Fault Tolerant Quantum Computation with Very High Threshold for Loss Errors

Many proposals for fault tolerant quantum computation (FTQC) suffer detectable loss processes. Here we show that topological FTQC schemes, which are known to have high error thresholds, are also extremely robust against losses. We demonstrate that these schemes tolerate loss rates up to 24.9%, determined by bond percolation on a cubic lattice. Our numerical results show that these schemes retain good performance when loss and computational errors are simultaneously present.

Figure 1

(a) Unit cell showing qubits centered on faces and edges of the primal lattice. Heavy lines indicateCPHASE gates. (b) A primal identity gate showing logical input operators ${\overline{X}}_{\mathrm{in}}$ and ${\overline{Z}}_{\mathrm{in}}$ and output operators ${\overline{X}}_{\mathrm{out}}$ and ${\overline{Z}}_{\mathrm{out}}$, and the correlation surfaces ${\sigma }_{\overline{X}}$ and ${\sigma }_{\overline{Z}}$, that interpolate between them. Also shown are physical qubits (spheres) living on the correlation surface between the $\overline{Z}$ operators, and the defect regions $D$ embedded in the bulk vacuum $V$. (c) Failed syndromes for error chains on the primal $E$ and dual $E^{*}$ lattices, and candidate correction chains $C$ and $C^{*}$.

Figure 2

(a) A correlation surface suffering loss is repaired by deforming around lost qubits (white spheres), by multiplying by the corresponding check operator. An affected logical input or output operator is similarly deformed. (b) In the bulk, supercheck operators are formed by products of elementary check operators that are affected by a loss. Error chains terminate within a supercheck operator.

Figure 3

Phase diagram showing correctable region of parameter space. Red circles are numerically calculated computational error thresholds, $p_t$ at different loss rates, for an error model in which $p_P=p_S=p_M=p_2=p_{\mathrm{comp}}$. The green diamond is the 3D bond percolation threshold [17]. The blue square is the threshold computed in [7], which ignores degeneracy in the matching algorithm.