Abstract
Faulttolerant quantum memory plays a key role in interfacing quantum computers with quantum networks to construct quantum computer networks. Manipulation of spin quantum memory generally requires a magnetic field, which hinders the integration with superconducting qubits. Completely zerofield operation is desirable for scaling up a quantum computer based on superconducting qubits. Here we demonstrate quantum error correction to protect the nuclear spin of the nitrogen as a quantum memory in a diamond nitrogenvacancy center with two nuclear spins of the surrounding carbon isotopes under a zero magnetic field. The quantum error correction makes quantum memory resilient against operational or environmental errors without the need for magnetic fields and opens a way toward distributed quantum computation and a quantum internet with memorybased quantum interfaces or quantum repeaters.
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Introduction
Nuclear spins around a nitrogenvacancy (NV) center^{1,2,3} in diamond (Fig. 1a) have long coherence times: over 10 seconds for nitrogen and carbon nuclear spins^{4,5} and ~1.9 min for carbon nuclear spin pairs^{6}. Thus, such spins are promising as quantum memories for quantum repeaters^{7} and quantum computers^{8}. However, changes in magnetic fields such as external noise fields or hyperfine fields by the electron spin due to optical excitation cause either a bitflip or phaseflip error. To avoid such errors, various kinds of quantum error correction (QEC)^{9} have been demonstrated^{10,11,12}. Waldherr et al.^{10} demonstrated QEC with nitrogen and two carbon nuclear spins, Taminiau et al. demonstrated QEC with an NV electron and two carbon nuclear spins^{11}, and Cramer et al. demonstrated a stabilizer code with three carbon isotopes^{12}. However, these demonstrations were carried out under relatively strong magnetic fields such as 6200 Gauss with individual error corrections^{10} or 403 Gauss with repetitive error corrections based on dynamical decoupling^{11,12}. Under the magnetic fields, an increase in the fidelity of the controlledphase (CZ) gate by increasing the external magnetic field strongly depends on the carbon position relative to the NV center unless the field is well aligned to the hyperfine field (see discussion section and Supplementary Note 1) (Fig. 1b, c). In contrast, the fidelity can be increased independent of the carbon position under a zero magnetic field, where the hyperfine field uniquely defines the quantization axis, which allows timereversal operation in any timing, resulting in high fidelity single nuclear spin manipulation conditioned by the NV electron, as discussed later. The application of an external magnetic field also limits the integration with other physical qubits. For example, superconducting qubits, which we believe are the most promising candidates for quantum computers, become unstable by the application of a magnetic field due to the penetration of magnetic flux into the superconductor or the Josephson junction. The accessible magnetic field is ~100 Gauss even in the plane of the superconducting loop^{13,14,15,16,17} unless using exotic junctions such as semiconductor nanowires. Although a flux qubit relies on applying external magnetic fields, it is usually <1 Gauss to introduce a half flux in the loop to stabilize the qubit, and the application of higher magnetic fields beyond a halfflux field reduces the supercurrent in the junction^{18}. A zero magnetic field is also advantageous for qubit integration because a spatially uniform magnetic field can be easily obtained compared to individually tuned magnetic fields. A flux qubit working under a completely zero magnetic field with a ferromagnetic πshifted Josephson junction, which does not require any precise tuning of the flux in a qubit loop by individual coil currents, is under development toward largescale integration of quantum computers with flexible layout^{19,20}. To further extend the scale of quantum computers in size and that of quantum communications^{21} in distance, it is desirable to use spins in diamond, which serve as quantum interfaces^{22,23,24,25} with longtime memory and excellent optical accessibility for connecting quantum computers to optical quantum networks^{26}. Li et al.^{27} recently demonstrated coherent coupling of a single NV center spin to a superconducting flux qubit via a nanomechanical resonator and Scarlino et al. demonstrated coherent coupling of a semiconductor double quantum dot charge qubit to a superconducting transmon qubit via a microwave photon^{28}, which also operate under a zero magnetic field. Furthermore, it is essential to provide error tolerance not only to the NISQ (noisy intermediatescale quantum)^{29} based on stabilizer codes such as surface codes^{30,31,32,33,34,35} but also to the quantum interface, which should also work under a zero magnetic field for stable and reliable operations with flexible layout. In this report, we first demonstrate the quantum operation of the electron and nuclear spins and their correlated operations based on the geometric phase in the absence of a magnetic field. We then demonstrate the most fundamental threequbit QEC^{36} for either a bitflip or phaseflip error using three nuclear spins on nitrogen and two carbon isotopes in the vicinity of an NV center.
