Using Concatenated Quantum Codes for Universal Fault-Tolerant Quantum Gates

We propose a method for universal fault-tolerant quantum computation using concatenated quantum error correcting codes. The concatenation scheme exploits the transversal properties of two different codes, combining them to provide a means to protect against low-weight arbitrary errors. We give the required properties of the error correcting codes to ensure universal fault tolerance and discuss a particular example using the 7-qubit Steane and 15-qubit Reed-Muller codes. Namely, other than computational basis state preparation as required by the DiVincenzo criteria, our scheme requires no special ancillary state preparation to achieve universality, as opposed to schemes such as magic state distillation. We believe that optimizing the codes used in such a scheme could provide a useful alternative to state distillation schemes that exhibit high overhead costs.

Figure 1

General construction of a logical gate for a concatenated error correction scheme. The qubit of information is encoded in a quantum error correcting code ${\mathcal{C}}_1$, whose qubits are in turn encoded into a code ${\mathcal{C}}_2$. As such, the logical gate $g_1$ (given by the boxed region) on the encoded space ${\mathcal{C}}_1$ will be composed of multiple logical gates $g_2$ on the ${\mathcal{C}}_2$ code blocks.

Figure 2

(a) Logical $T$ gate for the Steane code ${\mathcal{C}}_1$, composed of logical ${\text{CNOT}}_{15}$ and $T_{15}$ gates on the ${\mathcal{C}}_2$ code blocks. These gates are transversal in ${\mathcal{C}}_2$, and therefore only propagate errors to different code blocks, without propagating within a given ${\mathcal{C}}_2$ code block. (b) Logical $H$ gate for the Steane code ${\mathcal{C}}_1$, implemented using logical $H_{15}$ gates on each of the ${\mathcal{C}}_2$ code blocks. Note that the individual $H_{15}$ are nontransversal on each code block.