In Appendix F of this paper, the authors discuss simulating the Bosonic displacement operator on qubits. The generating Hamiltonian for this operation is $ H = a^\dagger + a. $
The leading-order cost of this arises from the need to compute square roots, as they note:
"This is because of the action of the creation and annihilation operators which involve a square root of the photon number. Realizing this (using a direct approach) requires Newton iterations in DV qubits which are very expensive."
Due to this, the leading order cost in terms of CNOT gates and qubits (detailed in Appendix F) largely stems from these Newton iterations.
Question:
Why is it necessary to compute these square roots explicitly on the quantum computer? Why can't one simply provide a qubit compiler with the matrix representation of $ H = a^\dagger + a $, expressed as a sum of Pauli strings, and perform Trotterization as usual? For example, in this work, a $ d $-level bosonic operator is converted into a sum of Pauli strings.
My assumption is that decomposing this Hamiltonian into Pauli strings is inefficient, but for this specific case, I don't immediately see why it would be infeasible for a reasonable bosonic Hilbert space cutoff.
