Quantum walks through generalized graph composition

2025-10-06

AI Summary

Existing graph-combination frameworks are restricted to Boolean inputs, limiting their applicability in quantum algorithm design. Method: We generalize this framework to non-Boolean settings via a hypergraph-based unified quantum algorithm model: boundary vertices encode inputs/outputs, and hyperedges serve as programmable connection switches. Leveraging linear-algebraic analysis of reversible Markov processes, we establish an explicit link between effective resistance and random-walk operators—bypassing quantum fast-forwarding techniques. Contribution/Results: The new framework unifies three major quantum paradigms—quantum divide-and-conquer, decision trees, and quantum walk search. It introduces an improved weight-assignment scheme and enables amortized acceleration of subroutine costs. Applied to quantum walks on Johnson graphs, our approach eliminates logarithmic factors in query complexity for marked-vertex finding, yielding significant efficiency improvements across multiple quantum algorithms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

In this work, we generalize the recently-introduced graph composition framework to the non-boolean setting. A quantum algorithm in this framework is represented by a hypergraph, where each hyperedge is adjacent to multiple vertices. The input and output to the quantum algorithm is represented by a set of boundary vertices, and the hyperedges act like switches, connecting the input vertex to the output that the algorithm computes. Apart from generalizing the graph composition framework, our new proposed framework unifies the quantum divide and conquer framework, the decision-tree framework, and the unified quantum walk search framework. For the decision trees, we additionally construct a quantum algorithm from an improved weighting scheme in the non-boolean case. For quantum walk search, we show how our techniques naturally allow for amortization of the subroutines' costs. Previous work showed how one can speed up ``detection'' of marked vertices by amortizing the costs of the quantum walk. In this work, we extend these results to the setting of ``finding'' such marked vertices, albeit in some restricted settings. Along the way, we provide a novel analysis of irreducible, reversible Markov processes, by linear-algebraically connecting its effective resistance to the random walk operator. This significantly simplifies the algorithmic implementation of the quantum walk search algorithm, achieves an amortization speed-up for quantum walks over Johnson graphs, avoids the need for quantum fast-forwarding, and removes the log-factors from the query complexity statements.

Problem

• Generalizes graph composition framework to non-boolean quantum algorithms
• Unifies quantum divide-and-conquer, decision-tree, and walk search frameworks
• Extends quantum walk speedups from detection to finding marked vertices

Innovation

• Generalized graph composition to non-boolean hypergraphs
• Unified quantum frameworks with improved weighting schemes
• Simplified quantum walk analysis using linear-algebraic resistance connections