Abstract
The scalable application of quantum information science will stand on reproducible and controllable highcoherence quantum bits (qubits). Here, we revisit the design and fabrication of the superconducting flux qubit, achieving a planar device with broadfrequency tunability, strong anharmonicity, high reproducibility and relaxation times in excess of 40 μs at its fluxinsensitive point. Qubit relaxation times T_{1} across 22 qubits are consistently matched with a single model involving resonator loss, ohmic charge noise and 1/fflux noise, a noise source previously considered primarily in the context of dephasing. We furthermore demonstrate that qubit dephasing at the fluxinsensitive point is dominated by residual thermalphotons in the readout resonator. The resulting photon shot noise is mitigated using a dynamical decoupling protocol, resulting in T_{2}≈85 μs, approximately the 2T_{1} limit. In addition to realizing an improved flux qubit, our results uniquely identify photon shot noise as limiting T_{2} in contemporary qubits based on transverse qubit–resonator interaction.
Introduction
Over the past 15 years, superconducting qubits have achieved a remarkable fiveorderofmagnitude increase in their fundamental coherence metrics, including the energydecay time T_{1} , the Ramsey freeinduction decay time T_{2}* , and the refocused Hahnecho decay time T_{2E}. This spectacular trajectory is traceable to two general strategies that improve performance: (1) reducing the level of noise in the qubit environment through materials and fabrication improvements, and (2) reducing the qubit sensitivity to that noise through design advancements^{1}.
The charge qubit evolution is a quintessential example^{2}. Early demonstrations (Cooperpair box) exhibited nanosecondscale coherence times^{3}. Since then, operation at noiseinsensitive bias points (quantronium)^{4}, the introduction of capacitive shunting (transmon)^{5}, the use of twodimensional^{6} and threedimensional (3D)^{7} resonators to modify the qubit electromagnetic environment, the development of highQ capacitor materials and fabrication techniques^{8,9}, and the introduction of alternative capacitor geometries (Xmon)^{10} have incrementally and collectively raised coherence times to the 10–100 μs range^{10,11} and beyond^{12,13}. In addition, the capacitive shunt has generally improved devicetodevice reproducibility. The tradeoff, however, is a significant reduction in the charge qubit intrinsic anharmonicity (that is, the difference in transition frequencies f_{01} and f_{12} between qubit states 0, 1 and 1, 2) to 200–300 MHz for contemporary transmons, complicating highfidelity control and exacerbating frequency crowding in multiqubit systems^{14}.
In contrast, the performance of the persistentcurrent flux qubit^{15,16} has progressed more slowly over the past decade. Device asymmetry was identified early on to limit flux qubit coherence^{17} and, since 2005, symmetric designs have generally achieved 0.5–5 μs (refs 18, 19) with a singular report of T_{2E}=23 μs ≈2T_{1} (ref. 20). Despite respectable performance for individual flux qubits, however, devicetodevice reproducibility has remained poor. An early attempt at capacitive shunting^{21} improved reproducibility, but coherence remained limited to 1–6 μs (refs 22, 23). Recently, flux qubits embedded in 3D (ref. 24) and coplanar^{25} resonators exhibited more reproducible and generally improved relaxation and coherence times: T_{1}=6–20 μs, T_{2}*=2–8 μs. Nonetheless, further improvements in these times and in reproducibility are necessary if the flux qubit is to be a competitive option for quantum information applications.
