Fault tolerance with bare ancillary qubits for a [[7,1,3]] code

We present a $\left[\right[7,1,3\left]\right]$ quantum error-correcting code that is able to achieve fault-tolerant syndrome measurement using one ancillary qubit per stabilizer for an error model of independent single-qubit Pauli errors. All single-qubit Pauli errors on the ancillary qubits propagate to form exclusively correctable errors on the data qubits. The situation changes for error models with two-qubit Pauli errors. We compare the level-1 logical error rates under two noise models: the standard Pauli symmetric depolarizing error model and an anisotropic error model. The anisotropic model is motivated by control errors on two-qubit gates commonly applied to trapped ion qubits. We find that one ancillary qubit per syndrome measurement is sufficient for fault-tolerance for the anisotropic error, but is not sufficient for the standard depolarizing errors. We then show how to achieve fault tolerance for the standard depolarizing errors by adding flag qubits to check for errors on select ancillary qubits. Our results on this $\left[\right[7,1,3\left]\right]$ code demonstrates how physically motivated noise models may simplify fault-tolerant protocols.

Figure 1

The bare $\left[\right[7,1,3\left]\right]$ code embedded in a plane. Each vertex represents a data qubit, and each stabilizer generator of the code corresponds to a face in this graph.

Figure 2

For the Steane $\left[\right[7,1,3\left]\right]$ code with bare ancillary qubits, a single-qubit error on the ancillary qubit would lead to a logical error.

Figure 3

Two examples of two-qubit $XX$ errors that propagate to become uncorrectable errors. In (a) the error leads to an $X_L$, while in (b) the error results in a $Z_L$.

Figure 4

Flagged syndrome measurement for the bare $\left[\right[7,1,3\left]\right]$ code. (a) Circuit to fault-tolerantly measure the syndrome of the stabilizer $Z_2{Z}_3{Y}_5{Y}_6$ using a flag qubit. (b) Circuit to fault-tolerantly measure the syndrome of the stabilizer $Z_0{Z}_1{Z}_2{X}_3{Z}_4{Z}_5$ using a flag qubit. Notice the order of gates is different from what we used in the bare method.

Figure 5

The circuit simulating the bare $\left[\right[7,1,3\left]\right]$ code and its measurement steps. The circuit is constructed with only Clifford gates, and errors (single-qubit and two-qubit) are inserted after ancillary qubit preparation, controlled-not (cnot), controlled-$Z$, controlled-$Y$, and measurements.

Figure 6

Logical error rate for the bare $\left[\right[7,1,3\left]\right]$ code under the standard error model. For the bare method, the logical error rate of the code remains parallel to the physical line for physical error rates below the intersection. This implies that the logical error rate is linear in $p$. For the flag method, we observe a level-1 pseudothreshold of $1.08\times {10}^{-3}$.

Figure 7

Logical error rate for the bare $\left[\right[7,1,3\left]\right]$ code under the anisotropic error model, $p_s=p_t$. For the bare method, the level-1 pseudothreshold is at $p=2.0\times {10}^{-4}$. For the flag method, the level-1 pseudothreshold is at $p=4.0\times {10}^{-4}$. The bare $\left[\right[7,1,3\left]\right]$ code is fault tolerant under the anisotropic error model with both syndrome measurement methods.

Figure 8

Logical error rate with the importance sampler and the traditional sampler for the Steane $\left[\right[7,1,3\left]\right]$ code and the five-qubit code with Shor-style ancillary qubits under the standard depolarizing error model. Up to about $p=2.0\times {10}^{-3}$, the two samplers result in essentially the same logical error rate.

Figure 9

Logical error rate with the importance sampler for the Steane $\left[\right[7,1,3\left]\right]$ code and the five qubit code with Shor-style ancillary qubits under the anisotropic error model. Around $p=1.0\times {10}^{-2}$, the probability of occurrence of the high-weight errors becomes significant and ignoring them causes a dip on the logical error rate. The dip occurs at a lower $p$ under the anisotropic error model because there are more faulty locations than under the standard model.