Surface Code Stabilizers for Defective Qubit Error Detection

The patent segments the quantum system into functional and non-functional qubit regions, and further segments gauge operators into X-type and Z-type categories. This segmentation allows independent optimization of measurement cycles for each type, resolving the interleaving complexity while maintaining dense packing for improved error detection efficiency.

The patent implements periodic measurement cycles where X-type and Z-type gauge operators are measured in alternating but synchronized periods. Each type of gauge operator is measured in every cycle, but with periodic alternation between types, achieving dense packing without requiring complex simultaneous interleaving of non-commuting operators.

The patent extracts non-functional qubits from the active measurement cycle by mapping gauge operators around them rather than through them. This allows measurement cycles to maintain their original timing while still achieving comprehensive error detection coverage, as the operators effectively skip over defective regions without extending the cycle duration.

The patent transitions from a two-dimensional spatial arrangement to a four-dimensional measurement space by adding temporal dimension (measurement cycles) and operator type dimension (X-type and Z-type). This allows gauge operators to be densely packed in the operator-type dimension and cycled through time, achieving comprehensive coverage without extending spatial measurement paths and thus avoiding increased measurement time.