In this context we revisit the design and fabrication of the flux qubit. Our implementation, a capacitively shunted (Cshunt) flux qubit^{21} coupled capacitively to a planar transmissionline resonator, exhibits significantly enhanced coherence and reproducibility, while retaining an anharmonicity varying from 500–910 MHz in the four devices with the highest relaxation times. We present a systematic study of 22 qubits of widely varying design parameters—shunt capacitances C_{sh}=9–51 fF and circulating currents I_{p}=44–275 nA—with lifetimes at the fluxinsensitive bias point ranging from T_{1}<1 μs (small C_{sh}, large I_{p}) to T_{1}=55 μs (large C_{sh}, small I_{p}). Over this entire range, the measured T_{1} values are consistent with a single model comprising ohmic charge noise, 1/fflux noise, and Purcellenhanced emission into the readout resonator. We furthermore investigated and identified quasiparticles as a likely source of observed T_{1} temporal variation. For the highest coherence devices, the Hahnecho decay time T_{2E}=40 μs<2T_{1} does not reach the 2T_{1} limit, as is also often observed with transmons coupled transversally to resonators^{7,10,26}. We demonstrate that this is due to dephasing caused by the shot noise of residual photons in the resonator (mean photon number ), observing a lorentzian noise spectrum with a cutoff frequency consistent with the resonator decay rate. We then use Carr–Purcell–Meiboom–Gill (CPMG) dynamical decoupling to recover T_{2CPMG}≈2T_{1} in a manner consistent with the measured noise spectrum.
Results
Cshunt flux qubit
Our circuits each contain two Cshunt flux qubits—with different frequencies—placed at opposite ends of a halfwavelength superconducting coplanar waveguide resonator (Fig. 1a). The resonator, ground plane and capacitors (Fig. 1b) were patterned from MBEgrown aluminium deposited on an annealed sapphire substrate^{8} (Supplementary Note 1). We used both square capacitors (Fig. 1b) and interdigital capacitors (IDCs, not shown) coupled capacitively to the centre trace of the coplanar waveguide resonator to enable qubit control and readout. In a second fabrication step, the qubit loop and its three Josephson junctions (Fig. 1c) were deposited using doubleangle, electronbeam, shadow evaporation of aluminium. One junction is smaller in area (critical current) by a factor α, and each of its leads contacts one electrode of the shunt capacitor. An equivalent circuit is illustrated in Fig. 1d (Supplementary Note 2).
Varying the qubit design enables us to explore a range of qubit susceptibilities to flux and charge noise with impact on both T_{1} and T_{2} (ref. 21). Compared with the conventional persistentcurrent flux qubit^{15,16}, our best Cshunt flux qubits have two key design enhancements. First, a smaller circulating current—achieved by reducing the area and critical current density of the Josephson junctions (Fig. 1c)—reduces the qubit sensitivity to flux noise, a dominant source of decoherence in flux qubits. Second, a larger effective junction capacitance—achieved by capacitively shunting the small junction (Fig. 1b)—reduces the qubit sensitivity to charge noise, and improves device reproducibility by reducing the impact of both junction fabrication variation and unwanted stray capacitance. Furthermore, the use of highquality fabrication techniques and physically large shunt capacitors reduces the density and electric participation of defects at the various metal and substrate interfaces^{1}.
The system is operated in the dispersive regime of circuit quantum electrodynamics and is described by the approximate Hamiltonian^{27}
where, the three terms are respectively the qubit (represented as a twolevel system), resonator and qubit–resonator interaction Hamiltonians, is the Pauli operator defined by the qubit energy eigenbasis, ω_{r} is the resonator angular frequency and is the resonator photonnumber operator. The qubit angular frequency ω_{q}(Φ_{b}) is set by the magnetic flux bias Φ_{b}, measured relative to an applied flux (m+1/2)Φ_{0} where m is an integer and Φ_{0} is the superconducting flux quantum, and attains its minimum value ω_{q}(0)≡Δ at the fluxinsensitive point Φ_{b}=0. The quantity χ(Φ_{b}) is the qubitstatedependent dispersive shift of the resonator frequency, which is used for qubit readout. In the Supplementary Notes 3–5, we discuss further the twolevel system approximation for the Cshunt flux qubit, an approximate analytic treatment which goes beyond equation (1), and the numerical simulation of the full qubit–resonator Hamiltonian used to make quantitative comparisons with our data.
