a sequence of unitary intermediate steps with a short
path integral formulation that expresses the wavefunction at each spatial point on the
most recent time level as a linear combination of values at immediately preceding time
levels and neighbouring spatial points. In one dimension, a particularly elegant
reformulation replaces two variables at two time levels with a single variable over three
time levels. The resulting algorithm is a variational integrator arising from a discrete
action principle, and coincides with the Ablowitz–Kruskal–Ladik finite difference scheme
for the Klein–Gordon equation. |
| first_indexed | 2024-04-11T03:27:45Z |
| format | Article |
| id | doaj.art-3df3069040b543e5897fab53e762709c |
| institution | Directory Open Access Journal |
| issn | 2267-3059 |
| language | English |
| last_indexed | 2024-04-11T03:27:45Z |
| publishDate | 2015-12-01 |
| publisher | EDP Sciences |
| record_format | Article |
| series | ESAIM: Proceedings and Surveys |
| spelling | doaj.art-3df3069040b543e5897fab53e762709c2023-01-02T06:58:01ZengEDP SciencesESAIM: Proceedings and Surveys2267-30592015-12-01527610410.1051/proc/201552005proc155205Quantum lattice algorithms: similarities and connections to some classic finite difference algorithmsDellar Paul J.0OCIAM, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory QuarterQuantum lattice algorithms originated with the Feynman checkerboard model for the one-dimensional Dirac equation. They offer discrete models of quantum mechanics in which the complex numbers representing wavefunction values on a discrete spatial lattice evolve through discrete unitary operations. This paper draws together some of the identical, or at least unitarily equivalent, algorithms that have appeared in three largely disconnected strands of research. Treated as conventional numerical algorithms, they are all only first order accurate under refinement of the discrete space/time grid, but may be raised to second order by a unitary change of variables. Much more efficient implementations arise from replacing the evolution through a sequence of unitary intermediate steps with a short path integral formulation that expresses the wavefunction at each spatial point on the most recent time level as a linear combination of values at immediately preceding time levels and neighbouring spatial points. In one dimension, a particularly elegant reformulation replaces two variables at two time levels with a single variable over three time levels. The resulting algorithm is a variational integrator arising from a discrete action principle, and coincides with the Ablowitz–Kruskal–Ladik finite difference scheme for the Klein–Gordon equation.http://dx.doi.org/10.1051/proc/201552005 |
| spellingShingle | Dellar Paul J. Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms ESAIM: Proceedings and Surveys |
| title | Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms |
| title_full | Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms |
| title_fullStr | Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms |
| title_full_unstemmed | Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms |
| title_short | Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms |
| title_sort | quantum lattice algorithms similarities and connections to some classic finite difference algorithms |
| url | http://dx.doi.org/10.1051/proc/201552005 |
| work_keys_str_mv | AT dellarpaulj quantumlatticealgorithmssimilaritiesandconnectionstosomeclassicfinitedifferencealgorithms |