Group Order is in QCMA
François Le Gall, Harumichi Nishimura, Dhara Thakkar
TL;DR
The authors resolve a long-standing open question by placing Group Order Verification in QCMA (indeed QCMA∩coQCMA), which immediately places Group Non-Membership in QCMA and yields stronger quantum upper bounds for a suite of group-theoretic problems in black-box groups. Their approach combines the Babai-Beals filtration with the construction of nice decompositions, enabling certificates that a quantum verifier can efficiently check using solvable-group techniques and isomorphism tests on composition factors. A central technical contribution is a QCMA protocol for testing isomorphism to Ree groups of rank one, alongside constructive membership procedures and completeness/soundness analyses. Collectively, these results deepen our understanding of the quantum-proof landscape for group problems and tighten the boundary between QCMA and QMA in the black-box setting, with potential implications for practical quantum verification of algebraic structures.
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.