T_{1} relaxation and noise modelling
We begin by presenting the T_{1} characterization protocol for the device in Fig. 1. We first identify the resonator transmission spectrum (Fig. 2a) by scanning the readoutpulse frequency ω_{ro} about the bare resonator frequency ω_{r}/2π≈8.27 GHz. Using standard circuit quantum electrodynamics readout, qubitstate discrimination is achieved by monitoring the qubitstatedependent transmission through the resonator^{27}. Next, we add a qubit driving pulse of sufficient duration to saturate the groundtoexcitedstate transition and sweep the pulse frequency ω_{d} (Fig. 2b). The resulting spectra for qubits A and B (Fig. 1a) exhibit minima Δ_{A}/2π≈4.4 GHz and Δ_{B}/2π≈4.7 GHz at the qubit fluxinsensitive points and increase with magnetic flux (bias current) away from these points. Finally, using a single πpulse to invert the qubit population, we measure the T_{1} relaxation of qubit A (T_{1}=44 μs) and qubit B (T_{1}=55 μs) at their fluxinsensitive points (Fig. 2c). Highpower spectroscopy (see Supplementary Note 6) reveals transitions among the first four qubit energy levels that are well matched by simulation, and identifies anharmonicities of 500 MHz in the two measured devices.
Using this protocol, we investigated 22 Cshunt flux qubits from five wafers (fabrication runs), spanning a range of capacitance values (C_{sh}=9–51 fF) and qubit persistent currents (I_{p}=44–275 nA) and featuring two capacitor geometries (interdigital and square). The junction critical currents were adjusted to maintain Δ/2π≈0.5–5 GHz (see Supplementary Note 7).
The data were analysed using simulations of the full system Hamiltonian and a Fermi’s golden rule expression for the exited state decay rate^{21},
where g〉(e〉) indicates the qubit ground (excited) states, and the sum is over four decay mechanisms: flux noise in the qubit loop, charge noise on the superconducting islands, Purcellenhanced emission to the resonator mode, and inelastic quasiparticle tunnelling through each of the three junctions. The operator is a transition dipole moment, and S_{λ}(ω_{q}) is the symmetrized noise power spectral density for the fluctuations which couple to it. For example, is a loop current operator for flux noise S_{Φ}(ω), and is an island voltage operator for charge noise S_{Q}(ω) (Supplementary Notes 8 and 9).
We considered both S_{λ}(ω)∝1/ω^{γ} (inversefrequency noise) and S_{λ}(ω)∝ω (ohmic noise)—the two archetypal functional forms of noise in superconducting qubits^{20,28,29,30,31,32,33}—for our magnetic flux and charge noise models, and used the frequency dependence of T_{1} for specifically designed devices to distinguish between them. While the following results are presented using symmetrized power spectral densities, we are careful to account for the distinction between classical and quantum noise processes in making this presentation (Supplementary Note 9).
For example, in Fig. 3a, Qubit C (C_{sh}=9 fF) has a large persistent current (I_{p}=275 nA) and a small qubit frequency (Δ_{c}/2π=0.82 GHz), making it highly sensitive to flux noise. Consequently, the measured T_{1} is predominantly limited by flux noise over a wide frequency range. This T_{1}trend constrains the flux noise model to the form S_{Φ}(ω)≡A_{Φ}^{2}(2π × 1 Hz/ω)^{γ} over the range 0.82–3 GHz (black dashed line, Fig. 3a). For comparison, the functional form for ohmic flux noise (grey dashed line), scaled to match T_{1} at Δ_{c}/2π=0.82 GHz (green dot), is clearly inconsistent with all other data over this frequency range. The noise parameters A_{Φ}^{2}=(1.4 μΦ_{0})^{2}/Hz and γ=0.9 used to match the data in Fig. 3a are derived from independent measurements—Ramsey interferometry^{31} and T_{1ρ} noise spectroscopy^{32} (Supplementary Note 10)—made at much lower frequencies in the context of classical noise related to qubit dephasing (Fig. 3b). These values are commensurate with earlier work on qubits^{20,31,32,33} and d.c. Superconducting QUantum Interference Devices (SQUIDs)^{34}. The consistency between the magnitude and slope of the flux noise power spectra, spanning more than twelve decades in frequency—millihertz to gigahertz—is remarkable, made even more so by the fact that the data in Fig. 3b were measured with a different device (qubit B, Fig. 3c).
