Abstract
Realistic quantum computing is subject to noise. Therefore, an important frontier in quantum computing is to implement noiseresilient quantum control over qubits. At the same time, dynamical decoupling can protect the coherence of qubits. Here we demonstrate nontrivial quantum evolution steered by dynamical decoupling control, which simultaneously suppresses noise effects. We design and implement a selfprotected controlledNOT gate on the electron spin of a nitrogenvacancy centre and a nearby carbon13 nuclear spin in diamond at room temperature, by employing an engineered dynamical decoupling control on the electron spin. Final state fidelity of 0.91(1) is observed in preparation of a Bell state using the gate. At the same time, the qubit coherence time is elongated at least 30 fold. The design scheme does not require the dynamical decoupling control to commute with the qubit interaction and therefore works for general qubit systems. This work marks a step towards implementing realistic quantum computing systems.
Introduction
To combat noise effects in quantum computing, there are three main strategies, namely, quantum error corrections^{1,2,3}, decoherencefree subspace^{4,5} and dynamical decoupling (DD)^{6,7,8,9}. DD, which originated from magnetic resonance spectroscopy, can average out the noise by flipping the qubit. DD has the merits of requiring no extra qubits and potential compatibility with quantum gates. It provides a potential method to realize fidelity of quantum gates above the threshold required by concatenated quantum error corrections for scalable faulttolerant quantum computing^{1,2,3}. Recent experiments have demonstrated the preservation of quantum coherence^{10,11,12,13,14,15}, or the NULL gate in terminology of quantum computing.
Integration of DD with quantum gates, however, is a nontrivial challenge because, in general, the quantum gates may not commute with the DD control and therefore can interfere with the DD. A straightforward approach is to insert the quantum gates in between the DD control sequences^{16,17}, which, however, significantly reduces the time windows for quantum gates. A clever solution is to design the DD sequences such that they commute with the qubit interaction, which can be realized either by encoding the qubits in decoherencefree subspaces^{18} or by choosing a certain qubit interaction that commutes with the DD sequences^{19}. It is also possible to apply control over one qubit while the other qubits are under DD control^{20,21,22}. These methods, however, require special types of interactions or hybridizing operations at very different timescales.
A more important question is whether DD, instead of locking the quantum states of qubits as previously demonstrated^{10,11,12,13,14}, can steer nontrivial and noiseresilient quantum evolutions of qubits with generic interactions (that is, not limited to interactions commutable with the DD control)^{23}. Recent theoretical studies established that arbitrarily accurate dynamical control could be designed for general quantum open systems via concatenated construction or pulse shaping^{24,25}.
Here we demonstrate the feasibility of integrating DD with quantum gates by steering the quantum evolution of a hybrid qubit system with numerically optimized DD control, to simultaneously realize a nontrivial twoqubit gate and coherence protection. We realize a selfprotected controlledNOT (C_{e}NOT_{n}) gate on the electron spin of a nitrogenvacancy (NV) centre and a nearby carbon13 nuclear spin in diamond, by employing an engineered DD control applied only on the electron spin.
Results
Design of quantum gates by DD
To demonstrate the concept of quantum steering by DD, we consider a negatively charged NV centre in a type IIa diamond (with nitrogen concentration <10 ppb) under an external magnetic field B. The scheme is motivated by the recent study of central spin decoherence in nuclear spin baths^{26,27,28,29,30}, which reveals that because of the quantum nature of the qubitbath coupling, the quantum evolution of nuclear spins is actively manipulated by flipping the central electron spin. Such quantum nature of qubitbath coupling has been previously utilized to realize control of nuclear spins by flipping the electron spin^{31,32,33,34}. By engineering the timing of the electron spin flipping, one can steer a noiseresilient quantum evolution of interacting qubits simply by DD control. Recent research on single nuclear spin sensing by central spin decoherence^{33,35,36,37,38,39} has already demonstrated the quantum nature of coupling between a NV centre spin and remote nuclear spins, and therefore the approach of quantum gates by DD may also be applied to those remote nuclear spins and this has the potential of extending the twoqubit system to fewqubit systems.
