Weak limits for quantum random walks

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We formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions. With Xₙ denoting position at time n, we show that Xₙ/n converges weakly as n → ∞ 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.