Abstract
We benchmark the decoherence of superconducting transmon qubits to examine the temporal stability of energy relaxation, dephasing, and qubit transition frequency. By collecting statistics during measurements spanning multiple days, we find the mean parameters \(\overline {T_1}\) = 49 μs and \(\overline {T_2^ \ast }\) = 95 μs; however, both of these quantities fluctuate, explaining the need for frequent recalibration in qubit setups. Our main finding is that fluctuations in qubit relaxation are local to the qubit and are caused by instabilities of nearresonant twolevelsystems (TLS). Through statistical analysis, we determine submillihertz switching rates of these TLS and observe the coherent coupling between an individual TLS and a transmon qubit. Finally, we find evidence that the qubit’s frequency stability produces a 0.8 ms limit on the pure dephasing which we also observe. These findings raise the need for performing qubit metrology to examine the reproducibility of qubit parameters, where these fluctuations could affect qubit gate fidelity.
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Introduction
Universal, faulttolerant quantum computers—a Holy Grail of quantum information processing—are currently being pursued by academia and industry alike. To achieve fault tolerance in a quantum information processor, a scheme for quantum error correction^{1} is needed due to the limited coherence lifetimes of its constituent qubits and the consequently imperfect quantumgate fidelities. Such schemes, e.g., the surface code,^{2} rely on gate fidelities exceeding a certain breakeven threshold. Adequately high fidelity was recently demonstrated with superconducting qubits;^{3} however, this breakeven represents a bestcase scenario without any temporal or devicetodevice variation in the coherence times or gate fidelities. Therefore, a faulttolerant quantum computer importantly requires not only improvements of the bestcase single^{4} and twoqubit^{3} gate fidelities: it actually requires the typical performance—in the presence of fluctuations—to exceed the error correction threshold. In the more immediate term, socalled Noisy IntermediateScale Quantum (NISQ)^{5} circuits will be operated without quantum error correction. In NISQ systems, gate fidelities and the fluctuations thereof directly limit the circuit depth, i.e., the number of consecutive gates in an algorithm that can be successfully implemented.
In experiments with superconducting qubits, it is usual to perform qubit metrology^{6} to benchmark the gate fidelity and quantify its error, although these benchmarks are not typically repeated in time to determine any temporal dependence. Since gate fidelities are at least partially limited by qubit T_{1} energy relaxation,^{4} one would expect a fluctuation in gate fidelity resulting from a fluctuation in the underlying decoherence parameters. However, benchmarking of decoherence, to quantify the mean lifetime together with its stability or variation, is also uncommon. Consequently, it is unclear whether reports on improvements in coherence times—cf. the review by Oliver and Welander^{7} and that by Gu and Frisk Kockum et al.^{8}—are reports of typical or of exceptional performance. Quantifying this difference is crucial for future work aimed at improving qubit coherence times.
In this paper, we benchmark the stability of decoherence properties of superconducting qubits: T_{1}, \(T_2^ \ast\) (freeinduction decay), T_{ϕ} (pure dephasing), and f_{01} (qubit frequency). This study is distinct from numerous studies that report on singular measurements of qubit lifetimes for different background conditions, such as temperature^{9} or magnetic flux.^{10,11} Some studies^{11,12,13,14,15,16,17} examine repeated measurements of qubit lifetimes under static conditions. However, when discussing these examples, it is important to quantify both the number of counts and the total duration of the measurement. Here, the number of counts relates to the statistical confidence, while the total duration relates to the timescale of fluctuations to which the study is sensitive. Therefore, to confidently report on fluctuations relevant to the calibration period of a quantum processor (for example a few times a day for the IBM Q Experience^{18}), we only discuss reports featuring both a large number of counts (N > 1000) and a total duration exceeding 5 h.
The first study to satisfy these requirements for relaxation measurements was that of Müller et al.,^{14} which revealed that unstable nearresonant twolevelsystems (TLS) can induce fluctuations in qubit T_{1}. They proposed a model in which the TLS produces a strongly peaked Lorentzian noise profile at the TLS frequency (which is near the qubit frequency). Under the separate model of interacting TLS,^{19,20} the frequency of this nearresonant TLS varies in time. Consequently, the qubit probes the different parts of the TLSbased Lorentzian noise profile, leading to variations in the qubit’s T_{1}. Although the mechanism was clearly demonstrated, this work^{14} was unable to determine properties of the TLS such as switching rates or dwell times of specific TLS frequency positions. Followup work by Klimov et al.^{17} used a tuneable qubit to map the trajectories of individual TLS. These findings^{17} supported the interactingTLS model and Müller’s findings, and were able to clearly determine TLS switching rates, as well as reveal additional diffusive motion of the TLS.