In contrast, Qubit B (C_{sh}=51 fF) has a much smaller persistent current (I_{p}=49 nA) and larger qubit frequency (Δ_{B}/2π=4.7 GHz). Its value of T_{1} is most strongly influencedby charge noise (magenta dashed line, Fig. 3c) in the 5.0–6.5 GHz range, consistent with an ohmic charge noise model of the form S_{Q}(ω)≡A_{Q}^{2}ω/(2π × 1 GHz) with the parameter A_{Q}^{2}=(5.2 × 10^{−9}e)^{2}/Hz adjusted to match the data. In addition to flux and charge noise, the predicted value of T_{1} due to Purcell loss (light blue dashed line) is also included in Fig. 3a,c and involves no free parameters (see Supplementary Note 8). The resulting net value of T_{1} due to all three mechanisms (inversefrequency flux noise, ohmic charge noise and Purcell loss) is indicated with a red solid line and is in relatively good agreement with the ceiling of measured T_{1} values. As we describe below, quasiparticles are responsible for reducing the T_{1} below this ceiling.
Using these models, Fig. 3d shows a comparison of the measured and predicted T_{1} values for all 22 qubits. The flux noise model (from Fig. 3a,b) is applied to all qubits, and the Purcell loss is included with no free parameters. For the charge noise model, to achieve agreement across all devices, it was necessary to use A_{Q,SQ}^{2}=(5.2 × 10^{−9}e)^{2}/Hz for square capacitors (from Fig. 3b) and A_{Q,IDC}^{2}=(11.0 × 10^{−9}e)^{2}/Hz for IDCs, presumably reflecting the larger electric participation of the surface and interface defects for the IDC geometry^{1}. The agreement is noteworthy, given that these qubits span a wide range of designs across five fabrication runs (see Supplementary Note 7).
We note that inversefrequency charge noise was incompatible with these data over the entire frequency range investigated (not shown), implying that the crossover between inversefrequency and ohmic charge noise occurred at a frequency below 0.82 GHz. However, while ohmic flux noise S_{Φ}(ω)∝ω was inconsistent with T_{1} over the frequency range 0.82–3 GHz, its functional form is plausibly consistent with data above 3 GHz when appropriately scaled (upper dashed grey line, Fig. 3a) and, therefore, cannot be conclusively distinguished from ohmic charge noise. Although the best agreement across all 22 qubits (Fig. 3d) did not require ohmic flux noise, we could not rule out its presence in the 3–7 GHz range. In Supplementary Note 11, we compare models that use ohmic charge noise (as in Fig. 3) and ohmic flux noise. Differentiating between such charge and flux noise at higher frequencies will be the subject of future work. Indeed, for both ohmic flux noise S_{Φ}(ω)∝ω and inversefrequency charge noise S_{Q}(ω)∝1/ω, it is certainly possible (even expected) that the former (latter) dominates the flux (charge) noise at sufficiently higher (lower) frequencies.
The measured data for qubit B (Fig. 3c) exhibit fluctuations in the range T_{1}=20–60 μs for qubit frequencies ω_{q}/2π=4.7–6.5 GHz. To investigate their temporal nature, we measured T_{1} repeatedly at the qubit fluxinsensitive point over a 10h period and collected the data into sets of 50 individual decay traces. Fig. 4a,b show the results of two such experiments, with set 2 being taken ∼17 h after set 1. The average of all traces from set 1 exhibits a purely exponential decay, whereas the corresponding average for set 2 exhibits a faster shorttime decay and clear nonexponential behaviour (Fig. 4a). Histograms of the T_{1} values for individual traces exhibit a tight, Gaussianshaped distribution centred at 55 μs for set 1 and a broader, quasiuniform distribution centred near 45 μs for set 2. Over the course of several weeks, we observed transitions between these two characteristic modes of behaviour every few days for this device^{35}.
