Desirability of terminal X-stabilizer extraction in iceberg-code readout

Determine whether performing non-destructive extraction of the X-type stabilizer (S_X) at the end of circuits implemented with iceberg I{k} quantum error-detecting codes is beneficial, given that the stabilizer-extraction gadget can introduce coherent idling errors that suppress acceptance rates and may add logical errors. Characterize the conditions and regimes under which terminal S_X extraction improves, degrades, or leaves unchanged the overall logical fidelity and acceptance of computations using iceberg codes.

Background

In the iceberg I{k} codes, mid-circuit and final readout can include non-destructive extraction of stabilizers to detect faults. While Z-stabilizers can be reconstructed from destructive Z-basis measurements at the end of a circuit, the X-stabilizer (S_X) can be optionally extracted non-destructively by a dedicated gadget.

The authors note that the S_X readout gadget itself can introduce coherent idling errors that are not visible in destructive Z-basis measurements but do reduce acceptance (and possibly increase logical errors). Empirically, state-preparation-and-measurement experiments that included terminal S_X extraction showed increased discard rates and higher logical error relative to runs without terminal S_X extraction, motivating a systematic determination of when this readout step is advantageous.

References

In general, it is not clear that extracting $S_X$ at the end of a circuit is always desirable, as the gadget itself can introduce coherent idling errors in the $Z$ basis that do not affect the destructive measurements of the qubits but suppress the acceptance rate since they are detected by $S_X$ (and potentially introduce extra logical errors).

— Computing with many encoded logical qubits beyond break-even  (2602.22211 - Dasu et al., 25 Feb 2026) in Supplementary Information, Section III (GHZ-state based syndrome extraction and readout in I{k} codes)