Nonuniform code concatenation for universal fault-tolerant quantum computing

Using transversal gates is a straightforward and efficient technique for fault-tolerant quantum computing. Since transversal gates alone cannot be computationally universal, they must be combined with other approaches such as magic state distillation, code switching, or code concatenation to achieve universality. In this paper we propose an alternative approach for universal fault-tolerant quantum computing, mainly based on the code concatenation approach proposed in [T. Jochym-O'Connor and R. Laflamme, Phys. Rev. Lett. 112, 010505 (2014)], but in a nonuniform fashion. The proposed approach is described based on nonuniform concatenation of the 7-qubit Steane code with the 15-qubit Reed-Muller code, as well as the 5-qubit code with the 15-qubit Reed-Muller code, which lead to two 49-qubit and 47-qubit codes, respectively. These codes can correct any arbitrary single physical error with the ability to perform a universal set of fault-tolerant gates, without using magic state distillation.

Figure 1

The schematic overview of the nonuniform code concatenation approach. Logical information is encoded by $C_1$. In the second level of concatenation each qubit of $B_1$ is, in turn, encoded into the code of $C_2$ and the $B_2$ qubits are left unchanged without encoding.

Figure 2

(a) Staircase of $\text{cnot}\mathrm{s}$. (b) Nontransversal application of $C^{k}Z\left(\theta \right)$ gate for a stabilizer code by involving only $d$ qubits of each code block, where $d$ is the code distance. $SC$ is an acronym for staircase of $\text{cnot}\mathrm{s}$ and $LC$ is a circuit containing only local Clifford gates which transform the original logical $Z$ operator into a form consisting of only Pauli $Z$'s and $I$'s. Note that only the qubits of $B_1$ are shown.

Figure 3

Fault-tolerant application of the $T$ gate for the proposed 49-qubit nonuniform concatenated code. A logical qubit is encoded by Steane where qubits 1, 2, and 7 are themselves encoded blocks of RM code and the other four qubits are left unchanged.

Figure 4

Nontransversal implementation of the $C^{k}Z\left(\theta \right)$ gate for the 5-qubit code. The circuit specified by the dotted box shows the implementation of this gate for the $5^{\prime }$-qubit code. Note that while this circuit generally holds for the 5-qubit code, not every $C^{k}Z\left(\theta \right)$ can be applied fault-tolerantly for the proposed 75- and 47-qubit concatenated codes, as they may be nontransversal in RM. Indeed, from the $C^{k}Z\left(\theta \right)$ gates, only the gates of $M=\{T,S,CZ,\text{ccz}\}$, which are transversal in RM, can be applied fault-tolerantly for the proposed 75- and 47-qubit concatenated codes.