The NV centre electron spin is coupled through hyperfine interaction to ^{13}C nuclear spins^{40}, which have a natural abundance of 1.1% (Fig. 1a). Lifting the degeneracy between m=+1 and m=−1 NV centre spin states by B, we encode the first qubit in the centre spin states 0≡m=0 and 1≡m=−1. As the hyperfine interaction strength decreases rapidly with the distance between a nuclear spin and the NV centre, a proximal ^{13}C spin can be identified by its strong hyperfine splitting in the optically detected magnetic resonance (ODMR) spectra. We encode another qubit in the ^{13}C nuclear spin1/2 states ↑ and ↓ (Fig. 1b), similar to the electronnuclear spin register studied in Gurudev Dutt et al.^{33} Thus, we have a welldefined twoqubit system.
The undesired coupling of this twoqubit system to the other ^{13}C spins in the bath leads to loss of quantum information and therefore reduces the fidelity of the quantum operations. In particular, because of the large difference between the gyromagnetic ratios of the two types of spins, the electron spin decoherence occurs in a timescale that is shorter than the typical operation timescale of the nuclear spin qubit, which results in difficulty in realizing highfidelity twoqubit operations.
The evolution of the twoqubit system is a path in the curved SU(4) operator space^{41}. In a free evolution, the system propagator follows the natural landscape in the operator space, but the uncertainty in the system propagator increases with time due to the coupling with the environment (Fig. 1c). Although a conventional DD scheme like the CarrPurcellMeiboomGill sequence or the Uhrig DD sequence can efficiently refocus the otherwise noncoherent evolution of the system propagator, the sequence in general corresponds to an unspecified twoqubit propagator unless resonant values of the total evolution time and the number of pulses are chosen^{21,38} or additional manipulation on the nuclear spin is employed^{21}. Simultaneously achieving coherence protection and gate implementation, our scheme can be intuitively understood as a systematic approach to identify a path in the operator space comprising segments of free evolution and centre spin πflips such that the path is selfprotected and guided from the identity to a desired twoqubit gate (Fig. 1d).
Because the timescale of interest is much shorter than the longitudinal relaxation time of the centre spin, the centre spin magnetic number m remains a good quantum number and the Hamiltonian of the twoqubit system can be expanded in the basis of the centre spin eigenstates and (Supplementary Note 1), given by , where E_{m} is the eigenenergy of the centre spin state , I represents the nuclear spin and ω_{m} is the local field for the nuclear spin conditioned on the electron spin state . In general, when the centre spin state is altered, the nuclear spin will evolve under a different local field, and therefore the nuclear spin evolution is conditional on the state of the centre spin^{33,34,37,40,41,42,43,44}. When the angle φ between ω_{1} and ω_{0} is nonzero (Fig. 1b), which is expected in a general setting (Supplementary Note 1), they represent different axes on the nuclear spin Bloch sphere and hence can be utilized to generate universal qubit operations conditioned on the electron spin qubit.
