Building compact superconducting microwave resonators with Hilbert space-filling curves

Superconducting quantum computing is currently one of the most promising platforms for universal quantum information processing. The readout resonator is an essential integral part of a superconducting qubit, while its size is much larger compared to the Josephson junction. We propose and realize a new readout resonator using space-filling curves, specifically Hilbert space-filling curves. We introduce the frequency analysis method and demonstrate a qubit sample, in which the Hilbert-space-filling-curves resonator (HSFCR) is used to read out the qubit states. We also propose to fabricate the HSFCRs and Josephson junctions simultaneously in the same processes of E-beam lithography and E-beam evaporation. Our design reduces the resonator area sufficiently and, thus, will help to improve the integration of superconducting qubits, as well as to design other superconducting quantum devices.