Deterministic Generation of Multidimensional Microwave Photonic Cluster States with a Single Quantum Emitter - Part 1*

Multidimensional photonic graph states, such as cluster states, are important resource states for a variety of protocols in quantum metrology, quantum communication, and measurement-based quantum computing. While previous works have demonstrated protocols for generating 1D photonic cluster states via quantum emitters, expanding such protocols for generation of multidimensional states requires an increasing number of emitters and control complexity. Here we present an implementation in the microwave regime of a scheme, originally proposed by Pichler et al. (PNAS 2017), for generating higher-dimensional cluster states using a single emitter in a resource-efficient manner via a time-delayed feedback operation between previously emitted itinerant photons and the emitter. Our implementation uses a conventional flux tunable transmon as emitter, a superconducting slow light waveguide to introduce large group delay for itinerant photons, and a second auxiliary transmon as a switchable mirror. We discuss how this scheme is realizable by using the first three levels of the emitter as a ladder system with only one transition damped to the finite-bandwidth waveguide, and by control of the decay rate into the waveguide via flux-modulation, which allows for emission of shaped wavepackets.