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arXiv:2508.13318 · Journal of Physics A Mathematical and Theoretical

Quantum walk on a line with absorbing boundaries

Ammara Ammara, Václav Potoček, M. Štefaňák, Francesco Vincenzo Pepe

Published Jul 23, 2026• 1 citation
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Abstract

Abstract Absorption of two-state coined quantum walks on a finite line with two sinks located at N and − N is investigated. Elaborating on the results of Konno et al (2003 J. Phys. A: Math. Gen. 36 241), we derive closed formulas for the absorption probabilities at the boundaries in the limit of large system size N . Two limiting cases are considered, with the starting position k being independent of N , or kept at a constant distance δ from one of absorbers. In the first scenario, the absorption probability is determined only by the coin parameter and polar angle of the initial coin state decomposed into the eigenbasis of the coin operator. In the second case, a correction depending exponentially on δ is introduced. Finally, we perform an extensive numerical investigation for small system size N , showing excellent agreement between numerical and analytical results.

DOI
10.1088/1751-8121/ae8f8b

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•Jan 1, 2017•1 citation
Abstract Absorption of two-state coined quantum walks on a finite line with two sinks located at N and − N is investigated. Elaborating on the results of Konno et al (2003 J. Phys. A: Math. Gen. 36 241), we derive closed formulas for the absorption probabilities at the boundaries in the limit of large system size N . Two limiting cases are considered, with the starting position k being independent of N , or kept at a constant distance δ from one of absorbers. In the first scenario, the absorption probability is determined only by the coin parameter and polar angle of the initial coin state decomposed into the eigenbasis of the coin operator. In the second case, a correction depending exponentially on δ is introduced. Finally, we perform an extensive numerical investigation for small system size N , showing excellent agreement between numerical and analytical results.
We consider a two-state quantum walk on a line where after the first step an absorbing sink is placed at the origin. The probability of finding the walker at position jjj, conditioned on that it has not returned to the origin, is investigated in the asymptotic limit. We prove a limit theorem for the conditional probability distribution and show that it is given by the Konno's density function modified by a prefactor ensuring that the distribution vanishes at the origin. In addition, we discuss the relation to the problem of recurrence of a quantum walk and determine the P\'olya number. Our approach is based on path counting and the stationary phase approximation.
We investigate state estimation in discrete-time quantum walks with a single absorbing boundary. Using a spectral approach, we obtain closed expressions for the escape probability as a function of the initial coin state and the boundary position, together with the corresponding classical Fisher information for a binary absorption readout. Comparison with the single-copy quantum Fisher information reveals a clear complementarity: near boundaries carry broad information about the polar (Bloch-sphere) angle of the coin state, whereas moderate or distant boundaries reveal phase-sensitive regions. Because a single boundary probes only one information direction, combining two boundary placements yields, generically, a full-rank Fisher matrix and tight joint Cramér-Rao bounds while retaining a binary measurement without mode-resolved tomography. We also discuss a restricted-readout photonic implementation in which an on-chip sink realizes the absorber, and we frame the resulting advantage as a potential reduction in measurement-setting and reconfiguration overhead for low-dimensional parameter estimation tasks in architectures where direct projective access to the coin is unavailable. Our results show that absorption in quantum walks defines an analytically tractable restricted-access primitive for coin-state estimation.
Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one-dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Qn{Q}_{n}Qn​, where nnn is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times t=O(n)t=O(n)t=O(n), fractional quantum revivals for t=O(n2)t=O({n}^{2})t=O(n2), and stability for t=O(n3)t=O({n}^{3})t=O(n3). At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.
In this paper, a study on discrete-time coined quantum walks on the line is presented. Clear mathematical foundations are still lacking for this quantum walk model. As a step toward this objective, the following question is being addressed: Given a graph, what is the probability that a quantum walk arrives at a given vertex after some number of steps? This is a very natural question, and for random walks it can be answered by several different combinatorial arguments. For quantum walks this is a highly non-trivial task. Furthermore, this was only achieved before for one specific coin operator (Hadamard operator) for walks on the line. Even considering only walks on lines, generalizing these computations to a general SU(2) coin operator is a complex task. The main contribution is a closed-form formula for the amplitudes of the state of the walk (which includes the question above) for a general symmetric SU(2) operator for walks on the line. To this end, a coin operator with parameters that alters the phase of the state of the walk is defined. Then, closed-form solutions are computed by means of Fourier analysis and asymptotic approximation methods. We also present some basic properties of the walk which can be deducted using weak convergence theorems for quantum walks. In particular, the support of the induced probability distribution of the walk is calculated. Then, it is shown how changing the parameters in the coin operator affects the resulting probability distribution.
Quantum walks are interesting models with potential applications to quantum algorithms and physical processes such as photosynthesis. In this paper, we study two models of one-dimensional quantum walks, namely, quantum walks with a moving absorbing wall and quantum walks with one stationary and one moving absorbing wall. For the former, we calculate numerically the survival probability, the rate of change of average position, and the rate of change of standard deviation of the particle's position in the long time limit for different wall velocities. Moreover, we also study the asymptotic behavior and the dependence of the survival probability on the initial particle's state. While for the latter, we compute the absorption probability of the right stationary wall for different velocities and initial positions of the left wall boundary. The results for these two models are compared with those obtained for the classical model. The difference between the results obtained for the quantum and classical models can be attributed to the difference in the probability distributions.
The quantum walk is a unitary analogue to the discrete random walk, and its properties have been increasingly studied since the turn of the millennium. In comparison with the classical random walk, the quantum walk exhibits linear spreading and initial condition dependent asymmetries. As noted early on in the conjecture and subsequent calculation of absorption probabilities in the one dimensional Hadamard walk, the interaction of the quantum walk with an absorbing boundary is fundamentally divergent from classical case. Here, we will survey absorption probabilities for a more general collection of one dimensional quantum walks and extend the method to consider d-dimensional walks in the presence of d-1 dimensional absorbing walls. However, these results are concerned only with local behavior at the boundary in the form of absorption probabilities. The main results of this thesis are concerned with the global behavior of finite quantum walks, which can be described by linear spreading in the short term, modal phenomena in the mid term, and stable distributions in the exceedingly long term. These theorems will be rigorously proved in the one-dimensional case and extrapolated to higher dimensional quantum walks. To this end we introduce QWSim, a new and robust computational engine for displaying finite two dimensional quantum walks.