We demonstrate that sufficient statistical analysis can reveal the TLSbased Lorentzian noise spectrum and allow for extraction of switching rates. Importantly, this method does not require a tuneable qubit or advanced reset protocols^{21} and is therefore general to any qubit or setup. Furthermore, the lack of tuning results in a more frequencystable qubit and consequently less dephasing. This enables us to go beyond the studies of Müller et al. and Klimov et al. by studying the qubit’s frequency instabilities due to other noise sources, which reveals a 1/f frequency noise that is remarkably similar to interactingTLSinduced 1/f capacitance noise found in superconducting resonators.^{20,22} This frequency instability produces a limit on pure dephasing which we observe through sequential interleaved measurements of qubit relaxation, dephasing, and frequency.
Results
Our circuit is made of aluminium on silicon and consists of a singlejunction Xmontype transmon qubit^{23} capacitively coupled to a microwave readout resonator (see the Methods section IV A for more details). The shunt capacitor and the absence of magneticflux tunability (absence of a SQUID) effectively decouple the qubit frequency from electrical charge and magnetic flux, reducing the sensitivity to these typical 1/f noise sources.^{24,25} Although these qubits lack frequency tunability, they remain suitable for multiqubit architectures using allmicrowavebased twoqubitgates.^{26,27,28,29} The circuit is intentionally kept simple so that the decoherence is dominated by intrinsic mechanisms and not external ones in the experimental setup. Therefore, there are no individual qubit drive lines, nor any qubittoqubit couplings. In addition, both the spectral linewidth of the resonator and the resonatorqubit coupling are kept small, such that photon emission into the resonator (Purcell effect) and dephasing induced by residual thermal population of the resonator are minimised.^{30} A detailed experimental setup together with all device parameters are found in the Methods and Table 1.
This study involves two qubits on separate chips which we name A and B. The main differences are their Josephson and charging energies and that the capacitor of qubit B was trenched to reduce the participation of dielectric loss.^{31}
First we assess the stability of the energyrelaxation time T_{1} by consecutive measurements. The transmon is driven from its ground to firstexcited state by a calibrated π pulse. The qubit state is then read out with a variable delay. The population of the excited state, as a function of the readout delay, is fit to a singleexponential decay to determine T_{1}. Figure 1 shows a 65h measurement of two separate qubits (in separate sample enclosures) that are measured simultaneously. The first observation is that the periods of lowT_{1} values are not synchronised between the two qubits, indicating that the dominant mechanism for T_{1} fluctuations is local to each qubit. (The lack of correlation is quantified in Supplementary Fig. 6). In Fig. 1b, we histogram the T_{1} data: this demonstrates that T_{1} can vary by more than a factor 2 for both qubits, similarly to previous studies.^{14,17}
To make a fair comparison of the mean T_{1} for two qubits with different frequencies, we can rescale to quality factors (Q = 2πf_{01}T_{1}). We see that qubit B (Q = 1.67 × 10^{6}) has a higher quality factor compared with qubit A (Q = 1.29 × 10^{6}). However, while the quality of qubit B is higher, qubit B has a lower ratio of Josephson to charging energy (see Table 1), resulting in a larger sensitivity to charge noise and parity effects.^{32} Consequently, qubit B exhibits switching between two different transition frequencies, which was not suitable for later dephasing and frequency instability studies. Therefore, most of the paper focuses on qubit A.
We continue by measuring T_{1} consecutively for approximately 128 h, and plot the decays in a colour map (Fig. 2a). Here, the colour map makes some features of the data simpler to visualise. Firstly, the fluctuations are comprised of a switching between different T_{1} values, where the switching is instantaneous, but the dwell time at a particular value is typically between 2 h and 12.5 h. This behaviour (also seen in Fig. 1a) resembles telegraphic noise with switching rates ranging from 20 to 140 μHz. Later, we quantify these rates and their reproducibility.
The white box of Fig. 2a and inset Fig. 2b show this switching behaviour occurring within a single iteration. The decay can be fit to two different values of T_{1}, one before the switch and one afterwards. This type of decay profile is found in approximately 3% of the iterations. In all presented T_{1} values (histograms or sequential plots), the lower T_{1} value is used. This is motivated by quantum algorithms being limited by the shortestlived qubit.
The black box and inset Fig. 2c highlight a decayprofile that is no longer purely exponential, but instead exhibits revivals. Similar revivals have been observed in both phase^{33} and flux^{34} qubits, and were attributed to coherently coupled TLS residing in one of the qubit junctions. From the oscillations we extract a qubittoTLS coupling of g_{TLS} = 4.8 kHz. Assuming a TLS dipole moment of 1 eÅ,^{35} the coupling corresponds to an electric field line of 39 μm (see the Supplementary section C for more details). This length is larger than the Josephson junction; therefore, we conclude that this particular TLS is located on one of the surfaces of the shunting capacitor (not within the junction). Since the invention of transmons and improvement in capacitor dielectrics, individual TLS have only been found to incoherently couple to a transmon,^{23} and the authors are not familiar with any examples of a coherent coupling between a TLS and a transmon.