Results
Physical system
The ground state electrons in a negatively charged NV center constitute a spin1 triplet system represented by m_{s} = 0, ±1, and form a Vtype threelevel system under a zero magnetic field. The degenerate m_{s} = ±1 levels are used as a processor qubit (this is called a geometric qubit^{37,38,39,40,41,42,43,44,45}) and m_{s} = 0 as an ancilla. Although a degenerate qubit based on m_{s} = ±1 cannot be manipulated by a microwave field, it can be manipulated by a polarized microwave field via the ancilla state m_{s} = 0^{43}. A nuclear spin of nitrogen impurity, which is a component of the NV center, also constitutes a spin1 system, where the m_{I} = ± 1 levels are degenerate, as is the electron spin under a zero magnetic field. The m_{I} = 0, −1 levels are used as a memory qubit represented by 0⟩ (m_{I} = 0) and 1⟩ (m_{I} = −1) in this demonstration. Carbon isotopes (^{13}C) with a spinhalf nuclear spin are quantized along the hyperfine field by the electron spin to form eigenstates represented by 0⟩ \(({m}_{{{{{{\rm{I}}}}}}}=1/2)\) and 1⟩ \(({m}_{{{{{{\rm{I}}}}}}}=1/2)\). We use two carbon nuclear spins coupled to the processor electron spin as auxiliary memory qubits. All nuclear spins can be individually manipulated with the help of hyperfine interactions with the electron spin, as indicated by the ODMR (optically detected magnetic resonance) measurement (Fig. 1d).
Entanglement generation of three qubits
We first evaluate the microwave intensity and the angle between microwaves irradiating through two crossed wires placed on the sample to perform universal quantum operations of a degenerate twolevel subsystem of spin1 triplet electrons called a geometric qubit^{43}. During nuclear spin manipulation, we set the electron spin state into +1⟩_{S} by using a microwave with a rightcircular polarization, which eases individual manipulation of the nitrogen and carbon nuclear spins. Figure 1c shows that the fidelity is increased by decreasing the Rabi frequency, although it is limited in practice since the manipulation time cannot exceed T_{2}*. In this demonstration, however, we obtained higher fidelity by extending the manipulation time to the effective T_{2}* induced by weakly coupled carbons with hyperfine couplings below 0.1 MHz by using the GRAPE (gradient ascent pulse engineering)optimized waveform^{46} considering two strongly coupled carbons.
Figure 2a shows the quantum circuit that generates entanglement between three nuclear spins (one nitrogen and two carbons) known as the GHZ state \(\frac{1}{\sqrt{2}}\) \(({000\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}+{111\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}})\). The necessary operational elements for the demonstration are state initialization, universal quantum gate, and state measurement of nuclear spins. The three nuclear spins are first initialized into \({000\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\) (see Supplementary Note 2). Carbon nuclear spins are initialized by probabilistic projection by conditional electron rotation (socalled measurementbased initialization) with a high fidelity of more than 99%. On the other hand, the nitrogen nuclear spin is deterministically initialized with about 95% fidelity since the measurementbased initialization allows only about 90% fidelity (see Supplementary Note 3, 4) due to the depolarization during the optical excitation for the electron spin measurement. The Hadamard gate, which makes a superposition of the basis states, of the nuclear spins is implemented by applying a π/2 pulse of a radio wave that resonates with the hyperfine splitting of the corresponding carbon nuclear spin (Fig. 2b). The correlated operation between the three nuclear spins is implemented by applying the holonomic CZ gate based on the geometric phase^{43,47,48,49} with the help of the electron spin because the direct interaction between nuclear spins is too weak. This is achieved by a 2π rotation in the Bloch sphere spanned by +1⟩_{S} and 0⟩_{S} of the electron spin conditioned by the nuclear spin state, adding a π phase only for \({110\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\) and \({101\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\) (e.g., \(({110\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\rightarrow {110\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}})\) (Fig. 2c). Finally, after the generation of the GHZ state, the joint states of the three nuclear spins are measured by a singleshot measurement (Fig. 2d), confirming the classical correlation of entanglement with 78% fidelity. Although the quantum correlation is not measured since the measurement time is unrealistic, it should have about the same fidelity as the classical correlation since we confirmed that the twospin entangled state has about the same fidelity as the classically correlated state (see Supplementary Note 6). The probability of \({111\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\) is smaller than that of \({000\rangle }_{{{{{{\rm{N}}}}}},{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\) because the nitrogen nuclear spin is likely depolarized into 0⟩_{N} during the optical excitation to read out the electron spin state.
