Error tolerance and tradeoffs in loss- and failure-tolerant quantum computing schemes

Qubit loss and gate failure are significant problems for the development of scalable quantum computing. Recently, various schemes have been proposed for tolerating qubit loss and gate failure. These include schemes based on cluster and parity states. We show that by designing such schemes specifically to tolerate these error types we cause an exponential blowout in depolarizing noise. We discuss several examples and propose techniques for minimizing this problem. In general, this introduces a tradeoff with other undesirable effects. In some cases this is physical resource requirements, while in others it is noise rates.

Figure 1

Gate-failure-tolerant approach to constructing cluster states. The fundamental building block is the +-cluster. This has a central node (shown in gray), which will ultimately belong to the constructed square lattice. The central node is bonded to four linear chain clusters, each of length $n_l$. These arms provide redundancy, allowing multiple bonding attempts. To grow a cluster, rather than bond two cluster qubits together directly, we utilize +-clusters and attempt bonding starting at the ends of the arms (a). If this fails we lose two qubits from the respective arms, but can recover the remainder of the cluster by measuring the neighboring qubits in the $Z$ eigenbasis. We can keep reattempting the gate until there are no qubits remaining in the arms. When bonding succeeds we have the two desired cluster nodes with some remaining arm qubits left between them. These are removed by measuring them in the $X$ eigenbasis (b).

Figure 2

Examples of different resource states that can be employed in the scalable construction of cluster states using nondeterministic gates.

Figure 3

Using a tree cluster to perform indirect $Z$ measurement of a lost qubit. The qubit below the lost qubit is measured in the $X$ eigenbasis, and each of the qubits below that in the $Z$ eigenbasis. If the $X$ measurement fails, we can make another attempt on the next branch. If any of the $Z$ measurements fail, they can be indirectly measured by moving further down the tree.