We attribute the temporal fluctuations and nonexponential decay function to excess quasiparticles—above the thermal equilibrium distribution—near the qubit junctions^{36,37,38,39}. Following ref. 40, we define as the average relaxation time associated with a single quasiparticle and take the quasiparticle number n_{qp} to be Poissondistributed with mean value . This results in a qubit polarization decay function,
where T_{1R} captures the residual exponential decay time in the absence of quasiparticles . The nonexponential decay function observed for set 2 is well described by equation (3) (black line in Fig. 4a) with fitting parameters , and T_{1R}=60 μs.
We use a quantum treatment of quasiparticle tunnelling to model the impact of single quasiparticles on the T_{1} of qubit B (Supplementary Note 8). Using a quasiparticle density x_{qp}=4 × 10^{−7} (per superconducting electron), the calculated recovers the fitted value μs at the fluxinsensitive point. Both and x_{qp} are comparable to the quasiparticleinduced relaxation rates and quasiparticle density reported for similar devices^{24,41}. The shaded region in Fig. 3a,c indicates the range of predicted T_{1} in the presence of quasiparticle. Most T_{1} data lie within this region, supporting the hypothesis that their scatter (particularly for qubit B in Fig. 3c) and the observed temporal T_{1} variation (Fig. 4b) arise from the common mechanism of quasiparticle tunnelling. In addition, the residual relaxation time T_{1R} for set 2 is similar to the exponential time constant obtained for set 1, indicating an underlying consistency in the noise models between the two data sets in the absence of quasiparticles. Unlike qubit B, qubit C consistently exhibited an exponential decay function (Fig. 4c) with little temporal variation (Figs 3a and 4d), indicating that quasiparticles did not strongly influence this device.
The results of Figs 3 and 4 demonstrate clearly that 1/ftype flux noise is the dominant source of qubit relaxation for frequencies below 3 GHz. To further strengthen this claim, it is instructive to compare relaxation times for qubits with similar frequencies and shunting capacitances, but where the persistent current (and thereby the sensitivity to flux noise) differs. We find that by reducing I_{p} from 170 nA to 60 nA, we improve the measured T_{1} from 2.3 to 12 μs (see qubits 11 and 13 in Supplementary Table 1 in Supplementary Note 7).
Pure dephasing and thermalphoton noise
We now address the transverse relaxation time T_{2} and our ability to refocus coherent dephasing errors. Efficient refocusing implies that T_{2} is limited entirely by T_{1}, since 1/T_{2}=1/2T_{1}+1/T_{ϕ}, where T_{ϕ} is the dephasing time. Generally, T_{2} is maximal at the fluxinsensitive point for conventional flux qubits^{18,19,20}, and the device reported in ref. 20 was efficiently refocused with a single echo pulse (T_{2E}=23 μs≈2T_{1}). In the current work, however, a single refocusing pulse is no longer completely efficient (T_{2E}<2T_{1}). This suggests that an additional, higherfrequency noise channel has been introduced. Unlike the device in ref. 20, which was coupled to a d.c. SQUID for readout, our Cshunt flux qubits are transversally coupled to a resonator (Fig. 1). Such inefficient refocusing is also reported for transmons similarly coupled to resonators^{7,10,26}.
As we show below, the main source of dephasing in Cshunt flux qubits biased at their fluxinsensitive point is photonnumber fluctuations (shot noise) in the resonator, which vary the qubit frequency via the a.c. Stark effect (as in the transmon case^{26,27}). Given a small thermalphoton population <<1 in the resonator (see Supplementary Note 12), the photoninduced frequency shift Δ_{Stark}^{th} and dephasing rate of the qubit are^{42}
The factor η=κ^{2}/(κ^{2}+4χ^{2}) effectively scales the photon population seen by the qubit due to the interplay between the qubitinduced dispersive shift of the resonator frequency χ and the resonator decay rate κ. Both the strong dispersive (2χ>>κ) and weak dispersive (2χ<<κ) regimes have been previously addressed^{26,43,44}. Here, we use qubit B to focus primarily on the intermediate dispersive regime (2χ/2π=0.9 MHz, κ/2π=1.5 MHz, see Fig. 5a) relevant for highfidelity qubit readout^{45}.
