Grover walks on a line with absorbing boundaries

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]

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“…al. [25] but we repeat the analysis here in a more concise framework. We refer back to the previous section for several of the proofs.…”

Section: (Zc 1 G 3 ) Quantum Walkmentioning

confidence: 99%

“…al. [25] where it was discovered that those absorption probabilities satisfy p n+1 = 2+3pn 3+4pn . Other releated papers consider the hitting time of the quantum walk, or the average time it takes for the quantum walk particle to be absorbed [26] [18] [20].…”

Section: Introductionmentioning

confidence: 99%

Absorption probabilities of quantum walks

Kuklinski

Kon

2018

Quantum Inf Process

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“…In addition to this result, Bach and Borisov also proved a third order linear recursion governing finite absorption probabilities with respect to initial position. These results were extended to the three-state Grover walk by Wang et al [8], and further extrapolated to arbitrary two-state quantum walks and d-dimensional walks with d − 1-dimensional absorbing boundaries by Kuklinski [5]. It should be noted that the quantum walk has been physically realized in several studies [9][10][11][12].…”

Section: Introductionmentioning

confidence: 84%

“…We hope in the future to compute a closed form limit of the Zeno absorption probabilities and to develop closed form recursions governing the CTQW on the line in the same way that this has been done for several 'coined' quantum walks on the line [7,8]. One may also wish to incorporate a continuous quantum measurement scheme [29] to calculate well defined moments for the absorption probability distribution.…”

Section: Orcid Idsmentioning

confidence: 99%

Absorption probabilities of discrete quantum mechanical systems

Kuklinski¹

2018

J. Phys. A: Math. Theor.

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Absorbing boundaries introduce non-unitary dynamics in quantum mechanical systems. In particular, it has been shown that absorbing boundaries partially reflect quantum information in a variety of one-dimensional quantum walks. The methods used to compute absorption probabilities for quantum walks can be extended to calculate absorption probabilities for general finite and discrete quantum mechanical systems. These methods can also be used to compute moments of the associated absorption time distributions. We conclude the paper by discussing a modification of the quantum Zeno effect in these absorption systems.

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“…Lackadaisical quantum walks (LQWs), first considered by Wong et al [1], are quantum analogous of lazy random walks. This model also generalizes three-state quantum walks on a line [19][20][21][22], which only have one self-loop at each vertex. In [1], the authors mainly investigate the effect of extra self-loops on Grover's algorithm when formulated as search for a marked vertex on complete graphs.…”

Section: Introductionmentioning

confidence: 90%

The Lackadaisical Quantum Walker is NOT Lazy at all

Wang,

Wu,

et al. 2016

Preprint

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In this paper, we study the properties of lackadaisical quantum walks on a line. This model is first proposed in [1] as a quantum analogue of lazy random walks where each vertex is attached τ self-loops. We derive an analytic expression for the localization probability of the walker at the origin after infinite steps, and obtain the peak velocities of the walker. We also calculate rigorously the wave function of the walker starting from the origin and obtain a long time approximation for the entire probability density function. As an application of the density function, we prove that lackadaisical quantum walks spread ballistically for arbitrary τ , and give an analytic solution for the variance of the walker's probability distribution.

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Grover walks on a line with absorbing boundaries

Cited by 6 publications

References 32 publications

Absorption probabilities of quantum walks

Absorption probabilities of quantum walks

Absorption probabilities of discrete quantum mechanical systems

The Lackadaisical Quantum Walker is NOT Lazy at all

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