Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Exact localization of Grover quantum walks on crystal lattices: transfer currents, a universal bound, and uniform-spanning-tree degree laws

Jacob Goodchild

Published Sep 28, 2026• 0 citations
Publisher

Abstract

Exact results for the localization ("trapping") of discrete-time Grover quantum walks on crystal lattices. A finite fraction of a Grover walker stays at its starting point forever, because the walk operator has flat eigenvalues. This paper computes the long-time return probability p̄ exactly for many lattices. Main results:• For the flip-flop Grover walk, the +1 and −1 flat-band projectors are I − Y and I − Y^Q, where Y and Y^Q are the Kirchhoff transfer-current matrices of the Laplacian and signless Laplacian. This reduces trapping to lattice Green functions. It gives closed forms in terms of Watson integrals (simple cubic, BCC, FCC), complete elliptic integrals (triangular, kagome, checkerboard, star, line graphs) and elementary functions of the bond anisotropy.• A universal lower bound p̄ ≥ (d−2)² / [2d(d−1)] for edge-transitive lattices of vertex degree d, attained exactly on the honeycomb, diamond and Laves graph lattices.• Exact trapping for lazy (Szegedy) walks on the square, triangular, honeycomb and diamond lattices as functions of the laziness.• For the moving-shift walk on Z^d, trapping is governed by A_d = ∫₀^∞ erfc(√s)^d ds, with elementary closed forms for d ≤ 5.• The degree of a vertex in the uniform spanning tree is exactly 1 + Binomial(z−1, 1/(z−1)) on 2-arc-transitive lattices, with closed-form degree distributions for several other lattices. All closed forms were identified by integer-relation (PSLQ) searches at 25–80 digits and checked by Brillouin-zone diagonalization and direct simulation of the walks. AI-use disclosure: this research, including choosing the problems, discovering the formulas, running all computations, and writing the manuscript, was carried out with extensive use of an AI system (Claude Code, by Anthropic) under the author's direction. It has not been peer reviewed. The author takes full responsibility for the content. Code: https://github.com/JacobGoodchild/findformula

DOI
10.5281/zenodo.23010605

Related papers

Localization of two-dimensional quantum walks

Norio Inui, Yoshinao Konishi, Norio Konno

The Grover walk, which is related to Grover's search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is strikingly different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.

arXiv:quant-ph/0311118•Physical Review A•May 19, 2004•153 citations

Spectral and asymptotic properties of Grover walks on crystal lattice

Yusuke Higuchi, Norio Konno, Iwao Sato +1 more

We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on $\mathbb{Z}^d$ with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the $d$-dimensional sphere of radius $1/\sqrt{d}$, and $2^d$ singular points reside on the sphere's surface.

arXiv:1401.0154•Dec 31, 2013•2 citations

Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks

J. Mareš, J. Novotný, M. Štefaňák +1 more

Quantum walks exhibit properties without classical analogues. One of those is the phenomenon of asymptotic trapping---there can be nonzero probability of the quantum walker being localized in a finite part of the underlying graph indefinitely even though locally all directions of movement are assigned nonzero amplitudes at each step. We study quantum walks with the flip-flop shift operator and the Grover coin, where this effect has been identified previously. For the version of the walk further modified by a random dynamical disruption of the graph (percolated quantum walks) we provide a recipe for the construction of a complete basis of the subspace of trapped states allowing to determine the asymptotic probability of trapping for arbitrary finite connected simple graphs, thus significantly generalizing the previously known result restricted to planar 3-regular graphs. We show how the position of the source and sink together with the graph geometry and its modifications affect the excitation transport. This gives us a deep insight into processes where elongation or addition of dead-end subgraphs may surprisingly result in enhanced transport and we design graphs exhibiting this pronounced behavior. In some cases this even provides closed-form formulas for the asymptotic transport probability in dependence on some structure parameters of the graphs.

arXiv:2202.09582•Physical Review A•Jun 10, 2022•2 citations

Percolated quantum walks with a general shift operator: From trapping to transport

