The ‘t/2 law’ for quantum random walks on the line starting in the classical state
Chaobin Liu · Journal of Physics A Mathematical and Theoretical · 2008
Based on the theory of unitary matrices, our treatment of the theory of quantum random walks simplifies and clarifies certain prior derivations based on Fourier transform methods. Given a quantum random walk on the line determined by a 2× 2 unitary matrix U , we show how the first two moments of the position probability distribution are determined by the eigenvalues of U . By varying the ‘coin operator’ A , we show that the leading term of the standard deviation of the position probability distribution is ct , where t denotes time and 0 ⩽ c ⩽ 1. However, it turns out that the maximum value of c , namely c = 1, is achievable when and only when the coin operator A is diagonal, and the initial state is unbiased. Starting in the classical state |0⟩ ⊗ |1⟩, our approach confirms that the maximum value of the leading term of the standard deviation of the position probability distribution is , which, by way of known examples, is verified to be achievable.