We begin by intentionally injecting additional thermalphotons into the resonator from an external noise generator with power P_{add} (Fig. 5b and Supplementary Note 2). In the small limit, the measured qubit spectrum exhibits a linear relationship between the effective qubit frequency and the generator power P_{add} (Fig. 5c,d). For completeness, we have included the Lamb shift Δ_{Lamb}, a fixed frequency offset due to the resonator zeropoint energy. Combining the extracted slope with equation (4), we calibrate the dependence of the addedphoton population (in the resonator) on the generator power P_{add}.
Next, we measure the Hahnecho dephasing rate for several photon populations using the calibrated . All echo traces (Fig. 5e) feature exponential decay rates Γ_{2E}=1/T_{2E}, indicating little (if any) impact from 1/f noise (charge, flux and so on) and consistent with photon shot noise featuring a short correlation time 1/κ<<T_{2E}. The extracted pure dephasing rate scales linearly with photon population (Fig. 5f). The extracted slope agrees with equation (5) to within 5%. The nonzero dephasing rate at corresponds to a residual photon population , equivalent to an effective temperature T_{eff}=80 mK. By comparison, the qubit effective temperature determined from its first excitedstate population is 35 mK (ref. 13).
To confirm that the noise arises from residual thermalphotons, we directly measure the noise power spectral density (PSD) using the T_{1ρ} (spinlocking) method^{32}. This method (inset Fig. 6a) collinearly drives the qubit along the yaxis with a long Y pulse, which ‘locks’ the qubit state in the rotating frame. Measuring the qubit relaxation rate in the rotating frame, Γ_{1ρ}(Ω_{Rabi})=S_{z}(Ω_{Rabi})/2+Γ_{1}/2, effectively samples the noise PSD S_{z}(ω) seen by the qubit at the locking (Rabi) frequency Ω_{Rabi} (see Supplementary Note 10). By varying the locking drive amplitude, which is proportional to Ω_{Rabi}, we sample the noise spectrum over the range ω/2π=0.1–100 MHz (Fig. 6a). Below 10 MHz, the resolved noise spectra for all (including ) have similar shapes: flat (white) at low frequencies with a 3dB highfrequency cutoff at the resonator decay rate ω=κ. This form is consistent with the expected lorenzian PSD for thermalphotons in a resonator as seen by the qubit (see Supplementary Note 10),
which includes the dispersive coupling χ and the filtering factor η [see equations (4 and 5)]. Equation (6) agrees with the measured PSDs for all photon populations , with the residual photon number extracted from equation (6). This agreement eliminates the driving or readout field as the source of the residual photons, because such coherentstate photons follow Poisson statistics with a resulting cutoff frequency κ/2 (half the observed value)^{46,47}.
Finally, we apply dynamical decoupling techniques to validate the functional form of the measured noise PSD and to recover T_{2}≈2T_{1}. We use the CPMG (inset Fig. 6b) pulse sequence, comprising a number N_{π} of equally spaced πpulses. The application of πpulses in the time domain can be viewed as a bandpass filter in the frequency domain which shapes the noise spectra seen by the qubit^{21,48,49,50}. Since the filter passband is centred at a frequency inversely related to the temporal spacing Δτ between adjacent pulses, increasing N_{π} for a fixed sequence length will shift this passband to higher frequencies (see Supplementary Note 13).