Approximately 5% of decay profiles show a clear revival structure, with a further 3% showing hints of it. Of these, some revival shapes (such as the one shown in the black box) remained stable and persisted for approximately 10 h, whereas others lasted for only 2–3 traces (around 10 min). Since the qubit here is fixed in frequency, these appearances/disappearances of the coherent TLS arise due to the TLS shifting in frequency^{14,17,19,20} relative to the static qubit. The observation of coherent oscillations in the decay, and in particular that oscillation periods remained stable for hours (for the same duration as the T_{1} fluctuations), constitutes clear evidence for TLS being the origin of the T_{1} fluctuations, in agreement with both the Müller^{14} and Klimov^{17} results.
To gain further insight into these fluctuations we perform statistical analysis commonly used in the field of frequency metrology. In parallel, we examine both the overlapping Allan deviation (Fig. 3a) and the spectral properties (Fig. 3b) of the T_{1} fluctuations. Allan deviation is a standard tool for identifying different noise sources in e.g. clocks and oscillators.^{36} Here, we introduce the Allan deviation as a tool to identify the cause of fluctuations in qubits. The most striking feature in Fig. 3a is the peak and subsequent decay around τ = 10^{4} s. Importantly, no powerlaw noise process can produce such a peak; instead, it is an unambiguous sign of a Lorentzian noise process. Such Lorentzianlike switching was observed in the T_{1}vs.time measurement in Fig. 1a. In Fig. 3, we model the noise with two Lorentzians with a white noise floor, and apply the modelled noise to both the spectrum and the Allan deviation. Therefore, the noise parameters are the same for both plots: the Methods section has more details on the scaling of Lorentzian noise between the Allan and spectral analysis methods. From Fig. 3, we obtain Lorentzian switching rates of 80.6 and 158.7 μHz.
Within Fig. 4 and Table 2, we show the reproducibility of these features across thermal cycles. Collectively, we find switching rates ranging from 71.4 μHz to 1.9 mHz—slower than those obtained by measurements of charge noise^{37} but similar to bulkTLS dynamics^{38,39} and in agreement with rates determined from measurements tracking the time evolution of individual TLS.^{17} These measurements demonstrate not only that superconducting qubits are useful probes of TLS, but unambiguously demonstrate the role of a TLSbased Lorentzian noise profile as a limiting factor to the temporal stability of qubit coherence.
In addition to studying T_{1} fluctuations, we also explore fluctuations in qubit frequency and dephasing. To this end, we measure the qubit frequency and the characteristic decay time \(T_2^ \ast\) by means of a detuned Ramsey fringe. We interleave the Ramsey sequence, pointbypoint, with the previously discussed T_{1} relaxation sequence. For clarity, if we consider the energyrelaxation measurements in Fig. 2, the main plot (a) represents the complete measurement set, which is formed from 2000 iterations. Each iteration (e.g., either inset) consists of data points which are themselves the averaged results of 1000 repeated measurements. In the interleaved sequence, we measure the data point in the T_{1} sequence and then the data point in the Ramsey sequence for each delay time (i.e., the time between the π pulse and readout, in the T_{1} measurement, and inbetween the π/2 pulses, in the Ramsey \(T_2^ \ast\) measurement). This sequence is then looped through all values of the delay time to map out the T_{1} and Ramsey decay profiles (i.e., the iteration). While averaging each point in the inner loop gives a longer iteration time, and increases the noise window to which the Ramsey fringe is sensitive,^{40} it allows for all qubit parameters to be known in each iteration. From the soobtained T_{1} and \(T_2^ \ast\) we calculate the puredephasing time T_{ϕ} from \(1/T_2^ \ast = 1/2T_1 + 1/T_\phi\). These values are shown in Fig. 5b, and the histogram of \(T_2^ \ast\) values is shown in Fig. 5c.
In Fig. 5a we have extracted, from the Ramsey fringes, the frequency motion of the qubit relative to its mean frequency (\(f_{01}  \overline {f_{01}}\)). In general, the observed frequency shifts are on the order of 1 to 3 kHz, with infrequent shifts of up to 20 kHz. A histogram of the qubit frequency (Fig. 5d) reveals a main peak with a fullwidth at halfmaximum of approximately 2 kHz. These frequency shifts are significantly smaller than the approximately 500 kHz frequency instability found in fluxtuneable qubits.^{17} From the perspective of gate fidelity, a 1kHz frequency shift should have negligible effect, meaning that our qubits are well suited for quantum information processing since no recalibration of the qubit frequency is needed. However, a fluctuating qubit frequency necessarily leads to qubit dephasing so it is important to quantify this fluctuation and therefore aid in efforts to find, and mitigate, the noise source.
