|
| related topics |
| {classical, space, random} |
| {let, theorem, proof} |
| {equation, function, exp} |
| {qubit, qubits, gate} |
| {cos, sin, state} |
| {operator, operators, space} |
| {phase, path, phys} |
| {state, algorithm, problem} |
| {algorithm, log, probability} |
|
A New Type of Limit Theorems for the One-Dimensional Quantum Random Walk
Norio Konno
abstract: In this paper we consider the one-dimensional quantum random walk X^{varphi}
_n at time n starting from initial qubit state varphi determined by 2 times 2
unitary matrix U. We give a combinatorial expression for the characteristic
function of X^{varphi}_n. The expression clarifies the dependence of it on
components of unitary matrix U and initial qubit state varphi. As a consequence
of the above results, we present a new type of limit theorems for the quantum
random walk. In contrast with the de Moivre-Laplace limit theorem, our
symmetric case implies that X^{varphi}_n /n converges in distribution to a
limit Z^{varphi} as n to infty where Z^{varphi} has a density 1 / pi (1-x^2)
sqrt{1-2x^2} for x in (- 1/sqrt{2}, 1/sqrt{2}). Moreover we discuss some known
simulation results based on our limit theorems.
- oai_identifier:
- oai:arXiv.org:quant-ph/0206103
- categories:
- quant-ph
- comments:
- Final version; Journal-ref added; 14 pages; this arXiv version has no
figures
- arxiv_id:
- quant-ph/0206103
- journal_ref:
- Journal of the Mathematical Society of Japan, Vol.57, No.4,
pp.1179-1195 (2005)
- created:
- 2002-06-17
- updated:
- 2005-10-31
Full article ▸
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