[Paper Review] Transfiguration of Quantum Walks on a line

Mahesh N. Jayakody, Asiri Nanayakkara|arXiv (Cornell University)|Sep 3, 2018

Quantum Computing Algorithms and Architecture33 references4 citations

TL;DR

This paper introduces a spin decoherence model for one-dimensional quantum walks that exactly reproduces the position probability distribution of a classical random walk at all time scales. By applying a simultaneous bit and phase flip channel to the coin state, the model demonstrates that specific quantum channels can induce classical statistical behavior in quantum walks, offering potential algorithmic applications in quantum computing with non-local initial states.

ABSTRACT

We introduce an analytically treatable spin decoherence model for quantum walk on a line that yields the exact position probability distribution of an unbiased classical random walk at all-time scales. This spin decoherence model depicts a quantum channel in which simultaneous bit and phase flip operator is applied at random on the coin state. Based on this result we claim that there exist certain quantum channels that can produce exact classical statistical properties for a given one-dimensional quantum walk. Moreover, from the perspective of quantum computing, decoherence model introduced in this study may have useful algorithmic applications when it is applied on quantum walks with non-local initial states.

Motivation & Objective

• To develop an analytically tractable decoherence model for quantum walks on a line that reproduces classical random walk statistics.
• To investigate whether specific quantum channels can induce exact classical statistical behavior in quantum walks.
• To explore the implications of this model for quantum computing, particularly with non-local initial states.
• To establish a formal connection between quantum channels and classical stochastic processes in the context of quantum walks.

Proposed method

• The model applies a simultaneous bit and phase flip operation as a quantum channel on the coin qubit of a discrete-time quantum walk.
• The decoherence process is applied stochastically at each step, modeling noise that affects the coin state.
• The exact position probability distribution is derived analytically using the master equation formalism for the density matrix.
• The model is validated by showing that the resulting walk matches the second moment and distribution of a classical symmetric random walk.
• The analysis focuses on the unbiased quantum walk with a Hadamard coin, using the coin-space density matrix evolution.
• The key insight is that the specific structure of the noise channel leads to complete suppression of quantum interference, yielding classical statistics.

Experimental results

Research questions

• RQ1 Can a quantum channel be designed such that a quantum walk exhibits the exact position probability distribution of a classical random walk?
• RQ2 What type of decoherence mechanism leads to the emergence of classical statistical behavior in quantum walks?
• RQ3 Does the proposed channel preserve the structure of the quantum walk while inducing classicality?
• RQ4 What are the implications of this channel for quantum algorithms using non-local initial states?

Key findings

• The proposed quantum channel—simultaneous bit and phase flip on the coin state—produces the exact position probability distribution of a classical symmetric random walk at all time scales.
• The second moment of the position distribution matches that of a classical random walk, confirming classical diffusive scaling.
• The model achieves exact classical statistics without approximation, even in the long-time limit.
• The decoherence process fully suppresses quantum interference, leading to a classical diffusion process.
• The result demonstrates that specific quantum channels can induce classical behavior in quantum walks, challenging the assumption that quantum walks always exhibit non-classical features.