Feynman formula links quantum walks to Markov processes

What the study found: The paper explicitly connects discrete-time quantum walks on the integer line to a four-state Markov additive process using a Feynman-type formula. It also derives a relation between the spectral decomposition of that Markov additive process and the limiting density of the homogeneous quantum walk.
Why the authors say this matters: The authors state that their probabilistic representation for the resulting continuous-time, continuous-space quantum transport partial differential equations (PDEs) offers an efficient Monte Carlo computational technique.
What the researchers tested: The researchers developed a Feynman-type formula for quantum walks on mathbb{Z} and used it to study the link between the walk and a four-state Markov additive process. They also considered a space-time rescaling of quantum walks, which led to a system of quantum transport PDEs with a phase interaction term.
What worked and what didn't: The representation was used to derive the stated connection between spectral decomposition and limiting density, and it produced a probabilistic representation for the PDE system. The abstract does not describe any failed approach or negative result.
What to keep in mind: The available summary does not describe limitations, assumptions, or bounds of applicability beyond the systems named in the abstract.

Key points

• Discrete-time quantum walks on the integer line are explicitly connected to a four-state Markov additive process.
• A Feynman-type formula is used for this connection.
• The spectral decomposition of the Markov additive process is related to the limiting density of the homogeneous quantum walk.
• A space-time rescaling leads to quantum transport PDEs with a phase interaction term.
• The authors say the probabilistic representation offers an efficient Monte Carlo computational technique.