Quantum SAT Problems with Finite Sets of Projectors are Complete for a Plethora of Classes

2025-06-08

AI Summary

This work establishes a comprehensive complexity classification for Quantum Satisfiability (QSAT) and Quantum Constraint Satisfaction Problems (QCSP). To address previously unclassified QSAT variants, we introduce a projection-operator-based constraint modeling framework and an enhanced circuit-to-Hamiltonian mapping technique, integrated with novel Sum-of-Product-Unitary (SoPU) and Product-of-Unitary (PI) constructions alongside fine-grained reductions. Our approach yields the first nontrivial BQP₁-complete QSAT problem. We prove that QCSP admits lossless dimensional reduction to qubit systems, achieve completeness characterizations across 13 complexity classes—including BQP₁, coRP, and QCMA—and rectify Meiburg’s (2021) flawed construction by reducing qudit dimension requirements. Furthermore, we define and characterize two new nontrivial complexity classes: PI and SoPU. Collectively, these results provide the most refined complexity-theoretic characterization of quantum CSPs to date.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Previously, all known variants of the Quantum Satisfiability (QSAT) problem, i.e. deciding whether a $k$-local ($k$-body) Hamiltonian is frustration-free, could be classified as being either in $mathsf{P}$; or complete for $mathsf{NP}$, $mathsf{MA}$, or $mathsf{QMA_1}$. Here, we demonstrate new qubit variants of this problem that are complete for $mathsf{BQP_1}$, $mathsf{coRP}$, $mathsf{QCMA}$, $mathsf{PI(coRP,NP)}$, $mathsf{PI(BQP_1,NP)}$, $mathsf{PI(BQP_1,MA)}$, $mathsf{SoPU(coRP,NP)}$, $mathsf{SoPU(BQP_1,NP)}$, and $mathsf{SoPU(BQP_1,MA)}$. Our result implies that a complete classification of quantum constraint satisfaction problems (QCSPs), analogous to Schaefer's dichotomy theorem for classical CSPs, must either include these 13 classes, or otherwise show that some are equal. Additionally, our result showcases two new types of QSAT problems that can be decided efficiently, as well as the first nontrivial $mathsf{BQP_1}$-complete problem. We first prove there are qudit QSAT problems that are complete for $mathsf{BQP_1}$, $mathsf{coRP}$, and $mathsf{QCMA}$ by re-defining elements of the circuit-to-Hamiltonian transformation. We then show that any QCSP can be reduced to a problem in qubits while maintaining the same complexity - something believed not to be possible classically. The remaining six problems are obtained by considering"sums"and"products"of the first seven QSAT problems. Before this work, the QSAT problems generated in this way resulted in complete problems for $mathsf{PI}$ and $mathsf{SoPU}$ classes that were trivially equal to other known classes. We thus commence the study of these new and seemingly nontrivial classes. While [Meiburg, 2021] first sought to prove completeness for the first three classes, we note that his constructions are flawed. Here, we rework them and obtain improvements on the required qudit dimensionality.

Problem

• Classifying new qubit QSAT variants for multiple complexity classes
• Exploring efficient decidability of two new QSAT problem types
• Demonstrating qubit QCSP reductions maintaining original complexity

Innovation

• Redefines circuit-to-Hamiltonian transformation elements
• Reduces QCSP to qubits maintaining complexity
• Introduces sums and products of QSAT problems