Experimental Implementation of Encoded Logical Qubit Operations in a Perfect Quantum Error Correcting Code

Large-scale universal quantum computing requires the implementation of quantum error correction (QEC). While the implementation of QEC has already been demonstrated for quantum memories, reliable quantum computing requires also the application of nontrivial logical gate operations to the encoded qubits. Here, we present examples of such operations by implementing, in addition to the identity operation, the NOT and the Hadamard gate to a logical qubit encoded in a five qubit system that allows correction of arbitrary single-qubit errors. We perform quantum process tomography of the encoded gate operations, demonstrate the successful correction of all possible single-qubit errors, and measure the fidelity of the encoded logical gate operations.

Figure 1

Outline of the quantum algorithm.

Figure 2

Network representation of the encoding operation. The input state $|\phi ⟩=\alpha |0⟩+\beta |1⟩$ is initially encoded in qubit 1. Qubits 2–5 are the syndrome bits. $H$ denotes a Hadamard transform, and $Z$ the $\pi$ phase gate. The box or $\underset{}{\overset{}{⨁}}$ with a connected vertical line indicate controlled gates, where the filled or empty circle marks the control qubit. The operations are conditional on the control qubit being in state $|1⟩$ (filled circle) or $|0⟩$ (empty circle), respectively. The output of the circuit is the five-qubit state $\alpha |0{⟩}_L+\beta |1{⟩}_L$.

Figure 3

Parameters of the spin qubits in the molecule 1,2-difluoro-4-iodobenzene. The inset shows the structure, where the five qubits 1–5 are spins $F1$, $F2$, $H1$, $H2$, and $H3$, respectively. The chemical shifts with respect to the transmitter frequencies of the proton and fluorine spins are shown as the diagonal terms and the dipolar couplings between spins are shown as the off-diagonal terms in units of Hz. The effective relaxation times $T_2^{*}$ are determined by fitting the peaks in the spectra.

Figure 4

Bar plots of the real parts of the $\chi$ matrices for the identity, NOT and Hadamard gates in logical states. The rms values of the imaginary parts are 0.035, 0.0069, and 0.017, respectively.

Figure 5

Spectra of the fluorine spins obtained for different versions of the experiment. (a) Reference spectrum obtained by applying a $\pi /2$ pulse to the thermal equilibrium state. (b) Spectrum of the output state, starting with the initial state $Y$, encoding, applying the identity operation, $S1$ error and error correction. (c) Same as (b), but ${\rho }_{\mathrm{in}}=X$, NOT gate and $BS4$ error. (d) Same as (b), but ${\rho }_{\mathrm{in}}=Z$, Hadamard gate and $B5$ error. The dashed and solid curves indicate the experimental and simulated spectra, which are normalized to the initial pseudopure states. The lines with the main signal contributions are shown enlarged as insets.

Figure 6

Bar plots showing the experimentally determined fidelities for the identity, NOT and Hadamard gates applied to the encoded qubit after applying error operations and error correction. The average fidelity is shown as the solid line. Compared to the average fidelity without error correction, shown as the dashed line, the encoding (decoding) error correction scheme improves the fidelity.