Optimized Measurement Schedules for the Surface Code with Dropout

Recent work has shown that fabrication defects can be well-handled using a strategy relying on the mid-error-correction-cycle state. In this work we present two improvements to the original prescription. First, we quantify the impact of the choice of a more complete set of gauge operators originally proposed for the hex-grid surface code on the standard square-grid surface code, as well as a new method for excising effectively unused qubits. Second, we leverage the expressivity of the LUCI framework as an intermediate representation, using integer linear programming to find performant physical circuits from the large space of valid LUCI circuits. We show that on the d = 11 surface code at 1\%(3\%) dropout rate for qubits and couplers, these optimizations allow for a total improvement of 14.5\%(23.6\%) over 4d round of syndrome extraction using the SI1000 noise model at 0.1\% noise.