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Conjectured formula for the dimension of the dual-unitary manifold

Establish and prove the conjectured formula for the local dimension of the manifold of dual-unitary gates dualU(q,q') as a function of the local Hilbert space dimensions q and q', including the symmetric case q=q' and the case q≠q'.

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Background

Empirical tangent-space computations suggest a specific dimension for the manifold of dual-unitary gates when the even- and odd-site local dimensions may differ (q,q'). This led to a proposed closed-form expression for dim dualU(q,q').

While numerics support the formula, a rigorous proof for all q and q' is not yet available.

References

These empirical dimensions are described by the following conjectural formula

— Exactly solvable many-body dynamics from space-time duality (2505.11489 - Bertini et al., 16 May 2025) in Section 3.1. Parameterisation (Table 1 discussion)