Minimum alphabet size for optimal dimension–horizon laws

Determine whether fewer than four shared control generators—specifically, two or three letters, or more generally an alphabet budget q<4—attain the quantum and classical dimension–horizon laws established for four generators.

Background

The paper proves that four shared generators suffice to achieve the optimal asymptotic identification sizes: logarithmic code size of order min{d²,T} for quantum systems and min{d,T} for classical systems. The converse bounds apply to arbitrary fixed alphabet budgets, while the constructions use four reversible generators.

It remains unresolved whether the same scaling laws can be achieved with a smaller shared control interface. Resolving this would characterize the minimum control-alphabet cardinality needed to preserve the quadratic quantum versus linear classical separation, and may require combining semigroup growth restrictions with generation constraints for permutation or unitary groups.

References

The theorem establishes four generators as a sufficient constant-cardinality interface, with optimized generator matrices allowed to depend on the dimension $d$; the minimum alphabet size is an open problem. For every alphabet budget $q\ge4$, the same lower bounds hold by padding the four-letter construction with unused letters. Determining whether budgets $q<4$ attain these laws is an open alphabet-reduction problem.

— Controlled-Memory Identification with Shared Generators: Dimension--Horizon Tradeoffs and Quantum--Classical Separation  (2610.21797 - Zhao, 15 Oct 2026) in Theorem 1 (Four-generator dimension–horizon law), Section 2.2; also discussed in Section 3, Section 7, and Conclusion