Explicit upper bounds for arbitrary Hamiltonian pairs

Derive an explicit upper bound on the minimal number of alternating evolutions, or switches, required for a general pair of Hamiltonians on \(\mathbb{C}^d\) to generate or approximately reach every unitary in \(U(d)\).

Background

The paper studies the order of generation for two-Hamiltonian universality: the smallest number of alternating factors needed so that every unitary can be represented exactly or approximated arbitrarily well. For generic spectra, this problem is reduced to factoring arbitrary unitaries into alternating products of diagonal unitaries in two bases.

The authors note that, in dimensions greater than two, existing general bounds are either unavailable or ineffective for this purpose. In particular, D’Alessandro’s bound for generating a neighborhood of the identity grows exponentially in the dimension of the group, and the compactness argument used to extend it to the whole group does not provide an explicit bound. The unresolved problem is therefore to obtain a finite, explicit dimension-dependent bound for an arbitrary universal pair of Hamiltonians.