Energy Scale Degradation in Sparse Quantum Solvers: A Barrier to Quantum Utility

2025-03-11

AI Summary

In sparse-connectivity quantum annealers, minor-embedding induces energy-scale degradation: chain embeddings require strengthened intra-chain couplings to enforce logical consistency, yet hardware-imposed coupling bounds force global Hamiltonian rescaling—compressing the spectral gap and degrading solution fidelity. Method: We establish the first theoretical model of energy attenuation, identifying chain volume and chain connectivity as primary drivers; derive a worst-case energy attenuation bound via subgraph inverse conductance; and integrate graph-theoretic analysis, Ising Hamiltonian scaling, and empirical validation on D-Wave devices. Results: We prove that effective temperature grows polynomially with graph connectivity, while success probability decays exponentially. These findings rigorously demonstrate the necessity of both high-connectivity hardware architectures and scale-aware embedding algorithms to mitigate energy-scale collapse and preserve quantum annealing performance.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

Quantum computing offers a promising route for tackling hard optimization problems by encoding them as Ising models. However, sparse qubit connectivity requires the use of minor-embedding, mapping logical qubits onto chains of physical qubits, which necessitates stronger intra-chain coupling to maintain consistency. This elevated coupling strength forces a rescaling of the Hamiltonian due to hardware-imposed limits on the allowable ranges of coupling strengths, reducing the energy gaps between competing states, thus, degrading the solver's performance. Here, we introduce a theoretical model that quantifies this degradation. We show that as the connectivity degree increases, the effective temperature rises as a polynomial function, resulting in a success probability that decays exponentially. Our analysis further establishes worst-case bounds on the energy scale degradation based on the inverse conductance of chain subgraphs, revealing two most important drivers of chain strength, extit{chain volume} and extit{chain connectivity}. Our findings indicate that achieving quantum advantage is inherently challenging. Experiments on D-Wave quantum annealers validate these findings, highlighting the need for hardware with improved connectivity and optimized scale-aware embedding algorithms.

Problem

• Energy scale degradation in sparse quantum solvers limits performance.
• Increased connectivity raises effective temperature, reducing success probability.
• Hardware constraints necessitate improved connectivity and optimized embedding algorithms.

Innovation

• Theoretical model quantifies energy scale degradation.
• Identifies chain volume and connectivity as key factors.
• Highlights need for improved hardware and algorithms.