To be specific, we suppose the system is prepared in an initial state . When a sequence of N πpulses is applied to the centre spin, the nuclear spin state ψ_{m} evolves to ψ_{m}′=u_{m}{t_{α}}ψ_{m} with , where t_{α} is the time between the αth and the (α+1)th pulses, σ=0 for N being even and σ=1 for N being odd, and u_{1}{t_{α}} is similarly defined. This implies that the system propagator can be represented by . One can vary the timing parameters {t_{α}} and engineer the system evolution such that U{t_{α}}≈G for some desired twoqubit gates with the generic form , where O_{m}'s are nuclear spin operators. Important examples of gates with this form include the C_{e}NOT_{n} gate, nuclear spin singlequbit gates, centre spin phase gates and the twoqubit NULL gate. In general, it is nontrivial to exactly solve u_{m}{t_{α}}=O_{m} because of the nonlinearity and the large number of variables in the problem. Yet, one can recast the design protocol into a maximization problem through studying the average twoqubit gate fidelity {t_{α}}=∫dΨ Tr(U{t_{α}}ΨΨU^{†}{t_{α}}GΨΨG^{†}), where Ψ is a general pure twoqubit state and the integration is over the normalized uniform measure of the state space^{45}. As {t_{α}}=1 if and only if U{t_{α}}=G, the gate G is simulated by the system propagator when the timing parameters {t_{α}} are chosen to maximize {t_{α}}. Such gate design can be achieved by using only DD sequences, where the notion of DD can be understood as a set of criteria on the timing parameters {t_{α}} between the centre spin πpulses. Although a conventional Npulse DD sequence (say the CarrPurcellMeiboomGill sequence) is characterized by just one timing parameter, namely the pulse delay time, the general DD criteria can be derived by studying the expansion of the coherence function. Instead of using all the timing parameters to optimize protection of the centre spin coherence^{46}, one can relax some of the timing parameters by reducing the decoherence suppression order. The design procedure can be summarized as the maximization of [F{t_{α}}] with respect to an Npulse sequence {t_{α}} that complies with a set of DD constraints. The first order DD criterion requires t_{0}−t_{1}+t_{2}+⋯+(−1)^{N}t_{N}=0, which can be intuitively understood as the spin echo condition. The symmetric timing condition t_{n}=t_{N−n} automatically realizes the secondorder DD^{47,48}, and higherorder DD criteria can be similarly introduced. At least up to the second order, the DD constraints discussed above are independent of the detailed bath spectrum. DD constraints to higher orders may also be designed independent of the detailed noise spectrum, but a hard highfrequency cutoff (that is, slow noises) is required.
As a demonstration of principles, we used the echo condition and the symmetric requirement to realize the DD to the second order^{47,48}. The DD constraints are explicitly in the form {t_{α}}_{DD}={t_{0},t_{1},t_{2},…t_{2},t_{1},t_{0}} with the echo condition . With such, we designed DD sequences that execute the desired twoqubit gates by maximizing {t_{α}}_{DD} with respect to the Nindependent timing variables. It is straightforward to generalize the design to higherorder DD for better noise resilience.
We demonstrate the feasibility of the design by considering an experimentally identified target ^{13}C spin coupled to an NV centre spin. The experimental parameters were extracted from the ODMR spectra of the NV centre and the free precession signal of the nuclear spin, which were ω_{0}=0.256(2) MHz and ω_{1}=6.410(2) MHz (see Methods and Supplementary Fig. S1 for details). The magnetic field B was oriented such that φ was 90°.
Various DD gate sequences in Supplementary Table S1, were designed based upon the obtained experimental parameters, and their performances were assessed numerically (Fig. 2a–d). Five different twoqubit gates, namely, the C_{e}NOT_{n} gate (defined up to an additional π/2 phase shift of the centre spin)^{21}, the nuclear spin Hadamard (H_{n}), PauliX (X_{n}) and PauliZ (Z_{n}) gates, and the twoqubit NULL gate, were designed using sequences with 4–14 pulses. The gate fidelity {t_{α}} (ref. 45), numerically optimized, is at least 0.98 for the gates we considered, and the gate operation time T_{G} ranges from 1.4 to 5 μs. To incorporate the coupling between the twoqubit system and the environment, we simulated the environment by a small bath consisting of six ^{13}C spins aside from the ^{13}C qubit spin. The coupling to the spin bath causes centre spin decoherence in a timescale of ≈1.6(1) μs under free induction decay (FID), which is consistent with the experimental condition. To characterize the performance of the designed gates (G), we considered a typical system initial state and the bath in the thermal state ρ_{bath}. We then calculated the total propagator U_{T} under the pulse sequences by exact diagonalization, assuming perfect πpulses and interpulse timing. The state fidelity F is defined by (ref. 42), where is the ideal system final state and ρ_{s}=Tr_{bath}(U_{T}(Ψ_{0}Ψ_{0}⊗ρ_{bath})) is the simulated system density matrix. We found the final state fidelity to be at least 0.97 for the sequences we designed. We also characterized the coherence protection efficacy of the DD gate sequences by performing the clustercorrelation expansion calculation^{43} with a larger bath of 44 ^{13}C spins together with the ^{14}N nitrogen spin. Because of entanglement with the target spin, the centre spin coherence right after applying the DD gate sequence can be 0. For a fair comparison, therefore, we probe the centre spin coherence (ref. 44) after performing each gate sequence twice, such that ideally one would obtain for the gates we considered. The coherence function was found to be at least 0.92 even when the total evolution time exceeds the FID decoherence timescale. We note that although in principle the average twoqubit gate fidelity can be improved to almost unity by using a larger number of pulses, this does not automatically guarantee a higher resultant state fidelity when the spin bath is incorporated, and in the experimental setting, this can also introduce additional pulse errors.
