Thresholds for Topological Codes in the Presence of Loss

Many proposals for quantum information processing are subject to detectable loss errors. In this Letter, we show that topological error correcting codes, which protect against computational errors, are also extremely robust against losses. We present analytical results showing that the maximum tolerable loss rate is 50%, which is determined by the square-lattice bond percolation threshold. This saturates the bound set by the no-cloning theorem. Our numerical results support this and show a graceful trade-off between tolerable thresholds for computational and loss errors.

Figure 1

(a) Physical qubits (arrows) reside on the edges of a square lattice (dashed line). Also depicted are a plaquette operator, a star operator, the logical $\overline{Z}$ operator, and the logical $\overline{X}$ operator. (b) In the event of a qubit loss, an equivalent logical operator $\tilde{Z}$ can be routed around the loss.

Figure 2

(a) Lattice with two lost qubits (crosses). Representative qubits and plaquettes are labeled 1–8 and $\mathsf{A}–\mathsf{E}$, respectively. (b) Plaquettes sharing a lost qubit ($\{\mathsf{A},\mathsf{B}\}$ and $\{\mathsf{C},\mathsf{D}\}$) become superplaquettes $\mathsf{A}\mathsf{B}$ and $\mathsf{C}\mathsf{D}$ and may be multiply connected (i.e., share more than one qubit). (c) Degraded lattice showing superedges (thick lines). (d) Restored lattice with zero weight edges (dotted line) and irregular weights (thick line).

Figure 3

Correctability phase diagram. The shaded region is correctable in the limit $L\to \infty$. The threshold $p_t$ is calculated by fitting the universal scaling law $p_{\mathrm{fail}}=f[(p_{\mathrm{com}}-p_t)L^{1/{\nu }_0}]$. The curve is a quadratic fit to the points for which $p_{\mathrm{loss}}\le 0.4$ (where universal scaling is unaffected by the finite lattice size). It extrapolates through $(p_{\mathrm{loss}},p_t)=(0.5,0)$.