Fault-tolerant quantum computation with nondeterministic entangling gates

Performing entangling gates between physical qubits is necessary for building a large-scale universal quantum computer, but in some physical implementations—for example, those that are based on linear optics or networks of ion traps—entangling gates can only be implemented probabilistically. In this work, we study the fault-tolerant performance of a topological cluster state scheme with local nondeterministic entanglement generation, where failed entangling gates (which correspond to bonds on the lattice representation of the cluster state) lead to a defective three-dimensional lattice with missing bonds. We present two approaches for dealing with missing bonds; the first is a nonadaptive scheme that requires no additional quantum processing, and the second is an adaptive scheme in which qubits can be measured in an alternative basis to effectively remove them from the lattice, hence eliminating their damaging effect and leading to better threshold performance. We find that a fault-tolerance threshold can still be observed with a bond-loss rate of 6.5% for the nonadaptive scheme, and a bond-loss rate as high as 14.5% for the adaptive scheme.

Figure 1

The three-dimensional topological cluster state, constructed from cubic cells (inset); bulk qubits are hidden for clarity. Only the ends of strings of $Z$ errors (blue qubits surrounded by wire-frame cubes) are detected by check operators (highlighted cubes). The correlation surface (shaded surface) spans the lattice in the direction of the computation (shown by the arrow).

Figure 2

(a) Multiplying two cubes to form a supercheck removes the face qubit shared between them and results in a parity check involving the $X$ measurements associated with the ten remaining face qubits. (b) In the nonadaptive approach, all bulk qubits are measured in the $X$ basis in the presence of a failed bond. (c) In the adaptive approach, one of the qubits incident on the missing bond is randomly chosen to be measured in the $Z$ basis while the other is measured in the $X$ basis.

Figure 3

Thresholds in the presence of failed bonds. The shaded regions indicate correctable error rate combinations. In the absence of bond failures ($p_{\mathrm{bond}}=0$), the threshold (shown by the marker on the $y$ axis) agrees with [14].