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Weak limits for quantum random walks
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abstract
We formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions. With $X_n$ denoting position at time $n$, we show that $X_n/n$ converges weakly as $n \to \infty$ to a certain distribution which is absolutely continuous and of bounded support. The proof is rigorous and makes use of Fourier transform methods. This approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods.
Forward citations
Cited by 2 Pith papers
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Singular continuous Cantor spectrum for magnetic quantum walks
For irrational magnetic flux, the spectrum of the two-dimensional Hadamard magnetic quantum walk is a zero-measure Cantor set and the walk has no pure point spectrum, so the spectrum is purely singular continuous.
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A Probabilistic Representation for Multi-State Discrete-time Quantum Walks
Three-state discrete-time quantum walks on Z admit an exact Monte Carlo representation via Poisson-driven classical processes that converges to multi-state Dirac PDEs.
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