To provide more information on possible mechanisms for the frequency instability, we examine both the overlapping Allan deviation (Fig. 5e) and the spectrum of frequency fluctuations (Fig. 5f). In red, the frequency noise is modelled to A/f + B, where the exponent of f is 1. Similarly to the previous T_{1} analysis, the noise model is scaled so that the red line has the same amplitude in both Fig. 5e, f. In this model, the 1/f noise amplitude is A = 3.6 × 10^{5} Hz^{2}.
Discussion
For both qubits, across all cooldowns, we found fluctuations in T_{1} that could be described by Lorentzian noise with switching rates in the range from 75 μHz to 1 mHz. For all superconducting qubits, three relaxation channels are usually discussed: TLS, quasiparticles, and parasitic microwave modes. Of these, parasitic microwave modes should not cause fluctuation since they are defined by the physical geometry. For quasiparticles in aluminium, we can compare our observed slow fluctuations with quasiparticle mechanisms found in the qubit literature: the quasiparticle recombination rate is 1 kHz;^{41} the timescale of quasiparticle number fluctuations leads to rates in the range from 0.1 kHz to 10 kHz; and finally, quasiparticle tunnelling (parity switching events) in transmons have rates in the range 0.1 kHz to 30 kHz.^{32} Therefore, fluctuations in the properties of the superconductor occur over rates which differ by over six orders of magnitude compared to those found in our experiment. Instead, we highlight that, at low temperatures, bulkTLS dynamics^{38,39} and TLScharge noise^{34,37} vary over long timescales equivalent to rates in the range from 10 mHz to 100 Hz.
The observed coherent qubit–TLS coupling (Fig. 2c) is an unambiguous sign of the existence of nearresonant TLS. Its fluctuation follows similar time constants as the T_{1} fluctuations, which constitutes clear evidence of spectral instability, as expected from the interactingTLS model.^{19,20} We therefore attribute the origin of the T_{1} decay to nearresonant TLS, and the Lorentzian fluctuations in the qubit’s T_{1} (shown in Figs. 3 and 4g–l) arise due to spectral instabilities of the TLS as described by Müller et al.^{14} The extracted switching rates then represent the rate at which the nearresonant TLS is changing frequency. Similarly, the quality factor of superconducting resonators has also been found to vary^{42} due to spectrally unstable TLS.
In general, we find that two separate Lorentzians are required to describe the fluctuation. This does not necessarily imply the existence of two nearresonant TLS—instead it is a limitation of the analysis, as we cannot resolve the difference between, say, two nearresonant TLS, each with two preferential frequencies, vs. one nearresonant TLS that has four preferential frequencies. Such a difference could be inferred by measuring the local density of nearresonant TLS,^{17} although such a measurement has demonstrated that both scenarios above are possible.^{17} In addition, when repeating the measurements across multiple cooldowns, we consistently find a nearresonant TLS that follows similar switching statistics. Between each cooldown, the TLS configuration is expected to completely change. Essentially, this means that the detuning and coupling of the observed nearresonant TLS should vary for each cooldown. However, despite any expected reconfiguration, at least one spectrally unstable nearresonant TLS is always found to exist.
When examining the frequency stability of qubit A, we found a frequency noise of approximately 2 kHz, which was well described by a 1/f amplitude of 3.6 × 10^{5} Hz^{2} (Fig. 5d–f). Typically, dephasing is thought to arise due to excess photons within the cavity,^{9,43} flux noise,^{25} charge noise,^{34,37} quasiparticles tunnelling through the Josephson junctions,^{32} or the presence of excess quasiparticles.^{44} For qubit A, the charge dispersion is calculated to 524 Hz, much smaller than most of the observed frequency shifts. This rules out charge noise and tunnelling quasiparticles as the main source of the observed frequency fluctuations. Quasiparticle fluctuations have been extensively studied,^{41} where the magnitude of frequency shifts scales with the kinetic inductance. Therefore, while they can be of order 100 kHz in disordered superconductors,^{45} they are much smaller in elemental superconductors. In fact, recent experiments^{41} showed that the quasiparticles in aluminium produced an unmeasurably small frequency shift; instead, the quasiparticles’ influence was revealed only by examining correlated amplitude and frequency noise. Therefore, not only do quasiparticles produce immeasurably small frequency shifts, but, as noted earlier, they act over much shorter timescales (i.e., rates are equivalent to kHz^{32,41} rather than the submHz timescales observed here).
Instead we highlight two further TLSbased mechanisms. Firstly, TLS within the Josephson junction can cause criticalcurrent noise,^{46} which can produce a frequency noise by modulating the Josephson energy. Alternatively, superconducting resonators demonstrate that TLS can produce frequency instabilities^{20,22} (capacitance noise). Both of these mechanisms exhibit a 1/f noise, where the noise amplitude is close to that which we find here. One could distinguish between these two effects by examining the temperature dependence of the qubit’s frequency noise. Here, critical current noise^{46} scales ∝T while capacitance noise^{20,22} scales ∝1/T^{1.3}.
