[Paper Review] Analysis of Lackadaisical Quantum Walks

Peter Høyer, Zhan Yu|arXiv (Cornell University)|Feb 26, 2020

Quantum Computing Algorithms and Architecture23 references4 citations

TL;DR

This paper analytically proves that lackadaisical quantum walks—quantum walks with self-loops of weight ℓ = d/N on each vertex—find a unique marked vertex with constant success probability in O(√HT) steps on any regular, locally arc-transitive graph. The key contribution establishes a direct equivalence between lackadaisical quantum walks and quantum interpolated walks, validating prior numerical conjectures for graphs including the torus, cycle, Johnson graphs, and hypercube.

ABSTRACT

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any locally arc-transitive graph.

Motivation & Objective

• To resolve long-standing conjectures about the success probability and runtime of lackadaisical quantum walks on structured graphs.
• To provide an analytical foundation for numerical findings on the torus, cycle, Johnson graphs, and hypercube, where lackadaisical walks were observed to succeed with constant probability.
• To establish a formal connection between lackadaisical quantum walks and quantum interpolated walks on regular, locally arc-transitive graphs.
• To prove that self-loop weight ℓ = d/N optimally enables constant-success-probability search across a broad class of vertex-transitive and symmetric graphs.

Proposed method

• The authors define a lackadaisical quantum walk by adding a self-loop of weight ℓ to each vertex in a d-regular graph, generalizing the coined walk framework.
• They establish a mathematical equivalence between lackadaisical quantum walks and quantum interpolated walks via a parameter mapping s = 1 − ℓ/d.
• The proof relies on spectral analysis and the use of the quantum hitting time, showing that both walk types have the same asymptotic scaling.
• Lemmas and triangle inequality arguments are used to bound the ℓ2-norm difference between the state evolutions of the two walk models, proving convergence within O(1/N^{1/4}).
• The analysis leverages locally arc-transitive symmetry to ensure uniform behavior across neighbors of the marked vertex, enabling analytical tractability.
• The framework applies to any d-regular, locally arc-transitive graph, including vertex-transitive and symmetric graphs such as the torus, Johnson graphs, and hypercube.

Experimental results

Research questions

• RQ1 Can lackadaisical quantum walks achieve constant success probability for finding a unique marked vertex on regular, locally arc-transitive graphs?
• RQ2 Is there a formal equivalence between lackadaisical quantum walks and quantum interpolated walks on such graphs?
• RQ3 Does the self-loop weight ℓ = d/N yield optimal performance across multiple graph classes, as suggested by prior numerical studies?
• RQ4 Why do lackadaisical walks succeed with high probability on graphs like the torus and cycle, despite lacking prior analytical proof?

Key findings

• The lackadaisical quantum walk with self-loop weight ℓ = d/N finds a unique marked vertex with constant success probability on any d-regular, locally arc-transitive graph.
• The quantum hitting time of lackadaisical quantum walks is of the same order as that of quantum interpolated walks, proving equivalent asymptotic performance.
• The paper proves that the self-loop weight ℓ corresponds to an interpolation parameter s = 1 − ℓ/d in quantum interpolated walks, establishing a precise mapping between the two models.
• The success probability remains bounded away from zero (constant) for all tested graph classes, including the √N × √N torus, cycle, Johnson graphs, and hypercube.
• The theoretical framework validates prior numerical conjectures for the torus (ℓ = 4/N), cycle (ℓ = 2/N), and Johnson graphs (ℓ = d/N), now proven analytically.
• The results extend to all regular, locally arc-transitive graphs, and the method provides a general analytical tool for analyzing lackadaisical walks beyond vertex-transitive graphs.