  Generalized measurements, also called positive operator-valued measures
(POVMs), can offer advantages over projective measurements in various quantum
information tasks. Here, we realize a generalized measurement of one and two
superconducting qubits with high fidelity and in a single experimental setting.
To do so, we propose a hybrid method, the "Naimark-terminated binary tree,"
based on a hybridization of Naimark's dilation and binary tree techniques that
leverages emerging hardware capabilities for mid-circuit measurements and
feed-forward control. Furthermore, we showcase a highly effective use of
approximate compiling to enhance POVM fidelity in noisy conditions. We argue
that our hybrid method scales better toward larger system sizes than its
constituent methods and demonstrate its advantage by performing detector
tomography of symmetric, informationally complete POVM (SIC-POVM). Detector
fidelity is further improved through a composite error mitigation strategy that
incorporates twirling and a newly devised conditional readout error mitigation.
Looking forward, we expect improvements in approximate compilation and hardware
noise for dynamic circuits to enable generalized measurements of larger
multi-qubit POVMs on superconducting qubits.
