How many switches does two-Hamiltonian universality require?

Abstract: Any pair of sufficiently generic Hamiltonians on $\mathbb{C}<sup>d$ generates the full unitary group, i.e., any $W\in U(d)$ can be synthesized to arbitrary precision by alternating between the two evolutions for suitably chosen durations. For any given pair of Hamiltonians, the minimal number of switches required to cover the entire group is at least $d$ and unbounded in general. Despite this, we show that it scales linearly in $d$ in expectation. Assuming that the two spectra are sufficiently generic, we derive constructive upper bounds on the minimal switching number as a function of the change-of-basis matrix between the two eigenbases. In particular, if this matrix has a column whose entries all have equal moduli, as for a complex Hadamard matrix, then $9d$ switches suffice to reach every unitary. To obtain these results, we construct a short exact representation of any unitary as a product of unitaries that are alternately diagonal in two given bases. All bounds are extended to non-generic spectra and bounded evolution times.