Irrespective of the origin of the frequency instability, the noise spectrum in Fig. 5f can be integrated to estimate the pure dephasing of the qubit.^{47} From this calculation, the expected T_{ϕ} is 0.8 ms. In Fig. 5g we histogram the T_{ϕ} to reveal a peak around 0.7 ms, with diminishing counts above 1 ms, in good agreement with the estimate from the integrated frequency noise.
In Fig. 5b, c, \(T_2^ \ast\) is almost always longer than T_{1}, implying that T_{ϕ} > 2T_{1}. In Fig. 5h this is quantified, as the histogram of the ratio of \(T_2^ \ast {\mathrm{/}}T_1\) reveals that the qubit dephasing is almost always near 2T_{1}. Therefore, the qubit’s \(T_2^ \ast\) is mainly limited by T_{1}. To the authors’ knowledge all other demonstrations of T_{1}limited T_{2} required dynamical decoupling by either a Hahnecho (spinecho)^{9,25} or CPMG^{11} sequence. However, neither of those works provide any statistics on whether the qubits were always T_{1}limited. The histogram in Fig. 5h also reveals counts where the ratio is above 2: these correspond to the instances where the T_{1} has fluctuated within an iteration, similar to that shown in Fig. 2b.
In summary, we have measured the stability of qubit lifetimes across more cooldowns and for measurement spans longer than previous studies. Collectively, this demonstrates that qubit fluctuations, due to spectrally unstable TLS, are consistently observed, even when T_{1} is high (approaching 100 μs). The spectral motion of TLS is particularly problematic for gates requiring frequency movement (e.g., CPHASE^{3}). Here, leakage to TLS during gate operations, and the optimum idling frequencies become timevarying on a perqubit basis. In addition, since gate fidelities contain a nonnegligible error contribution from T_{1},^{48} the optimal gate duration will be affected by T_{1} varying in time. Therefore, these fluctuations demonstrate why it is necessary to recalibrate qubits every few hours. Fundamentally, this also demonstrates that future reports on qubit coherence times require not only statistics for reproducibility, but also that the measurement duration should exceed several hours in order to adequately report the typical rather than exceptional coherence time.
Note added—Recently, a preprint on comparable observations was published by Schlör et al.,^{49} who independently demonstrated that single fluctuators (TLS) are responsible for frequency and dephasing fluctuations in superconducting qubits. In addition, another recent preprint by Hong et al.,^{50} specifically measures fluctuations in gate fidelity and independently identifies T_{1} fluctuation of the underlying qubits as the probable cause.
Methods
Experimental details
The qubits are fabricated out of electronbeam evaporated aluminium on a highresistivity intrinsic silicon substrate. Everything except the Josephson junction is defined using directwrite laser lithography and etched using wet chemistry. The Josephson junction is defined in a bilayer resist stack using electronbeam lithography, and later deposited using a twoangle evaporation technique that does not create any extra junctions or floating islands.^{51} An additional lithography step is included to ensure a superconducting contact between the junction and the rest of the circuit; after the lithography, but prior to deposition of aluminium, an argon ion mill is used to remove native aluminium oxide. This avoids milling underneath the junction, which has been shown to increase the density of TLS.^{10} Finally, the wafer is diced into individual chips and cleaned thoroughly using both wet and dry chemistry. Moreover, qubit B underwent a trenching step where approximately 1 μm of the silicon dielectric was removed from both the qubit and the resonator using an fluorine based reactiveion etch.^{52}
A simplified schematic of the experimental setup is shown in Fig. 6a. The samples sit within a superconducting enclosure, which itself is inside of an absorberlined radiation shield and a cryoperm layer. This is located within a further absorberlined radiation shield and a further superconducting layer which encloses the entire mixing chamber. Everything inside the cryoperm layer (screws, sample enclosures, and cables) is nonmagnetic. The setup, including absorber recipe, is similar to a typical qubit boxinabox setup.^{44} For the different cooldowns, two setups (labelled 1 and 2) were used. Setup 2 was as described above, whereas setup 1 lacked the absorber coating marked with a red asterisk in Fig. 6a.
