Cluster-based architecture for fault-tolerant quantum computation

We present a detailed description of an architecture for fault-tolerant quantum computation, which is based on the cluster model of encoded qubits. In this cluster-based architecture, concatenated computation is implemented in a quite different way from the usual circuit-based architecture where physical gates are recursively replaced by logical gates with error-correction gadgets. Instead, some relevant cluster states, say fundamental clusters, are recursively constructed through verification and postselection in advance for the higher-level one-way computation, which namely provides error-precorrection of gate operations. A suitable code such as the Steane seven-qubit code is adopted for transversal operations. This concatenated construction of verified fundamental clusters has a simple transversal structure of logical errors, and achieves a high noise threshold $~3\%$ for computation by using appropriate verification procedures. Since the postselection is localized within each fundamental cluster with the help of deterministic bare controlled-$Z$ gates without verification, divergence of resources is restrained, which reconciles postselection with scalability.

Figure 1

The error probabilities ${\epsilon }_A/(p_g/15)$ ($A=X,Y,Z$) for each level-0 qubit are plotted as functions of the physical error probability $p_g$ together with their leading values ${\epsilon }_X(p_g/15)={\epsilon }_Y/(p_g/15)=1$ and ${\epsilon }_Z/(p_g/15)=2$.

Figure 2

The error probability $p_q^{(1)}$ for measuring the level-1 qubit after the bare $\mathrm{C}Z$ connection is plotted as a function of the physical error probability $p_g$. The error probabilities $p_{\mathrm{Pauli}}^{(1)}$ for the Pauli frames ($X,Y,Z$) of the level-1 qubit are also plotted as functions of $p_g$ in comparison with $p_q^{(1)}$. The upper-most line indicates $p_q^{(0)}$ in comparison to infer the threshold.

Figure 3

The error probability $p_q^{(2)}$ for measuring the level-2 qubit after the bare $\mathrm{C}Z$ connection is plotted as a function of the physical error probability $p_g$, together with the leading term ${}_7{\mathrm{C}}_2(p_q^{(1)}){}^2$ (dotted line). The upper-most line indicates $p_q^{(1)}$ in comparison to infer the threshold.

Figure 4

The success probabilities $p_0^{(l)}$ are plotted as functions of the physical error probability $p_g$ for the levels $l=1,2,3,4$.

Figure 5

Resources for the present architecture of verified logical clusters (LC) with $p_g={10}^{-2}$ and ${10}^{-3}$, which are compared with those for the Steane’s QEC scheme with $p_g={10}^{-3}$.