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

arXiv:2609.13392

Orientation-dependent Pauli noise in one-dimensional discrete-time quantum walks

Alejandro Contreras Reynoso, Adrian Ortega, Thomas Gorin

quant-phSubmitted Sep 11, 2026• Announced Sep 15, 2026• 0 citations
arXivHTMLPDF

Abstract

Discrete-time quantum walks exhibit ballistic spreading that is highly sensitive to decoherence. We study the long-time dynamics of a one-dimensional discrete-time quantum walk under Pauli noise acting on the coin. Choosing the unitary coin such that its invariant Bloch-sphere axis is orthogonal to the axis fixed by the conditional shift, we identify the distinct effects of noise aligned with either dynamical axis or orthogonal to both. Using a Fourier-space Pauli-superoperator formalism, we derive expressions for the first two position moments and long-time asymptotics for X-, Y-, Z-, and depolarizing noise. All channels induce a crossover from ballistic to diffusive spreading, with a second moment growing linearly at long times. The approach to this regime is channel dependent: it is exponentially fast for X-, Z-, and depolarizing noise, but algebraic, with a leading $t^{-1/2}$ correction, for Y-noise. The position distributions retain distinct finite-noise signatures, including a Gaussian-like profile with a central depression for Y-noise. In the maximal-noise limit, depolarizing, Y-, and Z-noise yield the classical binomial distribution for arbitrary initial coin states, whereas X-noise does so only when the initial state has no component along the noise axis. Thus, the quantum-to-classical transition is governed by the relative alignment of the noise, the coherent walk dynamics, and the initial state.

Related papers

Quantum random walks with decoherent coins

Todd A. Brun, Hilary A. Carteret, Andris Ambainis

The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.

arXiv:quant-ph/0210180•Physical Review A•Mar 14, 2003•155 citations

On the spreading of quantum walks starting from local and delocalized states

Alexandre C. Orthey, Edgard P. M. Amorim

We investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-$1/2$ particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it depends on the sum of relative phases between spin states given by the quantum coin, being non-dispersive for a Fourier walk and large initial dispersion. We also perform numerical simulations of the average probability distribution and variance along the time to compare them with our analytical results.

arXiv:1706.06257•Jun 20, 2017•4 citations

Crossover dynamics of tunable quantum walk in noisy channels / Nur Izzati Ishak

Nur Izzati , Ishak

This study investigates the crossover dynamics of a one-dimensional discrete-time quantum walk (DTQW) in noisy environments using the density matrix formalism and Kraus operator representation of quantum operations. Decoherence is modeled by bit-flip, dephasing, and bit-phase noise channels, while dissipation is represented by generalized amplitude damping channels. The analysis of crossover dynamics involves examining the probability distribution, scaling exponent of variance, coherence measures, and entropy metrics such as Shannon and von Neumann entropy. The DTQW produces a symmetrical probability distribution of displacement under decoherence, with distinct features in scaling exponents of mean squared displacement or variance. Under dissipation, DTQW produces a non-symmetric probability distribution and spreads faster than classical walk under extreme decoherence. These suggest a broad class of crossover transport behavior, which can be classified as pure quantum walk, quantum-like walk, semi-classical-like walk, and classical-like walk. Remarkably, maximum Shannon entropy is achieved at lower degrees of decoherence. This study also analyzes the effect of temporal disorder in the coin operator modeled using fractional Gaussian noise (fGn). The crossover dynamic occurs in the form of weak localization, followed by enhanced entanglement and information backflow. Notably, the correlation of fGn is transferred to the coin's degree of entanglement and transpires in the two-point correlation of the von Neumann entropy fluctuation. These findings can be useful for implementing DTQW-based protocols in quantum error correction, quantum cryptography, and quantum memory on the noisy intermediate-scale quantum (NISQ) platform.

•Dec 1, 2023•0 citations

Incoherent tunneling effects in a one-dimensional quantum walk

Mostafa Annabestani, Seyed Javad Akhtarshenas, Mohamad Reza Abolhassani

Abstract In this article we investigate the effects of shifting position decoherence, arising from the incoherent tunneling effect in the experimental realization of the quantum walk, on the one-dimensional discrete time quantum walk. We show that in the regime of this type of noise the quantum behavior of the walker does not vanish, in contrast to the coin decoherence for which the walker undergoes a quantum-to-classical transition even for weak noise. In particular, we show that the quadratic dependence of the variance on the time and also the coin–position entanglement, i.e. two important quantum aspects of the coherent quantum walk, are preserved in the presence of tunneling decoherence. Furthermore, we present an explicit expression for the probability distribution of the decoherent one-dimensional quantum walk in terms of the corresponding coherent probabilities, and show that this type of decoherence smooths the probability distribution.

