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[[File:Brickwork State.jpg|center|thumb|500px|Figure 7: Brickwork State]]</div> | [[File:Brickwork State.jpg|center|thumb|500px|Figure 7: Brickwork State]]</div> | ||
'''Definition 1''' A brickwork state Gn×m, where m ≡ 5 (mod 8), is an entangled state of n × m qubits constructed as follows (see also Figure 7): | '''Definition 1''' ''A brickwork state Gn×m, where m ≡ 5 (mod 8), is an entangled state of n × m qubits constructed as follows (see also Figure 7):'' | ||
# Prepare all qubits in state |+i and assign to each qubit an index (i,j), i being a column (i ∈ [n]) and j being a row (j ∈ [m]). | # ''Prepare all qubits in state |+i and assign to each qubit an index (i,j), i being a column (i ∈ [n]) and j being a row (j ∈ [m]).'' | ||
# For each row, apply the operator c-Z on qubits (i,j) and (i,j + 1) where 1 ≤ j ≤ m − 1. | # ''For each row, apply the operator c-Z on qubits (i,j) and (i,j + 1) where 1 ≤ j ≤ m − 1.'' | ||
# For each column j ≡ 3 (mod 8) and each odd row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2). | # ''For each column j ≡ 3 (mod 8) and each odd row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2).'' | ||
# For each column j ≡ 7 (mod 8) and each even row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2). | # ''For each column j ≡ 7 (mod 8) and each even row i, apply the operator c-Z on qubits (i,j) and (i + 1,j) and also on qubits (i,j + 2) and (i + 1,j + 2).'' | ||