Circuit QED enables the combined use of qubits and oscillator modes. Despite
a variety of available gate sets, many hybrid qubit-boson (i.e., oscillator)
operations are realizable only through optimal control theory (OCT) which is
oftentimes intractable and uninterpretable. We introduce an analytic approach
with rigorously proven error bounds for realizing specific classes of
operations via two matrix product formulas commonly used in Hamiltonian
simulation, the Lie--Trotter and Baker--Campbell--Hausdorff product formulas.
We show how this technique can be used to realize a number of operations of
interest, including polynomials of annihilation and creation operators, i.e.,
apa†q for integer p,q. We show examples of this paradigm
including: obtaining universal control within a subspace of the entire Fock
space of an oscillator, state preparation of a fixed photon number in the
cavity, simulation of the Jaynes--Cummings Hamiltonian, simulation of the
Hong-Ou-Mandel effect and more. This work demonstrates how techniques from
Hamiltonian simulation can be applied to better control hybrid boson-qubit
devices.Comment: 48 pages, 5 figure