Data handling
The qubit decoherence data is processed in the following way. First, the digitiser signal is rotated to one quadrature. Next, the signal is normalised to the maximum visibility of the qubit 0〉 and 1〉 states. Then, for qubit relaxation data, a fit to a single exponential is performed. Within Fig. 4a–c, T_{1} data is presented with errorbars. These errorbars correspond to 1 standard deviation, determined from confidence intervals of the exponential fit. For the Ramsey measurements, the initial processing is as described above. However, the Ramsey frequency (f_{Ram}) is initially determined by FFT of the data. The resulting frequency from the FFT is used as an initial frequency guess to a model of the form:
where ϕ_{0} is a phase offset that is generally zero. Across all of the datasets examined for qubit A, the FFT reveals only one oscillation frequency, whereas for qubit B, two frequencies are observed due to a larger charge dispersion. Consequently, Eq. (1) does not fit well for qubit B, and we omit qubit B from the dephasing and frequency results.
For the qubit relaxation, we did attempt fits to a double exponential model.^{11,53} Within this model, an additional relaxation channel due to quasiparticles near the junction can lead to a skewing of the decayprofile. Here, we found the confidence interval for numerous parameters was unphysically large, indicating that the model overparameterised our data. Therefore, we continued to use the singleexponential model. However, this is not surprising as the doubleexponential is typically used for fluxqubits and fluxoniums, rather than the singlejunction transmontype qubit studied here.
Sample handling
Here, we clarify the sample handling across the entire experiment. For each qubit sample, after completing fabrication, they were covered in protective resist until the morning of their first cooldown (cooldown 1 for qubit B and cooldown 2 for qubit A). After removal of the resist, the samples were wirebonded within a sample enclosure. Once sealed, the samples remained within their enclosures and were kept attached to the fridge for the entire experimental run. Therefore, when the fridge was warm, the samples were kept at the ambient conditions of the lab. Qubit B was not measured between cooldown 2 and cooldown 5, although it was still cooled down. However, qubit B was examined again in cooldowns 5 and 6 to gather statistics on the reproducibility of parameter fluctuations.
Spectral and Allan analysis
Within the main text, information on TLS switching rates is inferred by examining the reproducibility of coherence parameters. Primarily, this is obtained by examining the Allan statistics and spectral properties of T_{1} fluctuations. Here, the same data set is used to produce a plot of a Welchmethod FFT (\(S_{T_1}(f)\)) and an overlapping Allan deviation (\(\sigma _{T_1}(\tau )\)). For the Welch analysis, the quantity analysed is \(T_1  \overline {T_1}\) (or for frequency it is \(f_{01}  \overline {f_{01}}\)). Therefore, the analysed quantity is not presented in fractional units. Consequently, the units of spectral analysis are μs^{2}/Hz for T_{1} fluctuations (Hz^{2}/Hz for frequency analysis). Equivalently, the Allan statistics are presented in units of μs for T_{1} fluctuation (Hz for frequency fluctuation).
The Allan deviation offers a few advantages compared with the spectrum. The method is directly traceable in that the Allan methods use simple mathematical functions that do not require any careful handling of window functions or overlap. When examining lowfrequency processes, this eases a considerable burden in FFTanalysis which is to distinguish real features from remnants of window functions. This traceability is core to the usage within the frequency metrology community. The Allan method also provides clear error bars (defined as equal to 1 standard deviation), which translate to an efficient use of the data with optimum averaging of all data that shares a common separation, that is, all data pairs for any separation (τ in the Allan plot) are averaged over. Moreover, the Allan method can distinguish linear drift from any other divergent noise processes. Within an FFT, a linear drift appears as a general 1/f^{a} slope where a is not unique compared with other noise sources. Within the Allan, a linear drift appears as τ^{a} where a is distinct and unique compared with other divergent noise types.
From here, beginning with the Allan deviation, we consider the standard powerlaw model^{36} of noise processes,
which can also be represented as spectral noise
where, in frequency metrology notation, h_{−2} is the amplitude of a random walk noise process, h_{−1} is the amplitude of a 1/f noise process and h_{0} is the amplitude of white noise.
In general terms, the powerlaw noise processes create a welllike shape in the Allan analysis, where, with the terms listed above, the walls have slopes of \(\propto \tau ^{  \frac{1}{2}}\) and \(\propto \tau ^{\frac{1}{2}}\). If more terms are included in the powerlaw noise model, the available slope gradients increase, but the welllike shape remains. When applied to the T_{1} fluctuations (Fig. 3), this model is not able to describe the most striking feature: the hilllike peak with subsequent second decreasing slope. Within Allan analysis, the rise and fall of a single peak can only be represented by a Lorentzian noise process. Therefore, starting from
where A represents the Lorentzian noise amplitude and τ_{0} is the characteristic timescale, Lorentzian noise can be represented in Allan deviation by ref. ^{54}.
From here, we model the T_{1} fluctuations by two separate Lorentzians and white noise. When plotted, the noise from these sources is identical (i.e., the same h_{0}, A, and τ_{0}) for both the WelchFFT and Allan deviation. For the rest of the data sets, we tabulate the Lorentzian parameters and white noise level in Table 2.
Data availability
The data that supports the findings of this study is available from the corresponding authors upon reasonable request.
