2019/05/24 by Li, Chi-Kwong, Lyles, Seth, Poon, Yiu-Tung
#81P45 #FOS: Physical sciences #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.1905.10228
We study efficient quantum error correction schemes for the fully correlated channel on an n-qubit system with error operators that assume the form σx⊗ n, σy⊗ n, σz⊗ n. Previous schemes are improved to facilitate implementation. In particular, when n is odd and equals 2k+1, we describe a quantum error correction scheme using one arbitrary qubit σ to protect the data state ρ in a 2k-qubit system. The encoding operation σ⊗ ρ↦ Φ(σ⊗ ρ) only requires 3k CNOT gates (each with one control bit and one target bit). After the encoded state Φ(σ⊗ ρ) goes through the channel, we can apply the inverse operation Φ-1 to produce σ⊗ ρ so that a partial trace operation can recover ρ. When n is even and equals 2k+2, we describe a hybrid quantum error correction scheme using any one of the two classical bits σ∈ \|ij⟩ ⟨ ij|: i, j ∈ \0,1\\ to protect a 2k-qubit state ρ and 2 classical bits. The encoding operation σ⊗ ρ↦ Φ(σ⊗ ρ) can be done by 3k+2 CNOT gates and a single quibt Hadamard gate. After the encoded state Φ(σ⊗ ρ) goes through the channel, we can apply the inverse operation Φ-1 to produce σ⊗ ρ so that a perfect protection of the two classical bits σ and the 2k-qubit state is achieved. If one uses an arbitrary 2-qubit state σ, the same scheme will protect 2k-qubit states. The scheme was implemented using Matlab, Mathematica, Python, and the IBM's quantum computing framework qiskit.