We first note that the code generators commute, so it is sufficient to measure each one separately in order to obtain the syndrome. A method for performing this measurement, which is due to Shor [51], is based on the circuit in figure 5.1 that measures a unitary operatorU with eigenvalues±1—since the eigenvalues of the code generators are±1, they can be measured using this circuit. For instance, consider a code generator,Ga, of weightw; if we view aas a length-2nbinary vector,
Ga = n O j=1
data block U .. . ... |+i • >= X
Figure 5.1. A measurement of U using one ancillary qubit. If the measurement outcome is +1 (respectively,−1), the data qubits are projected in the +1 (respectively,−1) eigenspace ofU. The controlled-U gate appliesU on the data block if the state of the ancillary qubit is|1i, and applies the identity if the state of the ancillary qubit is|0i.
Then, the circuit for measuring Ga will consist of an ancillary qubit, a sequence of w two-qubit gates (we apply w controlled- ia[j]·a[j+n]Xa[j]Za[j+n]
gates with control the ancillary qubit and target qubitj in the data block), and a final measurement of the ancillary qubit.
This circuit does notyet satisfy properties 1 because the ancillary qubit interacts with multiple qubits in the data block. This is a problem because a single fault acting on the ancillary qubit can give rise to an error that propagates to cause errors on several qubits in the data block; e.g., if the error acting on the ancillary qubit is X, then it propagates to become an X error on any qubits in the data block that interact with this ancillary qubit via cnot gates subsequent to the fault.1 To prevent such propagation of errors to multiple qubits in the code block, we can replace the one ancillary qubit in figure 5.1 by multiple ancillary qubits. In particular, we can prepare w ancillary qubits in the cat state, |+irep ∝ |0i⊗w+|1i⊗w, i.e., the logical |+i state of the w-qubit quantum repetition code [51, 87]. Now, the interaction of ancillary and data qubits is bitwise or
transversal: thejth ancillary qubit interacts via a controlled- ia[j]·a[j+n]Xa[j]Za[j+n]
gate with the jth qubit in the data block. The benefit of transversal interactions is that faults acting on, say, r qubits in the cat state can cause errors on at mostr qubits in the data block, thus solving the problem of propagation of errors. The measurement in figure 5.1 needs now to be replaced by a measurement on all wancillary qubits. This measurement can also be performedtransversally by separately measuring each of thewancillary qubits along the eigenbasis ofX; then we can calculate the eigenvalue ofGa by computing the parity of the transversal measurement outcomes. However, in general this eigenvalue cannot immediately be trusted: Since even a single fault might flip one of the tranversal measurements causing an error in the computation of the parity of the outcomes, each code generator must be measured repeatedly sufficiently many times depending on the distance of the code; the eigenvalue ofGa will then be obtained by taking a majority vote.
1It is important to note howX andZerrors are propagated by
cnotgates. Acnotpropagates anX error in its control qubit toX errors in both control and target qubits; it also propagates aZerror in its target qubit toZerrors in both control and target qubits.
data block |0i • • >= X |+i • • • >= X |0i • • >= X |0i • • >= X |0i >= Z+1
Figure 5.2. Example of syndrome measurement with cat states. In this case, the code generatorX⊗4 is measured. A four-qubit cat state is prepared and verified, and then it controls the application of the Pauli operator to the data block. Finally, all ancillary qubits are measured along the eigenbasis ofX, and the parity of the measurement outcomes equals the measured eigenvalue ofX⊗4.
To finish the construction of the 1-EC gadget, it remains to specify how to prepare the cat state. Constructing an encoding circuit for |+irep is straightforward: e.g., we may start with the state |+i ⊗ |0i⊗(w−1) and perform
cnot gates with control the first qubit and target every other qubit. To deal with the problem of the propagation of errors inside the encoding circuit, we may nextverify
the cat state [51]. Verification can be performed by preparing extra ancillary qubits and using them to measure operators in the stabilizer of the cat state in order to check for undesired fault patterns inside the encoding circuit—of concern are multipleXerrors since any pair ofZerrors acts trivially on|+irep. For instance, figure 5.2 shows the complete circuit for measuring the weight-4 operator X⊗4. The verification step is a measurement of the operatorZ⊗I⊗I⊗Z which belongs in the cat-state stabilizer. If the outcome of the verification measurement is−1, then a fault might have occurred in one of the two later cnot gates during cat-state encoding which might have caused X errors on two qubits of the cat state; in this case, the cat state isdiscardedand the encoding is repeated anew. If the outcome of the verification measurement is +1, at least two faults are required to introduce twoX errors in the cat state, which can therefore be accepted. It is staightforward to adapt the principles underlying the construction of this circuit to measure code generators of higher weight for any stabilizer code.
Due to the transversal interaction between the verified cat state and the data block, property 1(b) is satisfied: Indeed, if the input data block passes through an s-filter, then its state can be expanded as a sum of terms where, in each term, Pauli errors on at mostsqubits act on some state in the code space. Also, therfaults inside the 1-EC gadget cannot cause errors acting on more than r qubits in the data block output from 1-EC. Overall, the state of the data block that is output from 1-EC can be expanded as a sum of terms where, in each term, Pauli errors on at most s+r qubits act on some state in the code space. Sinces+r≤t, all these Pauli errors can be corrected, which implies that their linear sum will also be corrected. In§5.2.1.4, we will discuss why property
1(a) is also satisfied. Finally, let us note that cat-state verification isnotnecessary for obtaining a 1-EC gadget satisfying properties 1; [84] discusses a more efficient procedure in which verification is avoided and, instead, the cat state is decodedafter interacting with the data.