High-fidelity $Z$-measurement error encoding of optical qubits

We demonstrate a quantum error correction scheme that protects against accidental measurement, using a parity encoding where the logical state of a single qubit is encoded into two physical qubits using a nondeterministic photonic controlled-NOT gate. For the single qubit input states $\mid 0⟩$, $\mid 1⟩$, $\mid 0⟩\pm \mid 1⟩$, and $\mid 0⟩\pm i\mid 1⟩$ our encoder produces the appropriate two-qubit encoded state with an average fidelity of $0.88\pm 0.03$ and the single qubit decoded states have an average fidelity of $0.93\pm 0.05$ with the original state. We are able to decode the two-qubit state (up to a bit flip) by performing a measurement on one of the qubits in the logical basis; we find that the 64 one-qubit decoded states arising from 16 real and imaginary single-qubit superposition inputs have an average fidelity of $0.96\pm 0.03$.

Figure 1

Encoding against $Z$-measurement error. (a) A schematic of a circuit for performing the encoding and decoding of the state ${\mid \psi ⟩}_L$. The circuit consisits of a CNOT gate with the control input set to $\mid 0⟩+\mid 1⟩$ and the arbitrary, one-qubit state to be encoded $\mid \psi ⟩$ input into the target. Note that decoding can be achieved by measuring either of the encoded qubits. (b) Schematic of the experimental CNOT gate shown in (a), as originally demonstrated in [10].

Figure 2

Two-qubit encoded states. The real and imaginary parts of the density matrices are shown for the encoded states produced for the one-qubit inputs given in Table .

Figure 3

One-qubit decoded states. A table of density matrices for the six inputs and four decoding measurements listed in the figure. The imaginary parts are shown for the $\mid 0⟩+i\mid 1⟩$ and $\mid 0⟩-i\mid 1⟩$ inputs only.

Figure 4

One-qubit decoded state fidelities for the one-qubit input states $\cos \left(\theta \right)\mid 0⟩+\sin \left(\theta \right)\mid 1⟩$ and $\mid 0⟩+e^{i(90°-2\varphi )}\mid 1⟩$.