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Equation 5 · Part 6 · Error Correction Is the Whole Problem in Quantum Computing

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Si ∣ψL⟩=+∣ψL⟩,i=1,…,n−k.S_i \, |\psi_L\rangle = + |\psi_L\rangle , \qquad i = 1, \dots, n - k .
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The expressions on both sides represent the same quantity under the stated assumptions.

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The passage around this formula

The technique that makes this work is the stabiliser formalism, given its general fault-tolerant treatment by Gottesman [ 4 ] . A code is defined not by listing its states but by naming a set of commuting Pauli operators SiS_i — the stabiliser generators — whose joint +1 eigenspace is the code space: Si ∣ψL⟩=+∣ψL⟩,i=1,…,n−kS_i \, |\psi_L\rangle = + |\psi_L\rangle , \qquad i = 1, \dots, n - k . An [[n, k, d]] code encodes k logical qubits into n physical qubits with distance d . The stabiliser generators are measured repeatedly. Because each SiS_i commutes with every other and with the logical operators, measuring them extracts no information about which encoded state is present. What it extracts is whether an error has anticommuted with a given check — a single…

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An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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