Extended Data Fig. 6: Preparing a CZ state over two [[4, 1, 2]]-codes.

At step 1 the codes are prepared. The [[4, 2, 2]] code that encodes the two-qubit CZ-state is represented by the red square where its four qubits lie at the vertices of the square. This preparation is described in the main text. The code is prepared adjacent to a [[4, 1, 2]]-code that is initialized in an eigenstate of the \(\left|+\right\rangle \) state. The qubits in the figure are indexed according to the qubit-map shown in Extended Data Fig. 7. At step 2 the qubits are transported in order to perform a logical parity measurement in step 3 using the heavy-hex lattice geometry. Note that the qubit indices have changed. This step can be performed with swaps, for instance, as shown in Extended Data Fig. 7(top). At step 3 a logical parity measurement is made. It can be performed in a fault-tolerant manner using qubits 5, 10, and 16, as shown in the green box in Extended Data Fig. 7(bottom). We complete the operation by measuring the logical operator \({\overline{Z}}_{2}\) in step 4. This weight-two measurement can be repeated in two locations on the [[4, 2, 2]] code such that a single measurement error can be detected. This final measurement projects the [[4, 2, 2]] code onto the [[4, 1, 2]]-code by reassigning the \({\overline{Z}}_{B}\) logical operators as stabilizers of the system.