4.10 Stabilizer syndromes and Clifford packets
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4.10.1 Syndrome projectors
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Let \(S_1,\ldots,S_m\) be commuting Pauli stabilizers with eigenvalues \(\pm1\). A syndrome \(s\in\{0,1\}^m\) defines
These projectors are orthogonal and sum to the identity. Syndrome measurement selects one packet, while recovery applies a syndrome-dependent transfer toward the code packet.
4.10.2 The syndrome label is not the logical state
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If the code encodes \(k\) logical qubits, a fixed syndrome packet can have dimension \(2^k\). The syndrome has only \(m\) bits, but arbitrary logical information remains inside the packet. Syndrome packetization therefore does not by itself prove efficient simulation.
4.10.3 Tableau compression
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Clifford circuits provide a different route. A stabilizer state may occupy exponentially many computational-basis strings, yet it is represented by a polynomial-size stabilizer tableau. Clifford gates update the tableau in polynomial time, and Pauli measurements are efficiently computable. This is an example of the compressed-packet theorem rather than a small-dimensional packet theorem.
Derived theorem or corollary
Packet labels and internal encoding are complementary. A Clifford circuit restricted to selected syndromes may be simulated by a tableau augmented with the active syndrome graph, even when the packet dimensions are exponential.
4.10.4 Non-Clifford residues
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A few non-Clifford gates can be handled by stabilizer decompositions, quasiprobability expansions, or magic-state rank methods. The cost grows with a non-Clifford resource such as stabilizer rank or robustness. Symmetry packets may reduce the number of relevant branches, but they do not eliminate the intrinsic cost of the non-Clifford residue.
4.10.5 Error-correction interpretation
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The packet view separates three objects:
This separation is useful for compressed simulation of fault-tolerant protocols, but a full complexity statement must include repeated measurements, classical control, and the size of the logical state.