Local fault-tolerant quantum computation

We analyze and study the effects of locality on the fault-tolerance threshold for quantum computation. We analytically estimate how the threshold will depend on a scale parameter $r$ which characterizes the scale-up in the size of the circuit due to encoding. We carry out a detailed seminumerical threshold analysis for concatenated coding using the seven-qubit CSS code in the local and the ‘nonlocal’ setting. First, we find that the threshold in the local model for the ⟦7,1,3⟧ code has a $1∕r$ dependence, which is in correspondence with our analytical estimate. Second, the threshold, beyond the $1∕r$ dependence, does not depend too strongly on the noise levels for transporting qubits. Beyond these results, we find that it is important to look at more than one level of concatenation in order to estimate the threshold and that it may be beneficial in certain places, like in the transportation of qubits, to do error correction only infrequently.

Figure 1

Two-dimensional plane with the spatial layout of $M_1$ and $M_2$. The grayness of the area indicates the amount of moving the qubits potentially need to do.

Figure 2

The replacement rule for a two-qubit gate $U$. The dashed box represents a 1-rectangle. $\mathcal{E}$ represents the error-correction procedure. $\mathbf{U}$ represents the encoded, fault-tolerant implementation of $U$.

Figure 3

The replacement rule for a local two-qubit gate $U$. Each dashed box represents an elementary 1-rectangle. $\mathcal{E}$ represents the error-correction procedure. $\mathbf{U}$ represents the local fault-tolerant implementation of $U$. The replacement circuit, i.e., the composite 1-rectangle, contains five elementary 1-rectangles.

Figure 4

Flows of the one- and two-qubit failure rates under concatenation in the nonlocal model. We initially set ${\gamma }_1={\gamma }_2={\gamma }_p={\gamma }_m=10\times {\gamma }_w$. Four starting values are shown, two below threshold and two above. The initial flow is evidently very similar regardless of whether the map is above or below threshold. The hyperbolic structure of the flow is controlled by an unstable fixed point of the map at ${\gamma }_1={\gamma }_w={\gamma }_{1m}={\gamma }_p=0.69\times {10}^{-4}$ and ${\gamma }_2=1.50\times {10}^{-4}$, shown as the black “star” symbol. Note that the line onto which these flows asymptote has ${\gamma }_2$ very close to $2\times {\gamma }_1$.

Figure 5

Flows of the one- and two-qubit failure rates under concatenation in the nonlocal model. We initially set ${\gamma }_1=0.25\times {\gamma }_2={\gamma }_p={\gamma }_m=10\times {\gamma }_w$. Four starting values are shown, two below threshold and two above. The initial flow is evidently very similar regardless of whether the map is above or below threshold. The hyperbolic structure of the flow is controlled by an unstable fixed point of the map at ${\gamma }_1={\gamma }_w={\gamma }_{1m}={\gamma }_p=0.69\times {10}^{-4}$ and ${\gamma }_2=1.50\times {10}^{-4}$, shown as the black “star” symbol. Note that the line onto which these flows asymptote has ${\gamma }_2$ very close to $2\times {\gamma }_1$.

Figure 6

Flows of the one- and two-qubit failure rates under concatenation in the nonlocal model. We initially set ${\gamma }_1=2.0\times {\gamma }_2={\gamma }_p={\gamma }_m=10\times {\gamma }_w$. Four starting values are shown, two below threshold and two above. The initial flow is evidently very similar regardless of whether the map is above or below threshold. The hyperbolic structure of the flow is controlled by an unstable fixed point of the map at ${\gamma }_1={\gamma }_w={\gamma }_{1m}={\gamma }_p=0.69\times {10}^{-4}$ and ${\gamma }_2=1.50\times {10}^{-4}$, shown as the black “star” symbol. Note that the line onto which these flows asymptote has ${\gamma }_2$ very close to $2\times {\gamma }_1$.

Figure 7

The threshold line and pseudothreshold curves shown in the plane defined by the memory failure rate ${\gamma }_w$ and all other failure rates ${\gamma }_{\text{else}}={\gamma }_1={\gamma }_2={\gamma }_p={\gamma }_m$. The pseudothreshold is defined as the line along which one of the failure rates remains unchanged after the first iteration of the map. Closer to the origin, this failure rate decreases; farther away, it increases. The pseudothresholds for ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_w$ are shown. We note that along the line (dotted) for which ${\gamma }_w=0.1\times {\gamma }_{\text{else}}$, a popular condition in earlier studies, the gate pseudothresholds, particularly for the one-qubit gate failure rate, are much higher than the true threshold.

