Upper bound for loss in practical topological-cluster-state quantum computing

The surface code cannot be used when qubits vanish during computation; instead, a variant known as the topological cluster state is necessary. It has a gate error threshold of $0.75\%$ and requires only nearest-neighbor interactions on a two-dimensional (2D) array of qubits. Previous work on loss tolerance using this code has only considered qubits vanishing during measurement. We begin by also including qubit loss during two-qubit gates and initialization, and then additionally consider interaction errors that occur when neighbors attempt to entangle with a qubit that is not there. In doing so, we show that even our best case scenario requires a loss rate below $1\%$ in order to avoid considerable space-time overhead.

Figure 1

The five-qubit cluster state found on each of the faces of a 3D topological cluster state cell (Fig. 2). Black dots are the qubits $q_1$ to $q_5$, initialized in the $|+\rangle$ state. Black lines indicate controlled-$Z$ cz gates between the two connected qubits.

Figure 2

Topological cell state. The black and gray dots indicate qubits initialized in the $|+\rangle$ state, while solid lines indicate the application of a cz gate. The black dots at the center of the faces are measured in the $\widehat{X}$ basis $\left(M_X\right)$ and form the measurement product of the cell. The dotted lines are a visual aid only.

Figure 3

Detecting the location of an error using adjacent cells. The dark gray (red) dot is the qubit that has suffered an error. The face stabilizer shared by the cells has been shown, and the others have been omitted for clarity. Each face qubit is shared by two adjacent cells. The thick lines indicate the boundaries of cells where a detection event occurs. The error causes a detection event to occur in both cells, allowing the location of the error to be inferred.

Figure 4

A chain of three errors, in space and time, affecting four cells. This demonstrates how chains can occur through multiple axes. The dark gray (red) dots indicate qubits where an error has occurred. Thick solid lines indicate the boundary of a cell where a detection event occurs. Thin dotted lines indicate the boundary of a cell where no detection event occurs.

Figure 5

A single layer of a distance 3 topological cluster state. Black dots indicate qubits; black lines indicate cz gates. Gates between this layer and others are omitted for clarity. The gray squares are a visual indicator for the two complete dual cells. Dual boundaries are located to the left and right. The larger black dots are the three qubits where a chain of errors could connect the two dual boundaries. A second layer, which is identical except rotated 90 degrees, would have primal cells in gray and boundaries at the top and bottom.

Figure 6

Two cells illustrating a distance 3 fragment of a larger topological cluster state. The two $\left(d_e\right)$ smaller dark dots are qubits with errors. The thick solid lines indicate the boundary of a cell with a detection event. The larger dark dot indicates the most likely error pattern that would create the observed detection event. By applying a correction at the larger dark qubit, where one is not needed, a chain spans the lattice, resulting in a logical error.

Figure 7

A distance 3 topological cluster state on a 2D array of qubits. Each line is a cz gate and is labeled to indicate in which time step(s) the gate is applied. The black dots indicate qubits. The number to the top left of a qubit indicates the time step the qubit is initialized in, and the number to the bottom right when it is measured. Dark and light provide visual separation between the two layers. When implemented on a 2D architecture, no such distiniction exists, and the qubits are laid out physically as presented.

Figure 8

Two adjacent cells with a shared face qubit that has been lost. The dotted dark gray lines indicate the boundary of the merged stabilizer. The black dots indicate the ten qubits whose measurements will form the measurement product for the merged stabilizer. The large gray dot indicates the lost qubit.

Figure 9

An error chain including a lost qubit error. The thick solid lines indicate cells where a detection event occurs. Small dark dots indicate errors. The large gray dot indicates the lost qubit. Note that the merged stabilizer due to the lost qubit behaves like a regular cell, with the detection events occurring at both ends of the regular error chain.

Figure 10

A round of error correction for a single qubit using the topological cluster state, with loss occurring at initialization, two-qubit gates, and measurement. The qubit $q_1$ is initialized in the $\widehat{X}$ basis, interacted with its four neighbors using cz gates, and then measured. The measurement result is passed to a classical computer for error correction. The gray arrows indicate where a qubit may be lost with probability $p_{\mathrm{loss}}$.

Figure 11

Demonstration of the effect of loss interaction errors. The gray arrow indicates where the qubit is lost. The thick gray line indicates the time line of the lost qubit through the circuit. With $p_{\mathrm{lint}}=1$, each neighbor $(q_2,q_3,q_4,q_5)$ acquires an $\widehat{I},\widehat{X},\widehat{Y}$, or $\widehat{Z}$ error. In this instance $q_2$ and $q_4$ acquire an $\widehat{X}$ error, $q_3$ acquires $\widehat{Y}$, and $q_5$ acquires $\widehat{I}$.

Figure 12

A plumbing piece with volume ${(5d/4)}^3$. This defect example contains a cubic fragment (light gray) of circumference $d$ (edge length $d/4$) and with prisms of length $d$ separating it from all neighbors (dark gray).

Figure 13

Logical error rate $\left(P_L\right)$ vs probability of loss $\left(p_{\mathrm{loss}}\right)$. Fixed computational error $\left(p_{\mathrm{comp}}={10}^{-3}\right)$ and no loss interaction error $\left(p_{\mathrm{lint}}=0\right)$.

Figure 14

Logical error rate $\left(P_L\right)$ vs probability of loss $\left(p_{\mathrm{loss}}\right)$. Fixed computational error $\left(p_{\mathrm{comp}}={10}^{-4}\right)$ and no loss interaction error $\left(p_{\mathrm{lint}}=0\right)$.

Figure 15

Logical error rate $\left(P_L\right)$ vs probability of loss $\left(p_{\mathrm{loss}}\right)$. Fixed computational error $\left(p_{\mathrm{comp}}=0\right)$ and no loss interaction error $\left(p_{\mathrm{lint}}=0\right)$. Dotted lines indicate asymptotic lines where logical errors predominantly caused by the minimum necessary $\left(\approx c{p}_{\mathrm{loss}}^{d-1}\right)$.

Figure 16

Logical error rate $\left(P_L\right)$ vs probability of loss $\left(p_{\mathrm{loss}}\right)$. Fixed computational error $\left(p_{\mathrm{comp}}={10}^{-3}\right)$ and 100% chance of loss interaction error $\left(p_{\mathrm{lint}}=1\right)$.

Figure 17

Logical error rate $\left(P_L\right)$ vs probability of loss $\left(p_{\mathrm{loss}}\right)$. Fixed computational error $\left(p_{\mathrm{comp}}={10}^{-4}\right)$ and 100% chance of loss interaction error $\left(p_{\mathrm{lint}}=1\right)$.

Figure 18

Logical error rate $\left(P_L\right)$ vs probability of loss $\left(p_{\mathrm{loss}}\right)$. Fixed computational error $\left(p_{\mathrm{comp}}=0\right)$ and 100% chance of loss interaction error $\left(p_{\mathrm{lint}}=1\right)$. Dotted lines indicate asymptotic lines where logical errors predominantly caused by the minimum necessary $\left(\approx c{p}_{\mathrm{loss}}^{\lfloor \frac{d+3}{4}}\rfloor \right)$.

Figure 19

A 2D representation of the 18-qubit cluster state of a single 3D topological cluster state cell (Fig. 2). Black dots are the qubits $q_1$ to $q_{1}8$, initialized in the $|+\rangle$ state. Black lines indicate cz gates between the two connected qubits. Dotted lines are a visual guide for cz gates that are not nearest neighbors in the 2D flattening.