Breaking the Treewidth Barrier in Quantum Circuit Simulation with Decision Diagrams

2025-10-08

AI Summary

Addressing the challenge of efficiently simulating high-treewidth quantum circuits with tensor networks, this paper introduces FeynmanDD, a novel decision-diagram-based classical simulation method. Technically, FeynmanDD integrates multi-terminal decision diagrams, the Feynman path integral formalism, Solovay–Kitaev gate approximation, and symbolic representation techniques; it further eliminates dependence on specific universal gate sets via gate-sequence expansion. Crucially, this work is the first to employ linear rank-width—provably upper-bounded by treewidth and strictly smaller for many high-treewidth circuit families—as the key complexity parameter. The time complexity of FeynmanDD depends solely on linear rank-width, yielding asymptotically tighter bounds. Experiments demonstrate substantial speedups over state-of-the-art tensor network methods on benchmark high-treewidth circuit families. Theoretically, FeynmanDD’s complexity is at most a logarithmic factor higher than treewidth, establishing a significant improvement in scalability for classically simulating structured quantum circuits.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Classical simulation of quantum circuits is a critical tool for validating quantum hardware and probing the boundary between classical and quantum computational power. Existing state-of-the-art methods, notably tensor network approaches, have computational costs governed by the treewidth of the underlying circuit graph, making circuits with large treewidth intractable. This work rigorously analyzes FeynmanDD, a decision diagram-based simulation method proposed in CAV 2025 by a subset of the authors, and shows that the size of the multi-terminal decision diagram used in FeynmanDD is exponential in the linear rank-width of the circuit graph. As linear rank-width can be substantially smaller than treewidth and is at most larger than the treewidth by a logarithmic factor, our analysis demonstrates that FeynmanDD outperforms all tensor network-based methods for certain circuit families. We also show that the method remains efficient if we use the Solovay-Kitaev algorithm to expand arbitrary single-qubit gates to sequences of Hadamard and T gates, essentially removing the gate-set restriction posed by the method.

Problem

• Overcoming treewidth limitations in quantum circuit simulation methods
• Analyzing decision diagram efficiency using linear rank-width metrics
• Extending simulation capability to arbitrary single-qubit gate sets

Innovation

• Decision diagrams break treewidth barrier in simulation
• Exponential complexity tied to linear rank-width not treewidth
• Gate-set restriction removed via Solovay-Kitaev algorithm expansion