Search-Driven Clause Learning for Product-State Quantum $k$-SAT (PRODSAT-QSAT)

2026-03-20

AI Summary

This study addresses the problem of determining whether a product-state solution exists for quantum k-SAT instances (PRODSAT-QSAT). To this end, it introduces, for the first time, the conflict-driven clause learning (CDCL) paradigm into this domain, proposing a refutation framework that integrates a finite partitioning of the Bloch sphere with a geometric over-approximation theory solver. By verifying constraint feasibility within each partitioned region and designing a sound conflict clause learning mechanism, the approach enables effective decision-making regarding product-state satisfiability. The work formally defines the PRODSAT-QSAT problem, establishes the soundness of its clause learning rules, and presents the first practical algorithm capable of certifying product-state unsatisfiability (UN-PRODSAT).

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

We study PRODSAT-QSAT($k$): given rank-one $k$-local projectors, determine whether a quantum $k$-SAT instance admits a satisfying product state. We present a CDCL-style refutation framework that searches a finite partition of each qubit's Bloch sphere while a sound theory solver checks region feasibility using a geometric overapproximation of the projection amplitudes for each constraint. When the theory solver proves that no state in a region can satisfy a constraint, it produces a sound conflict clause that blocks that region; accumulated blocking clauses can yield a global result of product-state unsatisfiability (UN-PRODSAT). We formalise the problem, prove the soundness of the clause-learning rule, and describe a practical algorithm and implementation.

Problem

• quantum k-SAT
• product state
• satisfiability
• PRODSAT-QSAT
• rank-one projectors

Innovation

• CDCL
• quantum k-SAT
• product state
• geometric overapproximation
• conflict clause