Figure 6b shows the measured CPMG decay time T_{2CPMG} versus πpulse number N_{π} with no added noise . From N_{π}=1 (Hahnecho) to N_{π}=100, the decay time T_{2CPMG} remains near 40 μs, consistent with the whitenoise (flat) portion of the noise PSD in Fig. 6a. Above N_{π}=100, the passband frequency traverses the cutoff region of the PSD and, as the integrated noise level decreases, T_{2CPMG} rises. For N_{π}>1,000, the refocusing becomes efficient with . The close correspondence between the noise spectral density in Fig. 6a and the mitigation of that noise by CPMG in Fig. 6b strongly supports our methods and interpretations.
Discussion
The Cshunt flux qubit is a planar device with broadfrequency tunability, relatively strong anharmonicity and high reproducibility, making it well suited to both gatebased quantum computing and quantum annealing. The anharmonicity can be significantly higher than that of transmon qubits, allowing for faster (even subnanosecond^{51,52}) control pulses and reduced frequency crowding in multiqubit systems. The addition of a highqualityfactor shunt capacitance to the flux qubit, together with a reduced qubit persistent current, has enabled us to achieve values of T_{1} as high as 55 μs at the qubit fluxinsensitive point. We are able to account for measured T_{1} values across 22 qubits with a single model involving ohmic charge noise, 1/fflux noise, and the Purcell effect, with temporal variation in T_{1} explained by quasiparticle tunnelling. On the basis of this model, we anticipate further design optimization leading to even higher coherence will be possible. Finally, we used spinlocking to directly measure the photon shot noise spectral density, and we verified its functional form using a CPMG pulse sequence to reach a T_{2} of 85 μs—limited by 2T_{1}—at the fluxinsensitive point. These measurements identify photon shot noise as the dominant source of the observed dephasing, and have direct implications for any qubit in which the readout involves its transverse coupling to a resonator.
The role of highfrequency 1/fflux noise in qubit relaxation is intriguing. Our T_{1} data and their frequency dependence across 22 different qubits strongly support the conclusion that 1/fflux noise contributes to qubit relaxation up to at least 3 GHz in our devices. Above 3 GHz, there is some ambiguity between ohmic flux and ohmic charge noise, and clarifying the roles of these respective noise sources is the subject of future work. A detailed understanding of such a broadband 1/fflux noise mechanism and its transition from classical to quantum behaviour is of great practical interest and awaits theoretical explanation.
Data availability
The data that support the findings of this study may be made available from the corresponding author upon request and with the permission of the US Government sponsors who funded the work.
Additional information
How to cite this article: Yan, F. et al. The flux qubit revisited to enhance coherence and reproducibility. Nat. Commun. 7, 12964 doi: 10.1038/ncomms12964 (2016).
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Acknowledgements
We gratefully acknowledge A. Blais, A. Clerk, and Y. Nakamura for useful discussions and P. Baldo, V. Bolkhovsky, G. Fitch, P. Murphy, B. Osadchy, K. Magoon, R. Slattery and T. Weir at MIT Lincoln Laboratory for technical assistance. This research was funded in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) and by the Assistant Secretary of Defense for Research & Engineering via MIT Lincoln Laboratory under Air Force Contract No. FA872105C0002; by the U.S. Army Research Office Grant No. W911NF1410682; and by the National Science Foundation Grant No. PHY1415514. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA or the US Government.
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F.Y., A.K., S.G., J.B., A.P.S. and D.H. performed the experiments. S.G., A.P.S., A.K., D.H., A.J.K. and W.D.O. designed the devices.T.J.G., J.L.Y. and J.B. fabricated the devices. A.J.K., S.G., F.Y., D.H., D.R., G.S., S.W. performed device simulations. S.G., T.P.O., J.C., A.J.K. and W.D.O. supervised the project. All authors contributed to the conception, execution and interpretation of the experiments.
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Supplementary Figures 111, Supplementary Table 1, Supplementary Notes 113 and Supplementary References. (PDF 10005 kb)
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Yan, F., Gustavsson, S., Kamal, A. et al. The flux qubit revisited to enhance coherence and reproducibility. Nat Commun 7, 12964 (2016). https://doi.org/10.1038/ncomms12964
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