Figure 8

Gate failure rate threshold vs the scale parameter $r$ for the local model. We have taken ${\gamma }_1={\gamma }_2={\gamma }_m={\gamma }_p$, ${\gamma }_{w1}={\gamma }_{w2}=0.1\times {\gamma }_2$, ${\gamma }_{wd}=0.1\times {\gamma }_{md}$, and ${\gamma }_{md}=r∕\tau \times {\gamma }_2$. $\tau$, the frequency with which a qubit is error-corrected while being moved over distance $r$, is optimized in every case. The threshold follows very close to a $1∕r$ dependence.

Figure 9

Gate failure rate threshold vs $\tau$, the frequency of error correction of a transported qubit, for $r=50$. As in Fig. 8, we have taken ${\gamma }_1={\gamma }_2={\gamma }_m={\gamma }_p$, ${\gamma }_{w1}={\gamma }_{w2}=0.1\times {\gamma }_2$, ${\gamma }_{wd}=0.1\times {\gamma }_{md}$, and ${\gamma }_{md}=r∕\tau \times {\gamma }_2$. While not very strongly $\tau$-dependent, the optimal threshold occurs at $\tau =4$.

Figure 10

Gate failure rate threshold vs $ϵ$, a parameter that measures the relative noise rate per unit distance for a qubit being moved. We initially set ${\gamma }_1={\gamma }_2={\gamma }_m={\gamma }_p$, ${\gamma }_{w1}={\gamma }_{w2}=0.1\times {\gamma }_2$, ${\gamma }_{wd}=0.1\times r∕\tau \times {\gamma }_2$, and ${\gamma }_{md}=ϵr∕\tau \times {\gamma }_2$. Scale parameters $r=20$, 50, and 80 are studied. At every point, $\tau$ is reoptimized. The dependence on $ϵ$ is slow, evidently slower than $1∕ϵ$.

Figure 11

The replacement rule for a one-qubit gate location $U$ or a wait1 location. The dashed box represents a 1-rectangle. $\mathcal{E}$ represents the error-correction procedure. $\mathbf{U}$ represents the local fault-tolerant implementation of $U$. Note that in each figure, a qubit in $M_{n-1}$ is encoded as $m=7$ qubits in $M_n$.

Figure 12

The replacement rule for a one-qubit gate $U$ followed by a measurement.

Figure 13

The replacement rule for a wait2 (also called $w2$) location acting in parallel with a two-qubit gate. The replacement circuit contains three elementary 1-rectangles.

Figure 14

The replacement rule for a $\text{move}\left(\mathrm{r}\right)$ gate. The replacement circuit contains $r$ elementary 1-rectangles.

Figure 15

The replacement rule for a wait(r) gate. The replacement circuit contains $r$ elementary 1-rectangles.

Figure 16

The Steane $X$-error-correction protocol, $\mathcal{X}$ [5]. The black circle represents control on a nonzero result. A white circle represents control on a zero result. $s^{\prime }$ represents a classical procedure to check if $s^{\prime }$ of the $s$ syndromes agrees. The dashed box procedure is applied only if the controlling syndrome is not zero. There are $n_{\mathrm{rep}}$ prepared ancilla blocks. Each line represents seven qubits. After $\mathcal{V}$, $\alpha$ “good” verification blocks remain. $\mathcal{R}$ represents the recovery procedure.

Figure 17

The $\mathcal{G}$ network for $X$ or $Z$-error correction [3]. The network can be executed in five time steps. It produces the encoded ∣0⟩ state. The boxed zero represents preparation of a ∣0⟩ state.

Figure 18

The $\mathcal{V}$ network [3, 14], executable in six time steps. The boxed zero represents preparation of the ∣0⟩ state. The state ∣0⟩ is the seven-qubit encoded ∣0⟩ state. If each measurement output is 0, then the ancilla block is deemed “good,” that is, it has been checked for $X$ errors. The network is the same for the $Z$-error-correction procedure.

Figure 19

The syndrome network $\mathcal{S}$ for $X$-error correction [3]. This network can be executed in three time steps. Here EE represents classical error extraction. The network $\mathcal{S}$ for $Z$-error correction uses $C^X$ gates in place of $C^Z$ gates, with the ancillas acting as control and the data as target qubits.