Decoherence benchmarking of superconducting qubits

We benchmark the decoherence of superconducting qubits to examine the temporal stability of energy-relaxation and dephasing. By collecting statistics during measurements spanning multiple days, we find the mean parameters $\overline{T_{1}}$ = 49 $\mu$s and $\overline{T_{2}^{*}}$ = 95 $\mu$s, however, both of these quantities fluctuate explaining the need for frequent re-calibration in qubit setups. Our main finding is that fluctuations in qubit relaxation are local to the qubit and are caused by instabilities of near-resonant two-level-systems (TLS). Through statistical analysis, we determine 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 is limited by capacitance noise. Importantly, this produces a 0.8 ms limit on the pure dephasing which we also observe. Collectively, these findings raise the need for performing qubit metrology to examine the reproducibility of qubit parameters, where these fluctuations could affect qubit gate fidelity.

Quantum information processors based on superconducting qubits have demonstrated single-qubit gate fidelities in excess of 0.999 [1], two-qubit gate fidelities in excess of 0.99 [2] and both energy relaxation [3] (T 1 ) and dephasing [4] (T 2 ) times in excess of 100 µs.It is usual to perform qubit metrology [5] to benchmark the gate fidelity and quantify its error; however, benchmarking of decoherence, to quantify the mean lifetimes, together with the stability of the lifetimes, is far less common.Consequently, it is not clear whether reports on improvements in coherence times, cf. the review by Oliver and Welander [6], are reports of typical or of exceptional performance.Quantifying this difference is crucial for future work aimed at improving qubit coherence times.Further improvements are required due to gate fidelities approaching the T 1 limit [1], making them susceptible to fluctuations in T 1 .Therefore, not only are long lifetimes required, but those lifetimes must be stable for reliable operation of any quantum processor.
We benchmark the stability of decoherence properties of superconducting qubits: T 1 , T * 2 (free-induction 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 [7] or magnetic flux [3,8].Some studies [8][9][10][11][12][13][14] 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, where the number of counts relates to the statistical confidence of the findings, 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 [15]), we only discuss reports featuring both a large number of counts (N > 1000) and a total duration exceeding 5 hours.
The first study to satisfy these requirements was that of Müller et al. [11], which revealed that unstable nearresonant two-level-systems (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 [16,17], the frequency of this near-resonant 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 [11] was unable to determine properties of the TLS such as switching rates or dwell times of specific TLS frequency positions.Follow-up work by Klimov et al. [14] used a tuneable qubit to map the trajectories of individual TLS.These findings [14] supported the interacting-TLS 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 TLS-based Lorentzian noise spectrum and allow for extraction of switching rates.Importantly, this method does not require a tuneable qubit or advanced reset protocols [18] and is therefore general to any qubit or setup.Furthermore, the lack of tuning results in a more frequency-stable qubit and consequently less dephasing.This enables the further study of the qubit's frequency instabilities due to other noise sources, which reveals a 1/f frequency noise that is remarkably similar to interacting-TLS induced 1/f capacitance noise found in superconducting resonators [17,19].This frequency instability produces a limit on pure dephasing which we observe through sequential inter-leaved measurements of qubit relaxation, dephasing, and frequency.
The circuit is made of Al on Si (see the supplemental[20]) and consists of a single-junction Xmon-type (co-planar shunt capacitor) transmon [21] capacitively coupled to a microwave readout resonator.The shunt capacitor and the absence of a SQUID effectively decouples the qubit frequency from charge and magnetic flux, reducing the sensitivity to these typical 1/f noise sources [22,23].Therefore, these qubits have no frequency tunability.However, they remain suitable for multi-qubit architectures using all-microwave-based two-qubit-gates [24][25][26][27].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 qubit-to-qubit couplings.Additionally, both the spectral linewidth of the resonator and the resonator-qubit coupling are kept small, such that qubit decay through the resonator (Purcell effect) and dephasing induced by residual thermal population of the resonator are minimized [28].A detailed experimental setup together with all device parameters are found in the supplemental [20].
First we assess the stability of T 1 by consecutive measurements.The transmon is driven from its ground to first-excited 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 single-exponential decay to determine the relaxation time T 1 .Figure 1  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 the supplemental[20]).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 [11,14].