arXiv:1004.4352•Journal of Physics A Mathematical and Theoretical•Feb 2, 2016•10 citations

A two-dimensional quantum walk driven by a single two-side coin*

Quan Lin, Hao Qin, Kun‐Kun Wang +2 more

We study a two-dimensional quantum walk with only one walker alternatively walking along the horizontal and vertical directions driven by a single two-side coin. We find the analytical expressions of the first two moments of the walker’s position distribution in the long-time limit, which indicates that the variance of the position distribution grows quadratically with walking steps, showing a ballistic spreading typically for quantum walks. Besides, we analyze the correlation by calculating the quantum mutual information and the measurement-induced disturbance respectively as the outcome of the walk in one dimension is correlated to the other with the coin as a bridge. It is shown that the quantum correlation between walker spaces increases gradually with the walking steps.

Chinese Physics B•Aug 13, 2020•4 citations

Noisy cyclic quantum random walk

G. Juárez Rangel, B. M. Rodríguez-Lara

We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on the spectral and dynamical properties of the system. Using a three-parameter unitary coin, we control the spectral structure of the noiseless step operator on the unit circle in the complex plane. One parameter governs the probability amplitudes and induces two spectral bandgaps proportional to its value. The half-sum of the two phase parameters rotates the spectrum and induces twofold degeneracy under specific conditions. Degenerate spectra yield eigenstates with sinusoidal probability distributions, while non-degenerate spectra produce flat distributions. We introduce static phase noise in the sites and analyze its impact across two distinct propagation regimes. In the walk-on-the-line regime, which precedes a full traversal of the graph, we extract the spreading exponent β from the step-resolved mean squared displacement. We find that low participation ratios correlate with sub-diffusive spread, while high ratios correspond to ballistic or super-diffusive evolution. After the walker completes a cycle, finite-size effects dominate. In this walk-on-the-cycle regime, the spreading exponent no longer characterizes the behavior. We quantify localization using a convergence criterion based on the coefficient of variation of the mean squared displacement. Across both regimes, we identify a sharp crossover near static site noise strength ϕs = π/3. This transition coincides with a drop in the participation ratio, a transition from diffusive to sub-diffusive spread in the walk-on-the-line regime, and a reduction in the saturation level of the mean squared displacement in the walk-on-the-cycle regime. Our results demonstrate that the eigenstate participation ratio provides a computationally efficient spectral diagnostic that anticipates localization across both regimes, offering an alternative to full dynamical simulations.