As the gates realized by applying DD sequences are selfprotected, our method offers an integrated solution to achieving both control and noise tolerance in quantum information processing. We note that when the hyperfine interaction is moderately strong, the gate speed of the nuclear spin gates constructed this way can be significantly faster than the conventional control by radio frequency pulses^{31,32}.
Experimental implementation
We experimentally demonstrated our scheme by implementing the designed fourpulse C_{e}NOT_{n} gate (Fig. 3a). Because the nuclear spin of the nitrogen host was not polarized in the experiment, the system was first initialized into the superposition state within the m_{I}=0 subspace of the ^{14}N spin using selective weak microwave (MW) pulses. We used state tomography to measure the state fidelity, which is defined as with being the ideal state of the system and ρ_{E} being the measured reduced density matrix^{42}. The measured initial state fidelity was 0.99(1) (Fig. 3b). The particular gate demonstrated can be readily understood by studying the state trajectory on the Bloch sphere. As illustrated in Fig. 3c, the nuclear qubit traces out paths that would approximately leave it unchanged or flipped when the centre spin is prepared in the 0 or 1 states, respectively. State tomography revealed that the nuclear spin was flipped conditional to the state of the centre spin and the final state resembled the Bell state with fidelity of 0.91(1) (Fig. 3d). We also carried out experiments for the system initially prepared in the basis states. The final state fidelity, as shown in Supplementary Fig. S2, ranged from 0.86(1) to 0.93(1).
Having demonstrated the quantum gate aspect of the sequence, we now turn to illustrate the coherence protection aspect of our design by applying the C_{e}NOT_{n} gate DD sequence to an initial superposition state using stronger selective MW pulses that drive all the components of the ^{14}N spin. Without coherence protection, the electron spin coherence decays in a short time of μs in FID, and the coherence was completely lost in ~2 μs (Fig. 4a). With the DD gate sequence, however, the centre spin coherence was well protected during the course of gate operation, even though the duration of the gate was more than twice of (Fig. 4b). The coherence protection capacity of the DD gate sequence is further illustrated in Fig. 4c–f. The designed pulse sequence was repeated for N=2, 4, 8 and 12 times on the initial state Ψ_{0}, and the Ramsey interference signal was measured after the total sequence time of 8, 16, 32 and 48 μs, correspondingly. The presence of strong coherent oscillation after an interval longer than 30 times of demonstrates the robust protection effect of the DD gate design.
