Quantum walk: Difference between revisions
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Label the vertexes of arbitrary graph as independent quantum states. Then a quantum walk in continuous time is specified by using the [[adjacency matrix]] <math>H</math> for the graph as a [[Hamiltonian_(quantum_mechanics)|Hamiltonian]], so time evolution is specified by the unitary operator <math>e^{i H t}</math> (in non-dimensionalized units). |
Label the vertexes of arbitrary graph as independent quantum states. Then a quantum walk in continuous time is specified by using the [[adjacency matrix]] <math>H</math> for the graph as a [[Hamiltonian_(quantum_mechanics)|Hamiltonian]], so time evolution is specified by the unitary operator <math>e^{i H t}</math> (in non-dimensionalized units). |
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Continuous quantum walks provide a model for universal quantum computation.<ref>Andrew M. Childs, [http://arxiv.org/abs/0806.1972 "Universal Computation by Quantum Walk"].</ref> |
Continuous quantum walks also provide a model for universal quantum computation.<ref>Andrew M. Childs, [http://arxiv.org/abs/0806.1972 "Universal Computation by Quantum Walk"].</ref> |
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==Discrete time== |
==Discrete time== |
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Revision as of 00:26, 22 January 2009
Quantum random walks are quantum mechanical extension of classical random walks where the walker may be in a superposition of positions.
Quantum walks exhibit very different features from classical random walks. In particular, they do not converge to limiting distributions and due to the power of quantum interference they may spread significantly faster or slower than their classical equivalents.
They are largely of interest as source for quantum algorithms.
Continuous time
Label the vertexes of arbitrary graph as independent quantum states. Then a quantum walk in continuous time is specified by using the adjacency matrix for the graph as a Hamiltonian, so time evolution is specified by the unitary operator (in non-dimensionalized units).
Continuous quantum walks also provide a model for universal quantum computation.[1]
Discrete time
A quantum walk in discrete time is specified a coin and shift operator, which are applied repeatedly.
References
- Andris Ambainis (2003). "Quantum walks and their algorithmic applications". International Journal of Quantum Information. 1: 507–518.
- Julia Kempe (2003). "Quantum random walks - an introductory overview". Contemporary Physics. 44: 307–327.
- ^ Andrew M. Childs, "Universal Computation by Quantum Walk".