arXiv:2412.00536•APL Quantum•Aug 8, 2025•0 citations
Discrete-time quantum walks exhibit ballistic spreading that is highly sensitive to decoherence. We study the long-time dynamics of a one-dimensional discrete-time quantum walk under Pauli noise acting on the coin. Choosing the unitary coin such that its invariant Bloch-sphere axis is orthogonal to the axis fixed by the conditional shift, we identify the distinct effects of noise aligned with either dynamical axis or orthogonal to both. Using a Fourier-space Pauli-superoperator formalism, we derive expressions for the first two position moments and long-time asymptotics for X-, Y-, Z-, and depolarizing noise. All channels induce a crossover from ballistic to diffusive spreading, with a second moment growing linearly at long times. The approach to this regime is channel dependent: it is exponentially fast for X-, Z-, and depolarizing noise, but algebraic, with a leading t−1/2t^{-1/2}t−1/2 correction, for Y-noise. The position distributions retain distinct finite-noise signatures, including a Gaussian-like profile with a central depression for Y-noise. In the maximal-noise limit, depolarizing, Y-, and Z-noise yield the classical binomial distribution for arbitrary initial coin states, whereas X-noise does so only when the initial state has no component along the noise axis. Thus, the quantum-to-classical transition is governed by the relative alignment of the noise, the coherent walk dynamics, and the initial state.
The quantum random walk has been much studied recently, largely due to its highly nonclassical behavior. In this paper, we study one possible route to classical behavior for the discrete quantum walk on the line: the presence of decoherence in the quantum ``coin'' which drives the walk. We find exact analytical expressions for the time dependence of the first two moments of position, and show that in the long-time limit the variance grows linearly with time, unlike the unitary walk. We compare this to the results of direct numerical simulation, and see how the form of the position distribution changes from the unitary to the usual classical result as we increase the strength of the decoherence.
We investigate the ballistic spreading behavior of the one-dimensional discrete time quantum walks whose time evolution is driven by any balanced quantum coin. We obtain closed-form expressions for the long-time variance of position of quantum walks starting from any initial qubit (spin-1/21/21/2 particle) and position states following a delta-like (local), Gaussian and uniform probability distributions. By averaging over all spin states, we find out that the average variance of a quantum walk starting from a local state is independent of the quantum coin, while from Gaussian and uniform states it depends on the sum of relative phases between spin states given by the quantum coin, being non-dispersive for a Fourier walk and large initial dispersion. We also perform numerical simulations of the average probability distribution and variance along the time to compare them with our analytical results.
This study investigates the crossover dynamics of a one-dimensional discrete-time quantum walk (DTQW) in noisy environments using the density matrix formalism and Kraus operator representation of quantum operations. Decoherence is modeled by bit-flip, dephasing, and bit-phase noise channels, while dissipation is represented by generalized amplitude damping channels. The analysis of crossover dynamics involves examining the probability distribution, scaling exponent of variance, coherence measures, and entropy metrics such as Shannon and von Neumann entropy. The DTQW produces a symmetrical probability distribution of displacement under decoherence, with distinct features in scaling exponents of mean squared displacement or variance. Under dissipation, DTQW produces a non-symmetric probability distribution and spreads faster than classical walk under extreme decoherence. These suggest a broad class of crossover transport behavior, which can be classified as pure quantum walk, quantum-like walk, semi-classical-like walk, and classical-like walk. Remarkably, maximum Shannon entropy is achieved at lower degrees of decoherence. This study also analyzes the effect of temporal disorder in the coin operator modeled using fractional Gaussian noise (fGn). The crossover dynamic occurs in the form of weak localization, followed by enhanced entanglement and information backflow. Notably, the correlation of fGn is transferred to the coin's degree of entanglement and transpires in the two-point correlation of the von Neumann entropy fluctuation. These findings can be useful for implementing DTQW-based protocols in quantum error correction, quantum cryptography, and quantum memory on the noisy intermediate-scale quantum (NISQ) platform.
Abstract In this article we investigate the effects of shifting position decoherence, arising from the incoherent tunneling effect in the experimental realization of the quantum walk, on the one-dimensional discrete time quantum walk. We show that in the regime of this type of noise the quantum behavior of the walker does not vanish, in contrast to the coin decoherence for which the walker undergoes a quantum-to-classical transition even for weak noise. In particular, we show that the quadratic dependence of the variance on the time and also the coin–position entanglement, i.e. two important quantum aspects of the coherent quantum walk, are preserved in the presence of tunneling decoherence. Furthermore, we present an explicit expression for the probability distribution of the decoherent one-dimensional quantum walk in terms of the corresponding coherent probabilities, and show that this type of decoherence smooths the probability distribution.
We study a two-dimensional quantum walk with only one walker alternatively walking along the horizontal and vertical directions driven by a single two-side coin. We find the analytical expressions of the first two moments of the walker’s position distribution in the long-time limit, which indicates that the variance of the position distribution grows quadratically with walking steps, showing a ballistic spreading typically for quantum walks. Besides, we analyze the correlation by calculating the quantum mutual information and the measurement-induced disturbance respectively as the outcome of the walk in one dimension is correlated to the other with the coin as a bridge. It is shown that the quantum correlation between walker spaces increases gradually with the walking steps.
We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on the spectral and dynamical properties of the system. Using a three-parameter unitary coin, we control the spectral structure of the noiseless step operator on the unit circle in the complex plane. One parameter governs the probability amplitudes and induces two spectral bandgaps proportional to its value. The half-sum of the two phase parameters rotates the spectrum and induces twofold degeneracy under specific conditions. Degenerate spectra yield eigenstates with sinusoidal probability distributions, while non-degenerate spectra produce flat distributions. We introduce static phase noise in the sites and analyze its impact across two distinct propagation regimes. In the walk-on-the-line regime, which precedes a full traversal of the graph, we extract the spreading exponent β from the step-resolved mean squared displacement. We find that low participation ratios correlate with sub-diffusive spread, while high ratios correspond to ballistic or super-diffusive evolution. After the walker completes a cycle, finite-size effects dominate. In this walk-on-the-cycle regime, the spreading exponent no longer characterizes the behavior. We quantify localization using a convergence criterion based on the coefficient of variation of the mean squared displacement. Across both regimes, we identify a sharp crossover near static site noise strength ϕs = π/3. This transition coincides with a drop in the participation ratio, a transition from diffusive to sub-diffusive spread in the walk-on-the-line regime, and a reduction in the saturation level of the mean squared displacement in the walk-on-the-cycle regime. Our results demonstrate that the eigenstate participation ratio provides a computationally efficient spectral diagnostic that anticipates localization across both regimes, offering an alternative to full dynamical simulations.