By repeating this study across several cool-downs (see supplemental [20]), we are confident that the fluctuations shown here are typical for the devices measured.Here, we point out that the capacitor of qubit B was trenched to reduce its dielectric loss [29] and increase T 1 .However, while the T 1 was successfully raised, qubit B has a lower ratio of Josephson to charging energy (see Table I), resulting in a larger sensitivity to charge noise and parity effects [30].Therefore, while it may have the higher T 1 , it is not as suitable for studies of dephasing and qubit frequency which feature later in the paper.Therefore, the rest of the paper focuses on qubit A.
We continue by measuring T 1 consecutively for approximately 128 hours, and plot the decays in a colour-map (Fig. 2a).Here, the colour-map makes some features of the data simpler to visualize.Firstly, the fluctuations  In both plots there are fits to the total noise (red line) which is formed of white noise (green lines) and two different Lorentzians (blue lines).The amplitudes and time constants of all noise processes are the same for both types of analysis.
typically latch between different T 1 values, where the switching is instantaneous, but the dwell time at a particular value is much longer.This behaviour resembles telegraphic noise with some characteristic switching rate (also seen in Fig. 1a).The longest dwell time observed is nearly 12.5 hours, while the shortest is approximately 2 hours.This suggests a distribution of switching rates ranging from 20 µHz to 140 µHz.Later, we quantify these rates and their reproducibility.
These slow fluctuations can be compared with mechanisms in qubit literature: the quasiparticle recombination rate in Al is 1 kHz [31]; the timescale of quasiparticle number fluctuations in Al leads to rates in the range 0.1 kHz to 10 kHz; and finally quasiparticle tunneling (parity switching events) in Al transmons [30] have rates in the range 0.1 kHz to 30 kHz.Therefore, fluctuations in the properties of the superconductor occur over rates which differ by over 6 orders of magnitude compared to those found here.Instead, we highlight that at low temperatures, bulk-TLS dynamics [32,33] and TLScharge noise [34,35] vary over long timescales equivalent to rates in the range 100 Hz to 10 mHz.
To highlight other features of the colour-map we have added dashed boxes with insets that show a line-cut of the data.The white box of Fig. 2a and inset Fig. 2b show a rapid, step-like change in T 1 occurring within a single iteration.The resulting decay can be fit to two different values of T 1 , one before the step 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 shortest-lived qubit.
Another type of behaviour is highlighted in the black box and inset Fig. 2c.Here the decay-profile is no longer purely exponential, but instead exhibits revivals.Similar revivals have been observed in both phase [36] and flux [35] qubits, and were attributed to coherently coupled TLS residing in one of the qubit junctions.From the oscillations we extract a qubit-to-TLS coupling of g TLS = 4.8 kHz (see supplemental for more details).Assuming a TLS dipole moment of 1 e Å [37], the coupling corresponds to a electric field line at 13 µm.This length is larger than the Josephson junction; therefore, this particular TLS is located in the qubit capacitor.Since the invention of transmons and improvement in capacitor dielectrics, individual TLS have only been found to incoherently couple to a transmon [21] 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 hours, others lasted only 2-3 traces (around 10 minutes).Since the qubit here is fixed in frequency, these appearances/disappearances of the coherent TLS arise due to the TLS shifting in frequency [11,14,16,17] relative to the static qubit.Observing the coherent oscillations in the decay and in particular that oscillation periods remained stable for hours (same duration as T 1 fluctuations) is clear evidence for TLS being the origin of the T 1 fluctuations, in agreement with both the Müller and Klimov results [11,14].
To gain further insight into these fluctuations we perform additional statistical analysis.In parallel, we examine both the overlapping Allan deviation (Fig. 3a) and spectral properties (Fig. 3b) of T 1 fluctuations.Beginning with the Allan deviation, the most striking feature is peak and subsequent decay around τ = 10 4 seconds.Importantly, no power-law 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.-timemeasurement 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 Allan deviation.Therefore, the noise parameters are the same for both plots: the supplemental 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 158.7 µHz and 83.3 µHz.Within the supplemental, we show the reproducibility of these features across both qubits and across several thermal cycles.Collectively, we find switching rates ranging from 75 µHz to 2 mHz -slower than those obtained by measurements of charge noise [34] but similar to bulk-TLS dynamics [32,33]; however, in agreement with rates de-termined from measurements tracking the time evolution of individual TLS [14].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.
