Absorption probabilities of quantum walks

Abstract

Quantum walks are known to have nontrivial interaction with absorbing boundaries. In particular, Ambainis et. al. [2] showed that in the (Z, C 1 , H) quantum walk (one-dimensional Hadamard walk) an absorbing boundary partially reflects information. These authors also conjectured that the left absorption probabilities P (1) n (1, 0) related to the finite absorbing Hadamard walks (Z, C 1 , H, {0, n}) satisfy a linear fractional recurrence in n (here P n (1, 0) is the probability that a Hadamard walk particle initialized in |1 |R is eventually absorbed at |0 and not at |n ). This result, as well as a third order linear recurrence in initial position m of P (m) n (1, 0), was later proved by Bach and Borisov [3] using techniques from complex analysis. In this paper we extend these results to general two state quantum walks and three-state Grover walks, while providing a partial calculation for absorption in d-dimensional Grover walks by a d − 1-dimensional wall. In the one-dimensional cases, we prove partial reflection of information, a linear fractional recurrence in lattice size, and a linear recurrence in initial position. arXiv:1905.04239v1 [quant-ph]

Conditional probability distributions of finite absorbing quantum walks

“…Previous studies have illustrated that the quantum walk has wave-like behavior for t = O(n) [5], and in particular quantum walks appear to partially reflect off absorbing boundaries [22] [20]. In the previous section, we showed that the absorbing quantum walk reaches a stable state for t = O(n 3 ).…”

“…[1]. The following definitions have appeared in previous works by Kuklinski [18] [20] [19]. The pair (G, Σ) can be thought of as an undirected Cayley graph which admits loops [11].…”

“…Then v must also be an eigenvalue of U . From Kuklinski [18], the eigenvectors of the corresponding quantum walk operator with periodic boundary conditions satisfy P v < v where P is the projection associated with the absorbing boundaries at |1 and |n and • is the 2 norm. Therefore, all eigenvalues of the absorbing quantum walk operator must satisfy |λ| < 1.…”

“…From [?] [18], localized eigenvectors with eigenvalue λ = 1 take nonzero values on 2 × 2 areas in the interior of B x,y . Since these eigenvectors are the only eigenvectors of Q x,y with eigenvalues |λ| = 1, from Proposition 2.2 in Kuklinski [19] an initial condition φ 0 which is orthogonal to each of these eigenvectors will eventually decay in norm to zero under Q t x,y .…”

“…A sharper result conjectured by these authors and later proved by Bach and Borisov [6] states that p n+1 = 1+2pn 2+2pn . These results were extended to the three-state Grover walk [27], two-state quantum walks [20], and general discrete quantum mechanical systems [17] [19]. Absorption probabilities are concerned with local behavior at an absorbing boundary as opposed to global behavior we wish to study.…”