Quantum error correction
The QEC codes implemented in this demonstration are fundamental building blocks of Shor’s QEC code^{36}. Although complete QEC is achieved by a ninequbit code, we demonstrate a threequbit code that can correct either a bitflip error (Fig. 3a) or a phaseflip error (Fig. 3b), which are otherwise identical except for the Hadamard gates inserted immediately after encoding and before decoding. In this experiment, the error that occurs on the nitrogen nuclear spin is protected by two carbon nuclear spins. Although an error occurring on any single qubit after encoding is corrected, errors occurring on multiple qubits cannot be corrected.
Figure 3 shows whether the bitflip and phaseflip errors can be corrected by intentionally inserting an error in the encoded nitrogen nuclear spin. The carbon nuclear spins are prepared into \({00\rangle }_{{{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\) and the nitrogen nuclear spin is prepared into one of ψ⟩ = 0⟩, 1⟩, +⟩, −⟩, +i⟩, −i⟩. The encoding and decoding are configured with the same gates as the threequbit entanglement shown in Fig. 2. The quantum Toffoli gate for the QEC is also configured with the combination of Hadamard gates and the holonomic CZ gate that adds the π phase only for \({111\rangle }_{{{{{{{\rm{N}}}}}},{{{{{\rm{C}}}}}}}_{1},{{{{{{\rm{C}}}}}}}_{2}}\). Quantum tomography measurements on the nitrogen nuclear spin state after the QEC showed that the state fidelities against bitflip and phaseflip errors are, respectively, 75.4% and 74.6% on average (see Supplementary Note 5) (Fig. 3c, d).
Discussion
To evaluate the usefulness of the QEC, we measure the state fidelity of the nitrogen nuclear spin when the probability of a single intentionally generated error is varied, both with and without the QEC (Fig. 4a). The initial state of the nitrogen nuclear spin is fixed at +⟩_{N}, and only the nitrogen nuclear spin is subjected to phase errors. Without QEC, the fidelity degrades in proportion to the error probability. The 10% deviation from the ideally obtained fidelities with error probabilities of both 0 and 1 should be due to state preparation and measurement (SPAM) errors. On the other hand, with QEC, the fidelity is constant regardless of the error probability. Although the fidelity is inferior to that without QEC due to operational errors for an error probability below 0.15, it exceeds that without QEC as expected for an error probability over 0.15.
The operational error should be caused mainly by environmental carbons whose hyperfine couplings are smaller than 0.1 MHz, which cannot be distinguished by ODMR measurement (Fig. 1d). Figure 4b shows the dependence of the fidelity of the encoding (decoding) operation on the hyperfine coupling in the presence of another carbon in addition to the two carbons used in the experiment. From the results of the experiment such as ODMR measurement and Rabi oscillation (see Supplementary Note 7), the hyperfine coupling is estimated to be in the range of 0.06–0.12, resulting in fidelity degradation of ~10% in addition to a SPAM error of 10%. The fidelity can be improved by initializing undetected carbons using dynamical decoupling with radio frequency (DDRF)^{5,50,51}. Even higher fidelity can be obtained by optimizing the waveform of the microwaves to be robust against unknown hyperfine coupling in a specific range.
