Deterministic generation of two-dimensional cluster states of itinerant microwave photonic qubits, part 2: characterization.

Multidimensional tensor-network states, in particular cluster states, are a key resource for quantum communication, measurement-based quantum computing and quantum metrology. While the generation of such states has been demonstrated using continuous-variable encoding, deterministic protocols with discrete-variable encoding remain limited to only a few entangled photonic qubits. Here, we develop a protocol for generating and characterizing large-scale entangled two-dimensional cluster states of such qubits. We utilize a source of two-dimensional entangled microwave photons consisting of a pair of superconducting transmon qubits and a set of tunable couplers, enabling control of both qubit-qubit interactions and emission into a common waveguide. We calculate statistical moments of the measured photon fields to characterize the generated cluster states. Direct many-photon tomography is infeasible due to exponential growth in computational resources required for state reconstruction and signal-to-noise ratio scaling of the statistical moments. Leveraging the structure of cluster states, we calculate joint moments from sets of up to four nearest-neighbor qubits and employ a maximum likelihood estimation algorithm to obtain the matrix product operator representation of the global density matrix. Using the matrix product operator representation, we characterize the localizable entanglement for states with up to 18 photonic qubits.