Having demonstrated that T 1 fluctuations are due to interacting TLS, we turn our focus to qubit frequency and dephasing.In the original proposal for the transmon [38], it is stated that noise in the E c parameter, i.e. capacitance noise, had not yet been demonstrated.However since then, TLS induced capacitance noise has been seen to give frequency instabilities in the range from 1 kHz to 50 kHz in coplanar waveguide resonators [17,19] with a similar geometry and fabrication as the qubits here.Therefore, there is no fundamental reason why similar frequency fluctuations, and consequently qubit dephasing, should not be present in a similarly fabricated transmon qubit.
Additionally, due to the strong coupling to a spectrally unstable TLS, we could also expect frequency shifts up to 2g TLS = 9.6 kHz.To examine this, we measure the qubit frequency and dephasing by a de-tuned Ramsey sequence, which we interleave with the previously used relaxation sequence.While this gives a longer iteration time, and increases the noise window that the Ramsey sequence is sensitive to [39], it should allow for all qubit parameters to be known in each iteration.From T 1 and T * 2 we calculate the pure-dephasing time T φ from 1 These values are shown in Fig. 4b.In Fig. 4a we have extracted the frequency motion of the qubit relative to it's mean frequency (f 01 − f 01 ).In general, the observed frequency shifts are on the order of 1 kHz to 3 kHz, with infrequent shifts of up to 20 kHz.A histogram of the qubit frequency (Fig. 4d) reveals a main peak with a full-width-half-maximum of approximately 2 kHz.From the perspective of gate fidelity, a frequency shift of this order should have negligible effect, meaning that our qubits are well suited for quantum information processing since no re-calibration of the qubit frequency is needed.However, a fluctuating qubit frequency necessarily leads to qubit dephasing so it is important to quantify this dephasing and therefore aid in efforts to find, and mitigate, the noise source.
Typically, dephasing is thought to arise due to excess photons within the cavity [4,7], flux noise [23], charge noise [34,35] (including parity effects [30] due to quasiparticles tunneling through the junctions) or the presence of excess quasiparticles [40].For qubit A, the charge dispersion is calculated to 524 Hz.[20] Quasiparticle fluctuations have been extensively studied [31], where the magnitude of frequency shifts scales with the kinetic inductance.Therefore, while they can be of order 100 kHz in disordered superconductors [41], they are much smaller in elemental superconductors.In fact, recent experiments [31] showed that the quasiparticles in Al produced an un-measurably small frequency shift; instead, the quasiparticle's 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 (e.g.rates are equivalent to kHz [30,31] rather than the µHz observed here).
To provide more information on possible mechanisms for the frequency instability, we examine both the overlapping Allan deviation (Fig. 4e) and spectrum of frequency fluctuations (Fig. 4f).In red, the frequency noise is modelled to A/f + B, where the exponent of f is 1.Similar to the previous T 1 analysis, the noise model is scaled so that the red line has the same amplitude in both Fig. 4e and Fig. 4f.Here, A = 3.6×10 5 Hz 2 , which is remarkably close to the magnitude of 1/f noise found in superconducting resonators due to TLS-induced capacitance noise [17].In equivalent units to Fig. 4f, the Nb+Pt resonator noise from Ref. [17] is described by A = 1.2×10 5 Hz 2 .This similarity is suggestive that the frequency instability arises from TLS-induced capacitance noise.To verify this, the frequency noise could be measured as a function of temperature, where observation of frequency noise scaling as 1/T 1+µ [16,17] would clearly demonstrate TLS-induced capacitance noise.
Irrespective of the origin of the frequency instability, the noise spectrum in Fig. 4e can be integrated to estimate the pure dephasing of the qubit [38].From this calculation, the expected T φ is 0.8 ms.In Fig. 4f 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. 4b and Fig. 4c, it is clear that T * 2 is almost always longer than T 1 , implying a longer T φ .In Fig. 4h this is quantified as the histogram of the ratio of T * 2 /T 1 reveals that the qubit dephasing is almost always near 2T 1 .Therefore the qubit has a T * 2 that 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 CPMG [23] or Hahn-echo (spin-echo) [7] sequence.However, neither of those works provide any statistics on whether the qubits were always T 1 -limited.The histogram in Fig. 4d 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 during measurement spans longer than previous studies.This has clearly identified that qubit calibration is necessary every few hours.Fundamentally, this also demonstrates that future reports on qubit coherence times require not only statistics for reproducibility, but also that measurement durations must exceed 6 hours to adequately quantify the mean coherence time.
