Figure 1

A tree cluster with branching parameters $b_0=b_1=b_2=2$.Reuse & Permissions

Figure 2

Certain measurements on a graph qubit leave the remaining qubits in a new cluster state with a different layout: (a) A $Z$ eigenbasis measurement removes the qubit from the cluster and breaks all bonds between that qubit and the rest. (b) Two adjacent $X$ measurements on a linear cluster remove the qubits and form direct bonds between their neighbors. (c) Measuring a qubit in the $X$ basis and performing $Z$ measurements on all but one of the remaining directly bonded qubits deterministically reveals what the $Z$ measurement outcome would have been on the unmeasured qubit.Reuse & Permissions

Figure 3

An indirect $Z$ measurement will be performed on the indicated qubit if either the three measurements circled in red are successful, or if those circled in green are successful.Reuse & Permissions

Figure 4

A standard cluster computation requires measurement of observable $A$ on the indicated qubit, as indicated in (a). Because of the properties depicted in Fig. 2, either of the two measurement patterns depicted in (b) and (c) will implement the desired measurement of $A$.Reuse & Permissions

Figure 5

Numerical results showing the number of qubits required versus the desired effective loss rate. The different curves, from left to right, correspond to ${\epsilon }_0=0.2$, 0.3, 0.4, 0.49, respectively.Reuse & Permissions