Discussion
Although the scheme was designed and implemented for a specific system with interaction diagonal in the electron spin qubit basis, the design protocol of optimizing gate fidelity under the DD constraints can be applied to general systems provided that (1) the qubits are well defined with respect to the bath, (2) at least one of the qubits can be independently controlled by DD pulses and (3) the bath dynamics is slow as compared with the control and the interqubit interactions or the noise spectrum has a hard highfrequency cutoff. In principle, all quantum gates that are allowed by the interqubit interactions can be designed. Let us consider, for instance, a pqubit system with general interqubit interactions characterized by the system Hamiltonian H_{p}. When a total of N DD pulses are applied to more than one qubits, the system propagator is given by , where {t_{α}} specifies the interpulse timings and {q_{α}} specifies the qubit that the αth DD pulse flips. The gate fidelity can then be maximized with respect to both {t_{α}} and {q_{α}} subjected to the DD constraints. Alternatively, one may also apply DD pulses to only one qubit and derive DD constraints that can protect the coherence of the multiqubit system^{49}. To combat general qubitbath coupling (beyond the pure dephasing model), the qubit flips along different axes (such as and ) can be employed.
It should be noted that when the system is too large, the numerical optimization procedure can become intractable as the gate fidelity has to be maximized with respect to the highdimension system propagators over the large parameter space, and the numerical complexity increases exponentially with the number of qubits in the system^{24}. A set of one and twoqubit gates is sufficient for universal quantum computing. For multiple qubits coupled to common noise sources, however, collective design of multiplequbit control would still be needed. Given a realistic limitation on numerical resources, the achievable fidelities of the multiplequbit control would drop due to the exhaustion of optimization parameters. Such a limitation is a common issue for all numerical optimization schemes for noiseresilient qubit control. On the other hand, because the DD constraints depend only on the algebraic form of qubitbath couplings but not on the qubit–qubit couplings, the coherence protection capacity of the scheme should not suffer from a growth in the system size.
Although the gatebyDD scheme demonstrated in this work is based on optimization of a discrete set of timing parameters, we note that numerical pulse shaping has been proposed for dynamically protected quantum gates, in which both environmental noise and control errors can be corrected^{25}. In this paper, we have not considered the effects of control errors (which is actually the main source of infidelity in our experimental implementation). The gatebyDD scheme, however, can also be extended to employ robust DD sequences^{50} for tolerance of control errors. As the DD gate design does not depend on the phase of the DD pulses, the scheme can be extended to employ both X and Y DD pulses so as to protect the gates against pulse imperfection^{50}.
This work demonstrates a general approach to quantum information processing in which the quantum evolution of the system is engineered to perform decoupling from a large spin bath and execute a designated gate on the twoqubit system, and thereby simultaneously realize highfidelity twoqubit gates and coherence protection. The approach developed here is applicable to other systems under a general setting and therefore provides a new avenue towards achieving scalable and faulttolerant quantum computing.
Methods
Experimental setup
A permanent magnet was used to apply an external magnetic field on the system such that the field B can be tuned in both strength (0–650 Gauss) and orientation. A homebuilt scanning confocal microscope combined with integrated MW devices was employed to initialize, control and read out the electron spin state. The 532nm continuous wave laser beam was switched on and off by an acoustic optical modulator, and the position of the laser spot was controlled by an XY galvanometer before the beam was directed to the sample through an oil immersion lens. The spin statedependent fluorescence of the NV centre was collected by the same lens and filtered by a 532nm notch and a 650nm long path. Then, the weak light signal was translated into the electronic pulse signal by a singlephoton counting module and was subsequently counted by a pulse counter.
A total of four MW channels were timecontrolled by individual RF switches and were then combined and amplified by a 16Watt wideband MW amplifier before being delivered to the sample through a coplanar waveguide antenna close to the NV centre. One of the MW channels was used to execute the DD gate sequences in our scheme, and the others were used for the polarization and readout procedures.
The timing sequences for controlling the acoustic optical modulator, RF switches and counter were generated by a multichannel pulse generation card (with 2 ns resolution). In each of the measurement, we loaded the pulse sequence to the card memory, and other instruments were triggered and controlled by the generated digital logic signal. The sequence was repeated (typically for 400,000 times) until a reliable signaltonoise ratio was reached.
