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Feynman Formula for Discrete-time Quantum Walks

Jean-Pierre Fouque, Tomoyuki Ichiba, Ka Lok Lam

TL;DR

This work introduces a Feynman-type (discrete path integral) representation that connects one-dimensional discrete-time quantum walks on $\mathbb{Z}$ to a four-state Markov additive process. The authors show that quantum walk amplitudes can be expressed as conditional expectations over this MA process, enabling a spectral viewpoint via a tilted kernel and a clean derivation of the ballistic weak limit through spectral analysis. They also derive a hyperbolic (space-time) scaling limit that yields a quantum transport PDE system with a phase interaction term, for which a probabilistic Monte Carlo representation is available. The framework extends to general coins, site- and time-dependent variants, and even to analogous continuous-time Schrödinger-type dynamics, offering a robust, versatile toolkit for both theoretical insight and practical Monte Carlo computations of quantum transport phenomena.

Abstract

We explicitly connect (discrete-time) quantum walks on Z with a four-state Markov additive process via a Feynman-type formula (2.5). 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.

Feynman Formula for Discrete-time Quantum Walks

TL;DR

This work introduces a Feynman-type (discrete path integral) representation that connects one-dimensional discrete-time quantum walks on to a four-state Markov additive process. The authors show that quantum walk amplitudes can be expressed as conditional expectations over this MA process, enabling a spectral viewpoint via a tilted kernel and a clean derivation of the ballistic weak limit through spectral analysis. They also derive a hyperbolic (space-time) scaling limit that yields a quantum transport PDE system with a phase interaction term, for which a probabilistic Monte Carlo representation is available. The framework extends to general coins, site- and time-dependent variants, and even to analogous continuous-time Schrödinger-type dynamics, offering a robust, versatile toolkit for both theoretical insight and practical Monte Carlo computations of quantum transport phenomena.

Abstract

We explicitly connect (discrete-time) quantum walks on Z with a four-state Markov additive process via a Feynman-type formula (2.5). 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.
Paper Structure (34 sections, 21 theorems, 149 equations, 3 figures)

This paper contains 34 sections, 21 theorems, 149 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Feynman formula for one-dimensional quantum walks
  3. Notations
  4. One-dimensional quantum walks
  5. Deriving the Feynman formula
  6. Monte Carlo simulations
  7. Relation between quantum walks and spectral properties of Markov additive models
  8. Motivation - ballistic weak limit of quantum walk
  9. Joint distribution of (Sn,Tn)
  10. Derivation of asymptotics via spectral properties
  11. Quantum walk amplitudes
  12. Deriving quantum walks asymptotics
  13. Hyperbolic scaling limit and relation with the telegraph process
  14. Extensions to other quantum walks
  15. General unitary coins
  16. ...and 19 more sections

Key Result

Lemma 2.3

For a homogeneous quantum walk on the 1-D lattice with coin $C_x = e^{i\theta \sigma_1}$, we have where $(x_n, z_n)$ are being defined by $z_m:= (-1)^{k_m}z_{m-1}$ and $x_m:= x_0-\sum_{j=0}^{m-1} z_j$ for $m\geq 1$ with $(x_0, z_0):=(x, z)$.

Figures (3)

  • Figure 1: The mechanism of the one-dimensional quantum walk $(\psi_{n} = W^{n} \psi_{0}, n = 0,1,2 )$ with the rotation coin $e^{i (\pi/4) \sigma_{1}}$ and the initial condition $\psi_{0}(0,-1) = \psi_{0}(0, 1) = 1 / \sqrt{2}$ in Example \ref{['ex: QWex2']}. Two coin states $z = -1$ and $z=+1$ are assigned for each position $x \in \mathbb Z$. For each step, we apply the coin operator $C$ first and then the shift operator $S$. At $n = 2$, the state $C\psi_1$ has 4 non-zero components because the action of $C$ splits each component of $\psi_1$ into two.
  • Figure 2: (Left) Probability mass function $\mathbb P ( \Xi_{100} = x)$, $-100 \le x \le 100$ of the discrete-time quantum walk computed recursively from \ref{['eq: def2.1']} and then \ref{['eq: Psin']} with the balanced coin $e^{i (\pi/4)\sigma_1}$ and the initial condition described in Example \ref{['ex: QWex2']} and Figure \ref{['fig: quantum_walk_exmple']}. (Right) The limiting probability density function $f_{K}(y; \theta )$, $\lvert y \rvert < 1/\sqrt{2}$ in \ref{['eqns: Konnos_distribution']} with $\theta = \pi / 4$, $R = L = 1/ \sqrt{2}$.
  • Figure 3: Monte Carlo simulation of \ref{['eq: MonteCalro2']} for $t = 5$ (left) and $t = 10$ (right).

Theorems & Definitions (55)

  • Definition 2.1: Quantum walks on $\mathbb{Z}$
  • Example 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7: Feynman Formula
  • proof
  • Proposition 3.1
  • ...and 45 more