Finally, we compare our results with previous related experiments^{10,11,12}. The most relevant experiment was given by Waldherr et al.^{10}, which demonstrated QEC with nitrogen and two carbon nuclear spins as in our demonstration but at 6200 Gauss. The situation in Fig. 4a in our demonstration exactly corresponds to that in Fig. 3b in ref. ^{10}, and the obtained fidelities are about the same within error bars. This means that we successfully excluded magnetic fields to achieve the same level of QEC without the necessity of a precise field alignment along the NV axis. Other relevant experiments were given by Taminiau et al.^{10}, which demonstrated QEC with an NV electron and two carbon nuclear spins, and Cramer et al.^{12}, which demonstrated stabilizer code with three carbon isotopes, both with dynamical decoupling schemes at 403 Gauss. Although these experiments cannot be directly compared with our experiments since they protect different spins from ours, their fidelities are relatively lower than ours possibly because the carbon isotopes used in their experiments were not properly oriented along the external magnetic field. It is shown in Fig. 2a in Methods of ref. ^{10} that usable carbon isotopes are limited to around six at 403 Gauss and to ~9 at 10 T with hyperfine of over 20 kHz^{11,52}. To overcome this limitation, Bradley et al.^{5} developed a more elaborate DDRF method^{5,50,51}. Although this method enables the detection of carbon isotope spins with a weak or negligible perpendicular hyperfine component as shown in Fig. S3 in this reference, the angle between the external and hyperfine fields must be within around 10^{−2} (0.6 degrees) to achieve infidelity of 10^{−4} at 403 Gauss^{5}, which is still consistent with our simulation shown in Fig.1c and Fig. 1 in SI. On the other hand, although we can only manipulate strongly coupled carbons due to the limitation from T_{2}* in the current scheme, we can also manipulate weakly coupled carbons limited by T_{2} instead of T_{2}*. The combination of robust dynamical decoupling in threelevel systems under low magnetic fields developed by Vetter et al.^{53} and nuclear spin manipulation with radiofrequency synchronized with the decoupling developed by Bradley et al.^{5}, enables the implementation of twoqubit gates between an electron and a weakly coupled carbon under a zero magnetic field. With the increase in operatable carbon isotopes, we expect to realize Shor’s ninequbit QEC code even under a zero magnetic field.
In conclusion, we demonstrated a threequbit QEC against either a bitflip or phaseflip error by introducing the holonomic CZ gate of three nuclear spins around an NV center in diamond under a zero magnetic field. This demonstration is applicable to the construction of a largescale distributed quantum computer and a longhaul quantum communication network by connecting quantum systems vulnerable to a magnetic field, such as superconducting qubits with spinbased quantum memories.
Methods
We use a single naturally occurring NV center in a highpurity type IIa chemicalvapor depositiongrown diamond with crystal orientation of <100> produced by Element Six. The diamond is cooled to 5 K to prolong the electron spin coherence. To achieve a zero magnetic field, the residual magnetic field including the geomagnetic field is canceled out by a threedimensional coil. The currents of the three coils are adjusted by monitoring the spinecho coherence time, which reaches its maximum at a zero magnetic field. Two orthogonal copper wires are attached to the sample surface to apply microwaves with arbitrary polarization. The optical system consists of a homemade confocal microscope system. A green laser (515 nm) is used for charge and electron spin initialization by nonresonant excitation, and two red lasers (637 nm) are used for further electron spin initialization with the A_{1}⟩ state and spin measurement using the E_{y}⟩ state by resonant excitation.
Data availability
Data are available from the corresponding authors upon reasonable request.
Code availability
All codes used to produce the findings of this study are available from the corresponding author upon request.
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Acknowledgements
We thank Hiromitsu Kato, Toshiharu Makino, Tokuyuki Teraji, Yuichiro Matsuzaki for their discussions and experimental help. This work was supported by the Japan Society for the Promotion of Science (JSPS) GrantsinAid for Scientific Research (20H05661, 20K2044120); by Japan Science and Technology Agency (JST) CREST (JPMJCR1773); and by JST Moonshot R&D (JPMJMS2062). We also acknowledge the assistance of the Ministry of Internal Affairs and Communications (MIC) under the initiative Research and Development for Construction of a Global Quantum Cryptography Network (JPMI00316).
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T.N., R.R., N.I., and H.K. designed and analyzed the experiment. T.N. carried out the experiment. K.M., K.T., and Y.S. provided theoretical and technical support. H.K. supervised the project. All authors discussed the results. T.N. and H.K wrote the manuscript.
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Nakazato, T., Reyes, R., Imaike, N. et al. Quantum error correction of spin quantum memories in diamond under a zero magnetic field. Commun Phys 5, 102 (2022). https://doi.org/10.1038/s42005022008756
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DOI: https://doi.org/10.1038/s42005022008756
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