Direct Measurement of a Very Small Logical Qubit's Observables*

We extend recent longitudinal readout schemes to the very small logical qubit (VSLQ) architecture. The two photon X-operators in the VSLQ are read out by borrowing principles from the longitudinal approach, although the operator itself is transverse. Our results follow those of previous studies of coupling between a single cavity mode and superconducting qubit, where high measurement fidelities are realized for shorter measurement pulses compared to dispersive readout. In the interest of minimizing microwave circuit resources per VSLQ unit cell, we have investigated the use of the higher modes of the same resonator for error correction. We demonstrate use of the fundamental resonator mode for fast readout of an inductively coupled VSLQ with improved fidelities compared to the dispersive case. By measuring along the X-axis, we benefit from the added protection of the passive error correction during readout, reducing potential state preparation and measurement error.