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.