Topological one-way quantum computation on verified logical cluster states

We present a scheme to improve the noise threshold for fault-tolerant topological one-way computation with constant overhead. Certain cluster states of finite size, say star clusters, are constructed with logical qubits through an efficient verification process to achieve high fidelity. Then, the star clusters are connected near-deterministically with verification to form a three-dimensional cluster state to implement topological one-way computation. The necessary postselection for verification is localized within the star clusters, ensuring the scalability of computation. By using the Steane seven-qubit code for the logical qubits, this scheme works with a high error rate of $2\%$ and reasonable resources comparable to or less than those for the other fault-tolerant schemes. A higher noise threshold would be achieved by adopting a larger code.

Figure 1

(a) Star clusters to be connected. (b) Verification of the transversal CZ gate by measuring the qubits 1 and 3. If the connection succeeds, the redundant qubits 2 and 4 are removed (left). Otherwise, the connection is abandoned (right). (c) By repeating this process, the root nodes are connected to form a cluster state.

Figure 2

The double verification for the logical qubits in the construction of a star cluster.

Figure 3

A 3D cluster state for TOWC, where each qubit is connected to four neighboring qubits in a specific way [13]. In the present scheme, this cluster state is constructed by using the process in Fig. 1 so that each qubit is replaced by a logical one with high fidelity.

Figure 4

The resources per two-qubit gate for a computation size ${10}^{21}$ (gate accuracy $~$${10}^{-22}$) are plotted as functions of the error rate $p$ for the present scheme with double verification (red $♦$), the cluster-based architecture (green $◯$) [10] and the Knill’s error-correcting architecture (purple $\square$) [8].