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<p>We explicitly connect (discrete-time) quantum walks on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z} \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation> with a four-state Markov additive process via a Feynman-type formula (<InternalRef RefID="Equ13">13</InternalRef>). Using this representation, we derive a relation between the spectral decomposition of the Markov additive process and the limiting density of the homogeneous quantum walk. In addition, we consider a space-time rescaling of quantum walks, which leads to a system of quantum transport PDEs in continuous time and space with a phase interaction term. Our probabilistic representation for this type of PDE offers an efficient Monte Carlo computational technique.</p>

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Feynman Formula for Discrete-Time Quantum Walks

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月9日

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  • Jean-Pierre Fouque,
  • Tomoyuki Ichiba,
  • Ka Lok Lam

摘要

We explicitly connect (discrete-time) quantum walks on \(\mathbb {Z} \) Z with a four-state Markov additive process via a Feynman-type formula (13). Using this representation, we derive a relation between the spectral decomposition of the Markov additive process and the limiting density of the homogeneous quantum walk. In addition, we consider a space-time rescaling of quantum walks, which leads to a system of quantum transport PDEs in continuous time and space with a phase interaction term. Our probabilistic representation for this type of PDE offers an efficient Monte Carlo computational technique.