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. 2016 May 5;7:11526.
doi: 10.1038/ncomms11526.

Repeated Quantum Error Correction on a Continuously Encoded Qubit by Real-Time Feedback

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Repeated Quantum Error Correction on a Continuously Encoded Qubit by Real-Time Feedback

J Cramer et al. Nat Commun. .
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Abstract

Reliable quantum information processing in the face of errors is a major fundamental and technological challenge. Quantum error correction protects quantum states by encoding a logical quantum bit (qubit) in multiple physical qubits. To be compatible with universal fault-tolerant computations, it is essential that states remain encoded at all times and that errors are actively corrected. Here we demonstrate such active error correction on a continuously protected logical qubit using a diamond quantum processor. We encode the logical qubit in three long-lived nuclear spins, repeatedly detect phase errors by non-destructive measurements, and apply corrections by real-time feedback. The actively error-corrected qubit is robust against errors and encoded quantum superposition states are preserved beyond the natural dephasing time of the best physical qubit in the encoding. These results establish a powerful platform to investigate error correction under different types of noise and mark an important step towards fault-tolerant quantum information processing.

Figures

Figure 1
Figure 1. Quantum error correction and implementation of stabilizer measurements.
(a) A quantum state is encoded in a logical qubit consisting of three physical qubits. Errors inevitably occur, for example, during computations. An ancilla is used to repeatedly perform measurements that detect errors. Errors are corrected through classical logic and feedback, while the quantum state remains coherent and encoded. (b) Device: chemical-vapour-deposition-grown single-crystal diamond with a solid-immersion lens and on-chip lines for microwave control. Scale bar, 5 μm. Ancilla: the optically addressable electronic spin of a nitrogen vacancy (NV) centre. Qubits: three 13C nuclear spins that are controlled and measured through the hyperfine coupling to the ancilla (Methods). (c) Free induction decay (Ramsey) experiments. Gaussian fits yield dephasing times formula image=12.0(9), 9.1(6) and 18.2(9) ms for qubits 1, 2 and 3, respectively. (d) Deterministic entanglement of two qubits by XX stabilizer measurement and feedback. The ±x gates are π/2 rotations around x with the sign controlled by the ancilla state. The final X operations reset the ancilla and account for an additional X flip for the +1 outcome (Methods). (e) State tomography of the generated entangled state for qubits 2 and 3. The fidelity with the ideal state is F=0.824(7) (see Supplementary Fig. 6 for other qubit combinations and post-selected results). All error bars are one statistical s.d.
Figure 2
Figure 2. Encoding of the logical qubit.
(a) Encoding an arbitrary quantum state formula image prepared on the ancilla into formula image. Successful encoding is heralded by outcome formula image. (b) Characterization of the logical states formula image, formula image and formula image. Only the logical qubit operators and stabilizers are shown (see Supplementary Fig. 7 for complete tomography of all 6 logical basis states). The fidelities with the ideal three-qubit states are F=0.810(5), 0.759(5)and 0.739(5), respectively, demonstrating three-qubit entanglement. The logical state fidelities are formula image, formula image and formula image. Ideally, all the encoded states are +1 eigenstates of the stabilizers X1X2I3 and I1X2X3. The fidelity to this code space, formula image, is 0.839(3) averaged over all states and gives the probability that the starting state is free of detectable errors. All error bars are one statistical s.d.
Figure 3
Figure 3. Active quantum error correction by stabilizer measurements.
(a) All qubits are simultaneously subjected to uncorrelated phase errors E with probability pe. Errors are detected by measuring X1X2I3 and I1X2X3 and subsequently corrected by Z operations through feedback. Finally, we measure the process fidelity with the identity. (b) Process fidelities for: an unencoded qubit (averaged over the three qubits), the logical qubit without stabilizer measurements, the error-corrected logical qubit and the logical qubit without feedback (that is, errors are detected but not corrected). We average over the logical qubit permutations, for example, XL=X1I2I3, I1X2I3 and I1I2X3, and the four ways to assign the ancilla states to the error syndromes (see Supplementary Fig. 8 for individual curves). Inset: probabilities for the error syndromes with theoretically predicted curves based on the state tomography in Fig. 2b (Supplementary Note 2). (c) Comparison between the error-corrected logical qubit and the logical qubit with the stabilizer measurements replaced by an equivalent idle time (2.99 ms). Compared with b, the effective readout fidelity is optimized by associating syndrome +1, +1 (no error) to obtaining formula image for both stabilizer measurements. Curves in b,c are fits described in the Methods. All error bars are one statistical s.d.
Figure 4
Figure 4. Extending coherence by active quantum error correction.
(a) Three rounds of error correction on a logical qubit. The first two rounds of quantum error correction use stabilizer measurements and feedback. The final round is implemented by majority voting. (b) Average logical state fidelity for formula imageand formula image as a function of total error probability pe for n=1, 2 and 3 rounds of error correction compared with an unencoded qubit. The errors per round En occur with probability pn. Inset: probabilities that no error is detected (n=3). The similarity of the results for rounds A and B confirms that errors are corrected in between rounds. (c) Correcting natural dephasing. The storage time is defined from the end of the encoding until the start of the final measurements. (d) Dephasing of the logical qubit: without stabilizer measurements, with quantum error correction and without feedback, compared with the best unencoded qubit. The dashed lines indicate the times between which the actively error-corrected logical qubit gives the highest fidelity. The data without feedback (detecting errors without correcting) isolate the suppression of coherently evolving errors by projecting them. For long times, applying error correction lowers the fidelity because the stabilizer measurements extract no useful information about errors, but nevertheless preferentially suppress evolutions that result in phase errors at the end of the sequence (see Supplementary Fig. 10 for a detailed analysis). See Supplementary Fig. 9 for error syndrome probabilities. Solid curves in b,d are fits described in the Methods and Supplementary Notes 1 and 2. Dashed lines are a guide to the eye. All error bars are one statistical s.d.

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