Jan Mareš, Jaroslav Novotný, Igor Jex

We present an alternative definition of discrete-time coined quantum walks convenient for capturing a rather broad spectrum of a walker's behavior on arbitrary graphs. It includes and covers both the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find that for these walks with cyclic shift operators, the existence of an edge-3 coloring of the graph allows for nonstationary asymptotic behavior of the walk. For different shift operators, the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory, we treat a single-excitation transport on a percolated cube.

arXiv:1812.02519•Physical Review A•Apr 30, 2019•12 citations

The Fourier and Grover walks on the two-dimensional lattice and torus

Asano, Masahiro, Komatsu, Takashi, Norio Konno +1 more

In this paper, we consider discrete-time quantum walks with moving shift (MS) and flip-flop shift (FF) on two-dimensional lattice $\mathbb{Z}^2$ and torus $π_N^2=(\mathbb{Z}/N)^2$. Weak limit theorems for the Grover walks on $\mathbb{Z}^2$ with MS and FF were given by Watabe et al. and Higuchi et al., respectively. The existence of localization of the Grover walks on $\mathbb{Z}^2$ with MS and FF was shown by Inui et al. and Higuchi et al., respectively. Non-existence of localization of the Fourier walk with MS on $\mathbb{Z}^2$ was proved by Komatsu and Tate. Here our simple argument gave non-existence of localization of the Fourier walk with both MS and FF. Moreover we calculate eigenvalues and the corresponding eigenvectors of the $(k_1,k_2)$-space of the Fourier walks on $π_N^2$ with MS and FF for some special initial conditions. The probability distributions are also obtained. Finally, we compute amplitudes of the Grover and Fourier walks on $π_2^2$.

arXiv:1811.05302•Nov 13, 2018•1 citation

Coined quantum walks on weighted graphs

Thomas G. Wong

Abstract We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy’s quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has l integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight l . We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover’s algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when l < 3 + 2 2 ≈ 5.828 .

