The weak limit theorem (WLT), the quantum analog of the central limit theorem, is foundational to quantum walk (QW) theory. Unlike the universal Gaussian limit of classical walks, deriving analytical forms of the limiting probability density function (PDF) in higher dimensions has remained a challenge since the one-dimensional (1D) Konno distribution was established. Previous explicit PDFs for two-dimensional (2D) models were limited to specific cases whose fundamental nature was unclear. This paper resolves this long-standing gap by introducing the notion of maximal speed v max as a critical parameter. We demonstrate that all previous 2D solutions correspond to a degenerate regime where v max = 1 . We then present the first exact analytical representation of the limiting PDF for the physically richer, unexplored regime v max < 1 of a general class of 2D two-state QWs. Our result reveals 2D Konno functions that govern these dynamics. We establish these as the proper 2D generalization of the 1D Konno distribution by demonstrating their convergence to the 1D form in the appropriate limit. Furthermore, our derivation, based on spectral analysis of the group-velocity map, analytically resolves the singular asymptotic structure: we explicitly determine the caustics loci where the PDF diverges and prove they define the boundaries of the distribution's support. By also providing a closed-form expression for the weight functions, this work offers a complete description of the 2D WLT.The weak limit theorem (WLT), the quantum analog of the central limit theorem, is foundational to quantum walk (QW) theory. Unlike the universal Gaussian limit of classical walks, deriving analytical forms of the limiting probability density function (PDF) in higher dimensions has remained a challenge since the one-dimensional (1D) Konno distribution was established. Previous explicit PDFs for two-dimensional (2D) models were limited to specific cases whose fundamental nature was unclear. This paper resolves this long-standing gap by introducing the notion of maximal speed v max as a critical parameter. We demonstrate that all previous 2D solutions correspond to a degenerate regime where v max = 1 . We then present the first exact analytical representation of the limiting PDF for the physically richer, unexplored regime v max < 1 of a general class of 2D two-state QWs. Our result reveals 2D Konno functions that govern these dynamics. We establish these as the proper 2D generalization of the 1D Konno distribution by demonstrating their convergence to the 1D form in the appropriate limit. Furthermore, our derivation, based on spectral analysis of the group-velocity map, analytically resolves the singular asymptotic structure: we explicitly determine the caustics loci where the PDF diverges and prove they define the boundaries of the distribution's support. By also providing a closed-form expression for the weight functions, this work offers a complete description of the 2D WLT.

arXiv:2408.09578Physical Review AFeb 2, 20261 citation

A new type of quantum walks based on decomposing quantum states

Chusei Kiumi

In this paper, the 2-state decomposed-type quantum walk (DQW) on a line is introduced as an extension of the 2-state quantum walk (QW). The time evolution of the DQW is defined with two different matrices, one is assigned to a real component, and the other is assigned to an imaginary component of the quantum state. Unlike the ordinary 2-state QWs, localization and the spreading phenomenon can coincide in DQWs. Additionally, a DQW can always be converted to the corresponding 4-state QW with identical probability measures. In other words, a class of 4-state QWs can be realized by DQWs with 2 states. In this work, we reveal that there is a 2-state DQW corresponding to the 4-state Grover walk. Then, we derive the weak limit theorem of the class of DQWs corresponding to 4-state QWs which can be regarded as the generalized Grover walks.In this paper, the 2-state decomposed-type quantum walk (DQW) on a line is introduced as an extension of the 2-state quantum walk (QW). The time evolution of the DQW is defined with two different matrices, one is assigned to a real component, and the other is assigned to an imaginary component of the quantum state. Unlike the ordinary 2-state QWs, localization and the spreading phenomenon can coincide in DQWs. Additionally, a DQW can always be converted to the corresponding 4-state QW with identical probability measures. In other words, a class of 4-state QWs can be realized by DQWs with 2 states. In this work, we reveal that there is a 2-state DQW corresponding to the 4-state Grover walk. Then, we derive the weak limit theorem of the class of DQWs corresponding to 4-state QWs which can be regarded as the generalized Grover walks.

arXiv:2012.00327Quantum Information and ComputationApr 27, 20214 citations

Limit theorem for continuous-time quantum walk on the line

Norio Konno

Concerning a discrete-time quantum walk ${X}_{t}^{(d)}$ with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: ${X}_{t}^{(d)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}(1\ensuremath{-}{x}^{2})\sqrt{1\ensuremath{-}2{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk ${X}_{t}^{(c)}$ on the line as follows: ${X}_{t}^{(c)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}\sqrt{1\ensuremath{-}{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: ${Y}_{t}∕\sqrt{t}\ensuremath{\rightarrow}{e}^{\ensuremath{-}{x}^{2}∕2}dx∕\sqrt{2\ensuremath{\pi}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.Concerning a discrete-time quantum walk ${X}_{t}^{(d)}$ with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: ${X}_{t}^{(d)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}(1\ensuremath{-}{x}^{2})\sqrt{1\ensuremath{-}2{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk ${X}_{t}^{(c)}$ on the line as follows: ${X}_{t}^{(c)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}\sqrt{1\ensuremath{-}{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: ${Y}_{t}∕\sqrt{t}\ensuremath{\rightarrow}{e}^{\ensuremath{-}{x}^{2}∕2}dx∕\sqrt{2\ensuremath{\pi}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.

arXiv:quant-ph/0408140Physical Review EAug 12, 2005125 citations

Concerning a discrete-time quantum walk

{X}_{t}^{(d)}

with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: ${X}_{t}^{(d)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}(1\ensuremath{-}{x}^{2})\sqrt{1\ensuremath{-}2{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk

{X}_{t}^{(c)}

on the line as follows: ${X}_{t}^{(c)}∕t\ensuremath{\rightarrow}dx∕\ensuremath{\pi}\sqrt{1\ensuremath{-}{x}^{2}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: ${Y}_{t}∕\sqrt{t}\ensuremath{\rightarrow}{e}^{\ensuremath{-}{x}^{2}∕2}dx∕\sqrt{2\ensuremath{\pi}}$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.