The quantum algorithmic reach of a high-Q radio frequency cavity

While NISQ-era quantum computers based on superconducting technology are generally comprised of 2D qubits, high quality 3D resonator cavities coupled to a smaller number of qubits offer potential advantages such as: efficient error correction protocols, better connectivity, and potential quantum memory. To understand such systems' algorithmic capabilities and benchmark their performance both over time and against other platforms, it is important to consider a number of metrics and adapt existing ones as needed. Here, we study a realistic superconducting cavity coupled to a single noisy transmon. Generating Haar random circuits, compiling them into selective number-dependent arbitrary phase (SNAP) and displacement pulses followed by simulating the pulse-level dynamics allow us to extract a "qudit" analog of quantum volume, the cross-entropy benchmark of the cavity, as well as the KL-divergence from the ideal Haar distribution to the real one. We discuss how these metrics depend on dominant noise sources and comment on both the limitations they place on 3D systems as well as potential advantages.