Quantum-error correction on linear-nearest-neighbor qubit arrays

A quantum circuit implementing 5-qubit quantum-error correction on a linear-nearest-neighbor architecture is described. The canonical decomposition is used to construct fast and simple gates that incorporate the necessary swap operations allowing the circuit to achieve the same depth as the current least depth circuit. Simulations of the circuit’s performance when subjected to discrete and continuous errors are presented. The relationship between the error rate of a physical qubit and that of a logical qubit is investigated with emphasis on determining the concatenated error correction threshold.

Figure 1

Decomposition into physical operations of (a) CNOT gate, (b) Hadamard gate, CNOT gate, then swap gate. Note that the Kane architecture has been used for illustrative purposes. In addition to the clear speed advantage when implementing compound gates, the decomposed CNOT gate is faster than its adiabatic equivalent $\left(26\mu \mathrm{s}\right)$ [28].

Figure 2

5-qubit encoding circuit for general architecture and (b) equivalent circuit for linear-nearest-neighbor architecture with dashed boxes indicating compound gates. CNOT gates that must be performed sequentially are numbered.

Figure 3

A sequence of physical gates implementing the circuit of Fig. 2b. Note the Kane architecture has been used for illustrative purposes.

Figure 4

Circuit equivalence used to reduce the number of physical gates in Fig. 3.

Figure 5

A complete encode-wait-decode-measure-correct QEC cycle.