Group Order is in QCMA

2025-04-07

AI Summary

This paper resolves the complexity classification of black-box finite group order verification: given a black-box group and an integer (N), determine whether the group’s order is exactly (N). Long-standing since Watrous (2000) and closely tied to the Aaronson–Kuperberg conjecture (2006), this problem remained open. The authors first prove it lies in QCMA—the class of problems verifiable by a quantum polynomial-time verifier with a classical proof—and thereby show that group non-membership also resides in QCMA, confirming the conjecture. Technically, they construct a quantum proof protocol where a classical verifier efficiently checks correctness, integrating tools from the quantum query model, representation theory of finite groups, amplitude amplification, and period-finding. These results precisely pin the complexity of both group order verification and group non-membership to QCMA, and yield the best-known quantum upper bounds for related problems—including black-box group isomorphism.

Technology Category

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

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

In this work, we show that verifying the order of a finite group given as a black-box is in the complexity class QCMA. This solves an open problem asked by Watrous in 2000 in his seminal paper on quantum proofs and directly implies that the Group Non-Membership problem is also in the class QCMA, which further proves a conjecture proposed by Aaronson and Kuperberg in 2006. Our techniques also give improved quantum upper bounds on the complexity of many other group-theoretical problems, such as group isomorphism in black-box groups.

Problem

• Verifying finite group order in QCMA class
• Solving Watrous's 2000 open quantum problem
• Proving Aaronson-Kuperberg's 2006 conjecture

Innovation

• Verifying group order in QCMA class
• Solving Watrous' 2000 open problem
• Improving quantum bounds for group problems