Code availability
The code that supports the findings of this study is available from the corresponding authors upon reasonable request.
Change history
02 August 2019
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
References
Preskill, J. Reliable quantum computers. Proc. R. Soc. London Ser A 454, 385–410 (1998).
Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical largescale quantum computation. Phys. Rev. A 86, 032324 (2012).
Barends, R. et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508, 500–503 (2014).
Rol, M. A. et al. Restless tuneup of highfidelity qubit gates. Phys. Rev. Appl. 7, 041001 (2017).
Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).
Martinis, J. M. Qubit metrology for building a faulttolerant quantum computer. NPJ Quantum Inf. 1, 15005 (2015).
Oliver, W. D. & Welander, P. B. Materials in superconducting quantum bits. MRS Bull. 38, 816–825 (2013).
Gu, X., Kockum, A. F., Miranowicz, A., Liu, Y.x & Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. 718, 1–102 (2017).
Wang, Z. et al. Cavity attenuators for superconducting qubits. Phys. Rev. Appl. 11, 014031 (2019).
Dunsworth, A. et al. A method for building low loss multilayer wiring for superconducting microwave devices. Appl. Phys. Lett. 112, 063502 (2018).
Yan, F. et al. The flux qubit revisited to enhance coherence and reproducibility. Nat. Commun. 7, 12964 (2016).
Dial, O. et al. Bulk and surface loss in superconducting transmon qubits. Supercond. Sci. Technol. 29, 044001 (2016).
Rosenberg, D. et al. 3D integrated superconducting qubits. NPJ Quantum Inf. 3, 42 (2017).
Müller, C., Lisenfeld, J., Shnirman, A. & Poletto, S. Interacting twolevel defects as sources of fluctuating highfrequency noise in superconducting circuits. Phys. Rev. B 92, 035442 (2015).
Gustavsson, S. et al. Suppressing relaxation in superconducting qubits by quasiparticle pumping. Science 354, 1573–1577 (2016).
Chang, J. B. et al. Improved superconducting qubit coherence using titanium nitride. Appl. Phys. Lett. 103, 012602 (2013).
Klimov, P. V. et al. Fluctuations of energyrelaxation times in superconducting qubits. Phys. Rev. Lett. 121, 090502 (2018).
IBM Q. IBM Q Experience website. https://www.research.ibm.com/ibmq/. Accessed July 2018.
Faoro, L. & Ioffe, L. B. Interacting tunneling model for twolevel systems in amorphous materials and its predictions for their dephasing and noise in superconducting microresonators. Phys. Rev. B 91, 014201 (2015).
Burnett, J. et al. Evidence for interacting twolevel systems from the 1/f noise of a superconducting resonator. Nat. Commun. 5, 4119 (2014).
Geerlings, K. et al. Demonstrating a driven reset protocol for a superconducting qubit. Phys. Rev. Lett. 110, 120501 (2013).
de Graaf, S. E. et al. Suppression of lowfrequency charge noise in superconducting resonators by surface spin desorption. Nat. Commun. 9, 1143 (2018).
Barends, R. et al. Coherent Josephson qubit suitable for scalable quantum integrated circuits. Phys. Rev. Lett. 111, 080502 (2013).
Gustafsson, M. V., Pourkabirian, A., Johansson, G., Clarke, J. & Delsing, P. Thermal properties of charge noise sources. Phys. Rev. B 88, 245410 (2013).
Bylander, J. et al. Noise spectroscopy through dynamical decoupling with a superconducting flux qubit. Nat. Phys. 7, 565–570 (2011).
McKay, D. C. et al. Universal gate for fixedfrequency qubits via a tunable bus. Phys. Rev. Appl. 6, 064007 (2016).
Chow, J. M. et al. Simple allmicrowave entangling gate for fixedfrequency superconducting qubits. Phys. Rev. Lett. 107, 080502 (2011).
Economou, S. E. & Barnes, E. Analytical approach to swift nonleaky entangling gates in superconducting qubits. Phys. Rev. B 91, 161405(R) (2015).
Paik, H. et al. Experimental demonstration of a resonatorinduced phase gate in a multiqubit circuitQED system. Phys. Rev. Lett. 117, 250502 (2016).
Clerk, A. A. & Utami, D. W. Using a qubit to measure photonnumber statistics of a driven thermal oscillator. Phys. Rev. A 75, 042302 (2007).
Barends, R. et al. Minimal resonator loss for circuit quantum electrodynamics. Appl. Phys. Lett. 97, 023508 (2010).
Riste, D. et al. Millisecond chargeparity fluctuations and induced decoherence in a superconducting transmon qubit. Nat. Commun. 4, 1913 (2013).
Cooper, K. et al. Observation of quantum oscillations between a Josephson phase qubit and a microscopic resonator using fast readout. Phys. Rev. Lett. 93, 180401 (2004).
