Optimized Measurement Schedules for the Surface Code with Dropout

TL;DR

This paper tackles dropout in surface-code quantum error correction by treating LUCI mid-cycle diagrams as a flexible intermediate representation. It introduces two main advances: (i) expanding the gauge-operator set and removing unused qubits to strengthen the midpoint code, and (ii) an ILP-based optimization that designs performant LUCI circuit shapes and schedules without altering the code distance. The combined approach delivers substantial reductions in logical error rate for a distance-$d=11$ surface code at dropout rates of $1\%$ and $3\%$ under SI1000 noise, achieving about $14.5\%$ and $23.6\%$ improvements respectively, and demonstrates robust gains without extensive hyperparameter tuning. These results suggest a practical path to mitigating fabrication defects in scalable quantum memory implementations by leveraging a one-time compilation step and a versatile intermediate representation for circuit optimization.

Abstract

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.

Figures (18)

• Figure 1: Here we track each stabilizer of the rotated surface code plus syndrome qubit system. We see that after the first two layers of cnots we land in the unrotated surface code state before returning to the rotated surface code state. Note that the familiar stabilizers we end up with are actually the result of evolving the single-qubit stabilizers. We choose a basis of weight-five and weight-one stabilizers instead of weight-one and weight-four to make the transformation more obvious.
• Figure 2: An example of a $Z$-type and $X$-type shape. The light gray legs of the shape correspond to the first cnot layer, the dark gray crossbeam corresponds to the second layer, and the dot corresponds to the measurement. This circuit is then repeated in reverse (substituting a reset for the measurement) to create the new weight-four stabilizer. For the sake of clarity we have labeled the qubits in this example.
• Figure 3: Here we provide an example of how a missing qubit and missing coupler are treated using a LUCI diagram. This diagram corresponds to four mid-cycle to mid-cycle rounds.
• Figure 4: The stabilizers produced for a fixed dropout configuration when we disallow weight-one gauge operators as in the original prescription debroy2024lucisurfacecodedropouts (left) versus when we allow weight-one gauge operators as in more recent work higgott2025handlingfabricationdefectshexgrid. One minimum weight representative of logical $Z$ is marked in each case.
• Figure 5: Comparison of the same two-round solution for a given dropout configuration with and without removing unnecessary boundary qubits. The boundaries in \ref{['fig:removal']} are asymmetrical because the usage of the syndrome qubits on the boundary in \ref{['fig:no_removal']} is asymmetrical.