A circuit-diagrammatic treatment of the 7-Qbit code
N. David Mermin · 2007
As a further exercise in the use of circuit diagrams, we rederive the properties of the 7-Qbit error-correcting code, using the method developed in Chapter 5 to establish that the circuit in Figure 5.11 gives the 5-Qbit codewords. We start with the observation that the seven mutually commuting operators M i , N i ( i = 0, 1, 2) in (5.42), and in (5.49), each with eigenvalues ±1, have a set of 2 7 nondegenerate eigenvectors that form an orthonormal basis for the entire seven-dimensional codeword space. In particular the two codeword states and are the unique eigen states of all the M i and N i with eigenvalues 1, and of with eigenvalues 1 and –1, respectively. It follows from this that if a circuit produces a state |Ψ〉 that is invariant under all the M i and N i then |Ψ〉 must be a superposition of the codeword states and, and if |Ψ〉 is additionally an eigenstate of then, to within factors e i ϕ of modulus 1, |Ψ〉 must be or depending on whether the eigenvalue is 1 or –1. Figure O.1 shows that the state |Ψ〉 produced by the circuit in Figure 5.10 is indeed invariant under M 0 = X 0 X 4 X 5 X 6 . This figure demonstrates that when M 0 is brought to the left through all the gates in the circuit it acts directly as Z 0 on the input state on the left, which is invariant under Z 0 .