arXiv:1703.10134•Journal of Physics A Mathematical and Theoretical•Sep 13, 2017•36 citations
Exact results for the localization ("trapping") of discrete-time Grover quantum walks on crystal lattices. A finite fraction of a Grover walker stays at its starting point forever, because the walk operator has flat eigenvalues. This paper computes the long-time return probability p̄ exactly for many lattices. Main results:• For the flip-flop Grover walk, the +1 and −1 flat-band projectors are I − Y and I − Y^Q, where Y and Y^Q are the Kirchhoff transfer-current matrices of the Laplacian and signless Laplacian. This reduces trapping to lattice Green functions. It gives closed forms in terms of Watson integrals (simple cubic, BCC, FCC), complete elliptic integrals (triangular, kagome, checkerboard, star, line graphs) and elementary functions of the bond anisotropy.• A universal lower bound p̄ ≥ (d−2)² / [2d(d−1)] for edge-transitive lattices of vertex degree d, attained exactly on the honeycomb, diamond and Laves graph lattices.• Exact trapping for lazy (Szegedy) walks on the square, triangular, honeycomb and diamond lattices as functions of the laziness.• For the moving-shift walk on Z^d, trapping is governed by A_d = ∫₀^∞ erfc(√s)^d ds, with elementary closed forms for d ≤ 5.• The degree of a vertex in the uniform spanning tree is exactly 1 + Binomial(z−1, 1/(z−1)) on 2-arc-transitive lattices, with closed-form degree distributions for several other lattices. All closed forms were identified by integer-relation (PSLQ) searches at 25–80 digits and checked by Brillouin-zone diagonalization and direct simulation of the walks. AI-use disclosure: this research, including choosing the problems, discovering the formulas, running all computations, and writing the manuscript, was carried out with extensive use of an AI system (Claude Code, by Anthropic) under the author's direction. It has not been peer reviewed. The author takes full responsibility for the content. Code: https://github.com/JacobGoodchild/findformula
The Grover walk, which is related to Grover's search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is strikingly different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.
We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on Zd\mathbb{Z}^dZd with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the ddd-dimensional sphere of radius 1/d1/\sqrt{d}1/d​, and 2d2^d2d singular points reside on the sphere's surface.
Quantum walks exhibit properties without classical analogues. One of those is the phenomenon of asymptotic trapping---there can be nonzero probability of the quantum walker being localized in a finite part of the underlying graph indefinitely even though locally all directions of movement are assigned nonzero amplitudes at each step. We study quantum walks with the flip-flop shift operator and the Grover coin, where this effect has been identified previously. For the version of the walk further modified by a random dynamical disruption of the graph (percolated quantum walks) we provide a recipe for the construction of a complete basis of the subspace of trapped states allowing to determine the asymptotic probability of trapping for arbitrary finite connected simple graphs, thus significantly generalizing the previously known result restricted to planar 3-regular graphs. We show how the position of the source and sink together with the graph geometry and its modifications affect the excitation transport. This gives us a deep insight into processes where elongation or addition of dead-end subgraphs may surprisingly result in enhanced transport and we design graphs exhibiting this pronounced behavior. In some cases this even provides closed-form formulas for the asymptotic transport probability in dependence on some structure parameters of the graphs.
We present an alternative definition of discrete-time coined quantum walks convenient for capturing a rather broad spectrum of a walker's behavior on arbitrary graphs. It includes and covers both the geometry of possible walker's positions with interconnecting links and the prescribed rule in which directions the walker will move at each vertex. While the former allows for the analysis of inhomogeneous quantum walks on graphs with vertices of varying degree, the latter offers us to choose, investigate, and compare quantum walks with different shift operators. The synthesis of both key ingredients constitutes a well-suited playground for analyzing percolated quantum walks on a quite general class of graphs. Analytical treatment of the asymptotic behavior of percolated quantum walks is presented and worked out in details for the Grover walk on graphs with maximal degree 3. We find that for these walks with cyclic shift operators, the existence of an edge-3 coloring of the graph allows for nonstationary asymptotic behavior of the walk. For different shift operators, the general structure of localized attractors is investigated, which determines the overall efficiency of a source-to-sink quantum transport across a dynamically changing medium. As a simple nontrivial example of the theory, we treat a single-excitation transport on a percolated cube.
In this paper, we consider discrete-time quantum walks with moving shift (MS) and flip-flop shift (FF) on two-dimensional lattice Z2\mathbb{Z}^2Z2 and torus πN2=(Z/N)2π_N^2=(\mathbb{Z}/N)^2πN2​=(Z/N)2. Weak limit theorems for the Grover walks on Z2\mathbb{Z}^2Z2 with MS and FF were given by Watabe et al. and Higuchi et al., respectively. The existence of localization of the Grover walks on Z2\mathbb{Z}^2Z2 with MS and FF was shown by Inui et al. and Higuchi et al., respectively. Non-existence of localization of the Fourier walk with MS on Z2\mathbb{Z}^2Z2 was proved by Komatsu and Tate. Here our simple argument gave non-existence of localization of the Fourier walk with both MS and FF. Moreover we calculate eigenvalues and the corresponding eigenvectors of the (k1,k2)(k_1,k_2)(k1​,k2​)-space of the Fourier walks on πN2π_N^2πN2​ with MS and FF for some special initial conditions. The probability distributions are also obtained. Finally, we compute amplitudes of the Grover and Fourier walks on π22π_2^2π22​.
Abstract We define a discrete-time, coined quantum walk on weighted graphs that is inspired by Szegedy’s quantum walk. Using this, we prove that many lackadaisical quantum walks, where each vertex has l integer self-loops, can be generalized to a quantum walk where each vertex has a single self-loop of real-valued weight l . We apply this real-valued lackadaisical quantum walk to two problems. First, we analyze it on the line or one-dimensional lattice, showing that it is exactly equivalent to a continuous deformation of the three-state Grover walk with faster ballistic dispersion. Second, we generalize Grover’s algorithm, or search on the complete graph, to have a weighted self-loop at each vertex, yielding an improved success probability when l < 3 + 2 2 ≈ 5.828 .