Demonstration of Sufficient Control for Two Rounds of Quantum Error Correction in a Solid State Ensemble Quantum Information Processor

We report the implementation of a 3-qubit quantum error-correction code on a quantum information processor realized by the magnetic resonance of carbon nuclei in a single crystal of malonic acid. The code corrects for phase errors induced on the qubits due to imperfect decoupling of the magnetic environment represented by nearby spins, as well as unwanted evolution under the internal Hamiltonian. We also experimentally demonstrate sufficiently high-fidelity control to implement two rounds of quantum error correction. This is a demonstration of state-of-the-art control in solid state nuclear magnetic resonance, a leading test bed for the implementation of quantum algorithms.

Figure 1

Shown are the implemented quantum circuits for: (a) labeled PPS preparation procedure: a 3QCF is conjugated by a unitary operation that encodes (and decodes) the labeled pseudopure state $|00⟩⟨00|X$ in the triple quantum coherence $|000⟩⟨111|+|111⟩⟨000|$; (b) the implemented quantum circuit of a 3-qubit QECC, showing the encoding, decoding, and error-correction steps. The top two qubits are initialized to the $|00⟩$ state, and the bottom qubit carries the information to be encoded. After the decoding and correction operations, the bottom qubit is restored to its initial state, while the top two qubits carry information about which error had occurred; and (c) the procedure for two rounds: $U_p$ prepares $X$, $Y$, or $Z$ inputs, and $U_s=\{II,XI,IX,XX\}$ toggles between the different syndrome subspaces; i.e., the experiment is repeated 4 times, cycling through the different $U_s$, and then the results are added, similar to a standard phase cycling procedure.

Figure 2

Malonic acid (${\mathrm{C}}_3{\mathrm{H}}_4{\mathrm{O}}_4$) molecule and Hamiltonian parameters (all values in kHz). Elements along the diagonal represent chemical shifts, ${\omega }_i$, with respect to the transmitter frequency (with the Hamiltonian $\sum _i\pi {\omega }_i{Z}_i$). Above the diagonal are dipolar coupling constants [$\sum _{i<j}\pi D_{i,j}(2Z_i{Z}_j-X_i{X}_j-Y_i{Y}_j)$], and below the diagonal are $J$ coupling constants, [$\sum _{i<j}\frac{\pi }{2}{J}_{i,j}(Z_i{Z}_j+X_i{X}_j+Y_i{Y}_j)$]. An accurate natural Hamiltonian is necessary for high-fidelity control and is obtained from precise spectral fitting of (also shown) a proton-decoupled ${}^{13}\mathrm{C}$ spectrum following polarization transfer from the abundant protons. The central peak in each quintuplet is due to natural abundance ${}^{13}\mathrm{C}$ nuclei present in the crystal at $\sim 1\%$ (for more details, see 7, 10 and references therein).

Figure 3

Experimentally determined entanglement fidelities for unencoded (blue squares), before (green diamonds), and after (red circles) the correction step of one round of QEC. After encoding, a variable delay is implemented before the recovery process. During the delay, the system evolves under the heteronuclear and homonuclear terms in the natural Hamiltonian. The nonmonotonicity in the unencoded and decoded data is indicative of the presence of a coherent error. The inset shows the intensities measured for the different syndromes; the dominant error is a phase rotation on the bottom qubit ($C_m$).

Figure 4

Summary of experimental results for the partial decoupling map: the system evolves under the natural Hamiltonian as well as 70 kHz decoupling fields that partially modulate the heteronuclear interactions (between the carbons and protons). Shown (on left) are the single-qubit entanglement fidelities in the cases where no encoding is employed (blue dots); or one round of the 3-bit code (red crosses); or two rounds of the 3-bit code (black asterisks), where the interaction interval is split to two equal intervals. The dashed lines are quadratic fits to the data and are included to guide the eye. Also shown (on right) is the signal after one round of error correction as distributed over the various error-syndrome subspaces. In this case, the dominant errors are phase flips on the top and bottom qubits, which are encoded on $C_1$ and $C_m$, respectively.