System characterization and control
The energy splittings between different levels were first measured using the pulsed ODMR technique (Fig. 5a). MW pulses of various powers, frequencies and phases were employed to coherently control the NV centre electron spin (Fig. 5b). Figure 5c–d shows the response of the system under different MW pulses. Note that we took the quantization axis of the nuclear qubit to be along its local field when the electron spin was in the m=−1 state. MW_{0} was the strongest pulse and was tuned in resonance with the transition between the centre spin states and , which was used to execute the centre spin flips in the DD sequence. The Rabi frequency of this control pulse was chosen to be 25 MHz, much larger than the hyperfine coupling strength (6.41 MHz), so that the electron spin flip was completed in a short time (<20 ns) and therefore was independent of the nuclear spin states. The other three MW channels used relatively weaker MW power so that the power broadening was less than the hyperfine splitting due to the nuclear qubit to selectively drive Rabi oscillations. MW_{1} and MW_{2} were tuned, respectively, resonant with the transitions and so that they could be used to polarize and read out the nuclear spin state^{34}. MW pulses were set to be about 1.25 MHz to selectively drive the m_{I}=0 component of the ^{14}N spin in the state tomography experiments, and were set to be about 4 MHz to drive all the ^{14}N spin components in the Ramsey interference experiments. The additional MW_{3} channel was used for the ‘kick out’ pulses discussed below.
Polarization and readout of the nuclear qubit state
Figure 6a shows the polarization and readout schemes used to study the free precession of the target nuclear spin. As we encoded the centre spin qubit in the m=0 and m=−1 states, we enhanced the desired polarization by employing an extra MW channel (MW_{3}) to ‘kick out’ the unwanted component to the m=+1 subspace (Fig. 5b). For instance, to initialize the system to the 0↓ state, the frequency of MW_{3} was set to be in resonance with the transition , and a πpulse of this frequency was added after performing the initialization steps as in Gurudev Dutt et al.^{33} This modification helped improve the effective polarization to 90% in the m_{I}=0 subspace.
In Fig. 6b,c, we show the effect of the ‘kick out’ pulse on polarization of the target nuclear spin. When the ‘kick out’ pulse was not applied, both MW_{1} and MW_{2} drove significant Rabi nutations, which indicates incomplete polarization (Fig. 6b). By using the MW_{3}, the unwanted components were removed (Fig. 6c), leaving a high polarization in the subspace spanned by m=0 and m=−1. As the decay of the population in the m=+1 subspace to the m=−1 and m=0 subspaces was negligible within the timescale of interest, we neglected the contribution of the m=+1 population, and the centre spin state was read out by normalizing the NV centre fluorescence signal with respect to the full Rabi oscillation driven by MW_{0}.
Determination of ω_{0} and ω_{1}
As the hyperfine tensor characterizing the interaction between the centre spin and a nuclear spin is in general not known, we resorted to independent experimental methods to extract the value of ω_{0/1} of the target ^{13}C nuclear spin under varying magnetic field strength and orientation. To determine ω_{0}, we first initialized the system into the 0↓ or the 0↑ states. The nuclear qubit was then allowed to precess about ω_{0} for a variable time t before its state was read out. This gave both the magnitude and the direction of ω_{0}. Similar procedures with initial states of 1↓ or the 1↑ and precession about ω_{1} allow one to determine ω_{1} (Supplementary Methods).
State tomography
State tomography was performed by adopting the method detailed in Neumann et al.^{51} using the transitions , and as the ‘working transitions’. The first two transitions were driven by weak MW pulses, and the third one was realized by free precession of the nuclear qubit. The Rabi nutation and free precession signals were compared with the full oscillations in the m_{I}=0 subspace to extract the corresponding elements of the density matrix. The real and imaginary parts of the matrix elements were measured by using MW pulses with phases of 0 and π/2, respectively, and the required phase shift in the free precession signal was realized by transferring the populations to the m=−1 subspace and allowing the nuclear qubit to precess about ω_{1}. Other entries were obtained by transferring the corresponding populations to one of the working transitions. To map out all the offdiagonal elements, each of the diagonal elements in the density matrix was measured three times, and the data presented were taken as the mean of these measurements. The errors in the fidelity were calculated from the fitting errors of the Rabi nutations.
