[Paper Review] Quantum random walks in higher dimensions

Troy D. Mackay, Stephen D. Bartlett+2 more|arXiv (Cornell University)|Aug 1, 2001

Quantum Computing Algorithms and Architecture1 citations

TL;DR

This paper extends quantum random walks to higher spatial dimensions using tensor products of Hadamard transformations and discrete Fourier transforms as quantum coin operations, demonstrating faster spreading than classical random walks. It establishes a classical limit via random phase variables and shows that quantum walks exhibit ballistic spreading, contrasting with classical diffusive scaling.

ABSTRACT

We analyze the quantum random walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the quantum coin toss in the one-dimensional random walk simulation, and other illustrative transformations are also investigated. The classical limit is obtained by introducing a random phase variable.

Motivation & Objective

• To generalize one-dimensional quantum random walk models to higher spatial dimensions.
• To investigate how quantum coin operations such as tensor products of Hadamard and discrete Fourier transforms affect walk dynamics in higher dimensions.
• To compare quantum walk spreading with classical random walk behavior using a random phase variable to model the classical limit.
• To analyze the scaling of spreading with time in both quantum and classical regimes.

Proposed method

• Extends the quantum coin operation in one dimension to higher dimensions using tensor products of Hadamard matrices.
• Applies the discrete Fourier transform as an alternative quantum coin operation to explore different walk dynamics.
• Models the classical limit by introducing a random phase variable that suppresses quantum interference.
• Uses unitary evolution operators to simulate time evolution of the quantum walk on a d-dimensional lattice.
• Analyzes the variance of the walker's position over time to quantify spreading behavior.
• Compares the time evolution of position variance between quantum and classical models.

Experimental results

Research questions

• RQ1 How does the quantum random walk spread in higher dimensions compared to the classical random walk?
• RQ2 What are the effects of different quantum coin operations—such as tensor product of Hadamard and discrete Fourier transforms—on the walk's dynamics in higher dimensions?
• RQ3 How can the classical limit of the quantum walk be formally derived using a random phase variable?
• RQ4 What scaling laws govern the spreading of the quantum walk in d-dimensional lattices?

Key findings

• Quantum random walks in higher dimensions exhibit ballistic spreading, with position variance scaling as t², in contrast to classical diffusive spreading scaling as t.
• Tensor products of Hadamard transformations provide a natural and effective extension of the one-dimensional quantum coin to higher dimensions.
• The discrete Fourier transform as a quantum coin leads to more uniform spreading patterns across dimensions.
• The introduction of a random phase variable successfully reproduces classical random walk behavior, validating the classical limit.
• The quantum walk's spreading rate is significantly faster than classical walks, highlighting a key quantum advantage in higher-dimensional lattices.
• The use of unitary evolution with structured coin operations enables precise control and analysis of quantum walk dynamics in d-dimensional systems.