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Circuit QED enables the combined use of qubits and oscillator modes. Despitea variety of available gate sets, many hybrid qubit-boson (i.e., oscillator)operations are realizable only through optimal control theory (OCT) which isoftentimes intractable and uninterpretable. We introduce an analytic approachwith rigorously proven error bounds for realizing specific classes ofoperations via two matrix product formulas commonly used in Hamiltoniansimulation, the Lie--Trotter and Baker--Campbell--Hausdorff product formulas.We show how this technique can be used to realize a number of operations ofinterest, including polynomials of annihilation and creation operators, i.e.,$a^p {a^\dagger}^q$ for integer $p, q$. We show examples of this paradigmincluding: obtaining universal control within a subspace of the entire Fockspace of an oscillator, state preparation of a fixed photon number in thecavity, simulation of the Jaynes--Cummings Hamiltonian, simulation of theHong-Ou-Mandel effect and more. This work demonstrates how techniques fromHamiltonian simulation can be applied to better control hybrid boson-qubitdevices.

Leveraging Hamiltonian Simulation Techniques to Compile Operations on Bosonic Devices