Sources of errors
In the experimental implementation of the scheme, a dominant source of errors originated from the imperfect selective πpulses used. As the ^{14}N nitrogen spin was unpolarized in the experiment, the weak MW_{1} and MW_{2} channels, which were respectively used to drive selective Rabi oscillations between the and transitions in the m_{I}=0 subspace of the ^{14}N spin, would also partially drive the other ^{14}Ncomponents. This resulted in imperfect selective πpulses and limited the effective polarization to only 90% even after applying the ‘kick out’ pulse (Fig. 6c). This introduced errors in both the initialization of the system and the state tomography process.
Such imperfection, however, does not affect the performance of the DD gate as the scheme used only nonselective centre spin DD pulses. The Rabi frequency of the strong MW channel (MW_{0}) (25 MHz) was much higher than the hyperfine interaction strengths for both the ^{14}N and ^{13}C nuclear spins, and this allowed Rabi oscillation between regardless of the states of the nuclear spins. Therefore, the fidelity of the demonstrated quantum gate should still be higher than that inferred from the measured state fidelities, which are currently limited by the errors in the weak selective MW pulses used in spin initialization and state tomography.
Design of DD gates by numerical optimization
We adopted a numerical optimization protocol similar to the one adopted in Khaneja et al.^{52} for the optimization of the average twoqubit fidelity {t_{α}} subjected to the DD constraints (echo condition and symmetric sequence). In each optimization step, we updated t_{α}→t_{α}+ subjected to the DD constraints, where could be directly calculated by evaluating U (Supplementary Methods) and ε is a small numerical parameter.
Simulation of the spin bath
The spin bath was simulated by considering the ^{14}N nitrogen host nuclear spin together with ^{13}C nuclear spins randomly placed in the diamond lattice with the natural abundance of 1.1%. Although the hyperfine coupling between the centre spin and the ^{14}N was modelled by established parameters, the coupling between the centre spin and the rest of the bath spins, as well as the coupling between bath spins, was assumed to be dipolar^{30,44}. The simulated bath was selected to reproduce the value of in FID experiment.
Additional information
How to cite this article: Liu, G.Q. et al. Noiseresilient quantum evolution steered by dynamical decoupling. Nat. Commun. 4:2254 doi: 10.1038/ncomms3254 (2013).
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Acknowledgements
This work was supported by National Basic Research Program of China (973 Program project Nos. 2009CB929103 and 2013CB921800), the National Natural Science Foundation of China (project Nos. 10974251, 11028510, 11227901, 1161160553 and 91021005), Hong Kong Research Grants Council—National Natural Science Foundation of China Joint Project N_CUHK403/11, the Chinese University of Hong Kong Focused Investments Scheme and Hong Kong Research Grants Council—Collaborative Research Fund Project HKU8/CRF/11G.
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Author notes
 GangQin Liu
 & Hoi Chun Po
These authors contributed equally to this work
Affiliations
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
 GangQin Liu
 & XinYu Pan
Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
 Hoi Chun Po
 & RenBao Liu
Hefei National Laboratory for Physical Sciences, Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
 Jiangfeng Du
Centre for Quantum Coherence, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
 RenBao Liu
Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China
 RenBao Liu
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Contributions
R.B.L. proposed the project and conceived the idea. H.C.P. designed the scheme and carried out the theoretical study. X.Y.P. and G.Q.L. designed the experiment. G.Q.L. carried out the experimental study. H.C.P. and G.Q.L. wrote the paper. J.D. discussed the scheme and the results. All authors analysed the data and commented on the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to RenBao Liu or XinYu Pan.
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Supplementary Information
Supplementary Figures S1S2, Supplementary Table S1, Supplementary Note 1 and Supplementary Methods
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