[Paper Review] Properties of quantum stochastic walks from the asymptotic scaling exponent

Krzysztof Domino, Adam Glos+3 more|arXiv (Cornell University)|Nov 4, 2016

Quantum Computing Algorithms and Architecture24 references4 citations

TL;DR

This paper introduces a coherence measure for quantum stochastic walks using the asymptotic scaling exponent α of the second moment of the walker's position. By analyzing the mean squared displacement μ₂(t) ∝ t^α, it shows that the ballistic regime (α = 2) persists even under strong global dissipation, demonstrating robust quantum coherence in a generalized quantum-classical hybrid model governed by the Lindblad master equation with a tunable coherence-dissipation parameter ω.

ABSTRACT

This work focuses on the study of quantum stochastic walks, which are a generalization of coherent, i. e. unitary quantum walks. Our main goal is to present a measure of a coherence of the walk. To this end, we utilize the asymptotic scaling exponent of the second moment of the walk i. e. of the mean squared distance covered by a walk. As the quantum stochastic walk model encompasses both classical random walks and quantum walks, we are interested how the continuous change from one regime to the other influences the asymptotic scaling exponent. Moreover this model allows for behavior which is not found in any of the previously mentioned model -- the model with global dissipation. We derive the probability distribution for the walker, and determine the asymptotic scaling exponent analytically, showing that ballistic regime of the walk is maintained even at large dissipation strength.

Motivation & Objective

• To develop a quantitative measure of coherence in quantum stochastic walks using the asymptotic scaling exponent of the second moment.
• To investigate how continuous transitions from coherent quantum walks to classical random walks affect the scaling behavior of the mean squared displacement.
• To analyze the impact of global dissipation—unattainable in standard quantum or classical walks—on the long-time dynamics of the walk.
• To derive analytical expressions for the probability distribution and second moment under both local and global dissipation models.
• To demonstrate that the ballistic regime (α = 2) is preserved even at high dissipation strength, indicating sustained quantum coherence.

Proposed method

• The study employs the quantum stochastic walk model governed by the Lindblad master equation with a tunable parameter ω ∈ [0,1] that interpolates between unitary (ω=0) and purely dissipative (ω=1) dynamics.
• The second moment μ₂(t) = ⟨x²⟩ is computed as a function of time t to extract the asymptotic scaling exponent α via μ₂(t) ∝ t^α.
• For the line segment and infinite line, exact analytical expressions for the diagonal density matrix elements ⟨k|ρ(t)|k⟩ are derived using Fourier transforms and orthogonal polynomials.
• The central moments μₘ(t) are computed via Taylor series expansion of the probability distribution, with coefficients derived from integrals over momentum space and combinatorial identities.
• The analysis leverages generating functions and combinatorial identities (e.g., binomial sums and central moment identities) to simplify high-order moment expressions.
• The leading-order term in the moment expansion is isolated to determine the asymptotic scaling exponent, particularly for even moments.

Experimental results

Research questions

• RQ1 How does the asymptotic scaling exponent α of the second moment evolve as the system transitions from quantum to classical behavior via the parameter ω?
• RQ2 Can the ballistic scaling (α = 2) characteristic of unitary quantum walks be preserved under strong global dissipation?
• RQ3 What is the analytical form of the probability distribution for quantum stochastic walks on a line with global dissipation?
• RQ4 How do the higher-order moments μₘ(t) scale with time in the presence of global dissipation, and what does this imply for the walk’s dynamics?
• RQ5 What role does the global Lindblad operator LS play in sustaining quantum coherence beyond the standard local dissipation regime?

Key findings

• The asymptotic scaling exponent α remains equal to 2 (ballistic regime) even at full dissipation (ω = 1), indicating that global dissipation does not destroy quantum coherence.
• The second moment μ₂(t) scales as t² for all ω ∈ [0,1], with the leading-order coefficient being proportional to (1−ω)^m for the m-th moment, showing that coherence is preserved under global dissipation.
• For even moments, the leading-order term in the moment expansion is βₘ,ω t^m, where βₘ,ω = (1−ω)^m × (m choose m/2), confirming the t^m scaling for μₘ(t).
• The probability distribution on the line segment is derived exactly using orthogonal functions and Fourier integrals, enabling analytical computation of moments.
• The model with global dissipation (LS) produces dynamics not realizable in either standard classical or quantum walks, enabling new quantum behavior under environmental interaction.
• The analysis confirms that the asymptotic scaling exponent α = 2 is robust across the full range of ω, demonstrating that coherence is not destroyed by environmental coupling when the dissipation is global.