A Difference Operator Approach to Quantum Random Walks: Parseval Identity, Krawtchouk Matrices, and Hermite Limits
Source: Originally published on arXiv quant-ph on July 29, 2026.
arXiv:2607.24837v1 Announce Type: new Abstract: We introduce a discrete difference operator D_k to study the one-dimensional quantum random walk (QRW) with the Hadamard coin. Explicit combinatorial expressions are obtained for the probability amplitudes a(n,k) and b(n,k), which encode the final step direction and carry alternating signs that reflect the merging of leftward steps. Removing these signs and the coin-state distinction recovers the classical binomial distribution. The symmetric and antisymmetric combinations $a\pm b$ are shown to coincide with diagonal and sub-diagonal entries of the Krawtchouk matrix. Using cross identities among Krawtchouk matrix elements, we prove by induction that the amplitudes satisfy a Parseval identity sum (a^2+b^2)=2^n-1, establishing probability conse...
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