Gustavsson, S. et al. Dynamical decoupling and dephasing in interacting twolevel systems. Phys. Rev. Lett. 109, 010502 (2012).
Martinis, J. et al. Decoherence in Josephson qubits from dielectric loss. Phys. Rev. Lett. 95, 210503 (2005).
Rubiola, E. Phase noise and frequency stability in oscillators. In The Cambridge RF and Microwave Engineering Series (Cambridge University Press, 2009). https://www.cambridge.org/core/books/phasenoiseandfrequencystabilityinoscillators/445C12C4ECBFCD7765116E61561EC0FE https://doi.org/10.1017/CBO9780511812798.
Kafanov, S., Brenning, H., Duty, T. & Delsing, P. Charge noise in singleelectron transistors and charge qubits may be caused by metallic grains. Phys. Rev. B 78, 125411 (2008).
Salvino, D. J., Rogge, S., Tigner, B. & Osheroff, D. D. Low temperature ac dielectric response of glasses to high dc electric fields. Phys. Rev. Lett. 73, 268–271 (1994).
Ludwig, S., Nalbach, P., Rosenberg, D. & Osheroff, D. Dynamics of the destruction and rebuilding of a dipole gap in glasses. Phys. Rev. Lett. 90, 105501 (2003).
Martinis, J., Nam, S., Aumentado, J., Lang, K. & Urbina, C. Decoherence of a superconducting qubit due to bias noise. Phys. Rev. B 67, 094510 (2003).
de Visser, P. J. et al. Number fluctuations of sparse quasiparticles in a superconductor. Phys. Rev. Lett. 106, 167004 (2011).
Earnest, C. T. et al. Substrate surface engineering for highquality silicon/aluminum superconducting resonators. Supercond. Sci. Technol. 31, 125013 (2018).
Rigetti, C. et al. Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms. Phys. Rev. B 86, 100506(R) (2012).
Barends, R. et al. Minimizing quasiparticle generation from stray infrared light in superconducting quantum circuits. Appl. Phys. Lett. 99, 113507 (2011).
Grünhaupt, L. et al. Loss mechanisms and quasiparticle dynamics in superconducting microwave resonators made of thinfilm granular aluminum. Phys. Rev. Lett. 121, 117001 (2018).
Nugroho, C. D., Orlyanchik, V. & Van Harlingen, D. J. Low frequency resistance and critical current fluctuations in albased josephson junctions. Appl. Phys. Lett. 102, 142602 (2013).
Koch, J. et al. Chargeinsensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007).
O’Malley, P. J. J. et al. Qubit metrology of ultralow phase noise using randomized benchmarking. Phys. Rev. Appl. 3, 044009 (2015).
Schlör, S. et al. Correlating decoherence in transmon qubits: low frequency noise by single fluctuators. Preprint at arXiv:1901.05352 (2019).
Hong, S. et al. Demonstration of a parametricallyactivated entangling gate protected from flux noise. Preprint at arXiv:1901.08035 (2019).
Gladchenko, S. et al. Superconducting nanocircuits for topologically protected qubits. Nat. Phys. 5, 48–53 (2009).
Burnett, J., Bengtsson, A., Niepce, D. & Bylander, J. Noise and loss of superconducting aluminium resonators at single photon energies. J. Phys. Conf. Ser. 969, 012131 (2018).
Pop, I. M. et al. Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles. Nature 508, 369–372 (2014).
Van Vliet, C. M. & Handel, P. H. A new transform theorem for stochastic processes with special application to counting statistics. Phys. A 113, 261–276 (1982).
Acknowledgements
We wish to express our gratitude to Philip Krantz and Tobias Lindström for insightful discussions. We acknowledge financial support from the Knut and Alice Wallenberg Foundation, the Swedish Research Council, and the EU Flagship on Quantum Technology H2020FETFLAG201803 project 820363 OpenSuperQ. In addition, J.J.B acknowledges financial support from the Industrial Strategy Challenge Fund Metrology Fellowship as part of the UK government’s Department for Business, Energy and Industrial Strategy.
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J.J.B. and A.B. are considered cofirst authors. J.J.B., A.B., and J.B. planned the experiment, A.B. designed the samples, A.B. fabricated the samples with input from J.J.B. The experiments were mainly performed by J.J.B. and A.B. with help from M.S., D.N., and M.K. Analysis was performed by J.J.B. and A.B. with input from P.D. and J.B. The manuscript was written by J.J.B., A.B., and J.B. with input from all authors. J.B. and P.D. provided support for the work.
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Burnett, J.J., Bengtsson, A., Scigliuzzo, M. et al. Decoherence benchmarking of superconducting qubits. npj Quantum Inf 5, 54 (2019). https://doi.org/10.1038/s4153401901685
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DOI: https://doi.org/10.1038/s4153401901685
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