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arXiv:2011.12907 · Quantum Information Processing

An index theorem for one-dimensional gapless non-unitary quantum walks

Keisuke Asahara, Daiju Funakawa, Motoki Seki, Yohei Tanaka

Published Sep 1, 2021• 4 citations
arXivHTMLPDF
DOI
10.1007/s11128-021-03212-y

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It is recently shown that a split-step quantum walk possesses a chiral symmetry, and that a certain well-defined index can be naturally assigned to it. The index is a well-defined Fredholm index if and only if the associated unitary time-evolution operator has spectral gaps at both [Formula: see text] and [Formula: see text] In this paper, we extend the existing index formula for the Fredholm case to encompass the non-Fredholm case (i.e. gapless case). We make use of a natural extension of the Fredholm index to the non-Fredholm case, known as the Witten index. The aim of this paper is to fully classify the Witten index of the split-step quantum walk by employing the spectral shift function for a rank one perturbation of a fourth-order difference operator. It is also shown in this paper that the Witten index can take half-integer values in the non-Fredholm case.
In this paper we consider the one-dimensional quantum random walk X n ϕ at time n starting from initial qubit state ϕ determined by 2 × 2 unitary matrix U . We give a combinatorial expression for the characteristic function of X n ϕ . The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state ϕ . As a consequence, 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 n ϕ / n converges weakly to a limit Z ϕ as n → ∞ , where Z ϕ has a density 1 / π ( 1 - x 2 ) 1 - 2 x 2 for x ∈ ( - 1 / 2 , 1 / 2 ) . Moreover we discuss some known simulation results based on our limit theorems.