Reachability Constraints in Variational Quantum Circuits: Optimization within Polynomial Group Module

2026-04-15

AI Summary

Variational quantum algorithms generally lack guarantees of exact convergence to the ground state. This work establishes, for the first time, a necessary condition for such exact convergence: the input state and the ground state must exhibit matching projection norms onto polynomial group modules, which yields a priori constraints on the ansatz weights. Building upon this insight and leveraging group representation theory together with classical simulability analysis, the authors construct an efficient classical surrogate framework. This framework enables exact solutions to problems such as MaxCut with a per-iteration time complexity of $O(n^5)$, offering a provably accurate and computationally tractable alternative to conventional variational quantum approaches.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Security and Privacy: Large-scale security measurements

Abstract

This work identifies a necessary condition for any variational quantum approach to reach the exact ground state. Briefly, the norms of the projections of the input and the ground state onto each group module must match, implying that module weights of the solution state have to be known in advance in order to reach the exact ground state. An exemplary case is provided by matchgate circuits applied to problems whose solutions are classical bit strings, since all computational basis states share the same module-wise weights. Combined with the known classical simulability of quantum circuits for which observables lie in a small linear subspace, this implies that certain problems admit a classical surrogate for exact solution with each step taking $O(n^5)$ time. The Maximum Cut problem serves as an illustrative example.

Problem

• Reachability Constraints
• Variational Quantum Circuits
• Group Module
• Ground State
• Classical Simulability

Innovation

• variational quantum circuits
• reachability constraints
• group module
• matchgate circuits
• classical simulability