Combining long measurement periods with frequency metrology techniques revealed that the T 1 fluctuations followed a Lorentzian behaviour with a range of switching rates between 75 µHz and 2 mHz, where the switching rates describe the spectral instability of TLS about a stationary qubit.This further reveals both that a qubit remains an excellent probe of dielectrics and that the prevalence of TLS remain a problem for the stable operation of solid-state quantum processors.
Next we quantified the frequency stability of the qubit.Finding that the magnitude and behaviour of the qubit frequency noise was very similar to that of TLS-induced capacitance noise found in superconducting resonators, suggesting this as a probable origin.However, regardless of origin, when integrating the qubit frequency noise, we found T φ to be limited to approximately 1 ms, which we also found experimentally.Remarkably, we therefore observed with a T 1 -limited dephasing, even without dynamical decoupling.
Collectively, these measurements highlight that an improved understanding of interacting-TLS processes is directly relevant to quantum computing applications.Specifically, these fluctuations cause limits in tune-up protocols and ultimately determine the duty cycle of calibration periods.It is noteworthy, that recent experiments were able to reduce fluctuations of TLS [19].Therefore, it is promising that similar methods could be employed to increase the duty cycle of calibration periods in quantum processors based on superconducting qubits.The qubits are fabricated out of electron-beam evaporated aluminium on a high-resistivity intrinsic silicon substrate.Everything except the Josephson junction is defined using direct write laser lithography and etched using wet chemistry.The Josephson junction is defined in a bi-layer resist stack using electron-beam lithography, and later deposited using a two-angle evaporation technique that doesn't create any extra junctions or floating islands [42].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 [3].Finally, the wafer is diced into individual chips and cleaned thoroughly using both wet and dry chemistry.
A simplified schematic of the experimental setup is shown in Fig. 5a.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 absorber-lined radiation shield and a further superconducting layer which encloses the entire mixing chamber.Everything inside the cryoperm layer (screws, sample enclosures, and cables) is non-magnetic.The setup, including absorber recipe is similar to a typical qubit box-in-a-box setup [40].

Sample handling
The qubits featured within this study were examined across multiple measurement runs which were interrupted by thermal cycling of the fridge.During these interruptions, the samples were kept within their sample enclosures, at ambient conditions within the laboratory.Generally this meant that the samples remained attached to the dilution fridge while samples for other experiments were cycled.In these instances, the time between cooldowns was between one and five days.An exception to this was the change from setup 1 to setup 2, which took two weeks.During this period of time, the qubit samples were kept within a drawer in the laboratory at ambient conditions.Throughout the entire experiment (including the interruptions), the samples were kept within the same light-tight sample enclosure.Therefore, since the lid was not detached after the initial wire-bonding, we expect the air circulation between the sample enclosure and the laboratory to have been small.

A. Measurements of qubits A and B across multiple cooldowns
In the main text we show measurements spanning a total time duration that is representative of the operation of quantum processors.These measurements are repeated across multiple cooldowns.The results are shown below for qubit A (Figs. 6-9) and qubit B (Figs. 10-12).These summaries feature the T 1 values vs. measurement iteration; a histogram of T 1 values with a fit determining the mean and standard deviation; and a Welch FFT and overlapping Allan deviation of the T 1 series.

B. Qubit aging
Here, we collate the histograms of T 1 from section II A to show the effect of qubit aging across several cooldowns.
The qubit A sample was kept within its enclosure and mounted to the fridge throughout the measurement runs that span cooldown 2 through to cooldown 6.Additionally, the sample was present for cooldown 4, however, no T 1 statistics were gathered in that thermal cycle.Since the sample was thermally cycled, the counting includes cooldown 4 and therefore resumes at cooldown 5.
The qubit B sample was kept within its enclosure during cooldown 1 through to cooldown 6.However, during cooldown 2 through cooldown 4, it was kept at ambient conditions and not thermally cycled.From cooldown 5 the sample was kept mounted to the fridge through to the end of the measurement run at cooldown 6.The qubit decoherence data is processed in the following way.First the digitizer signal is rotated to one quadrature.Next the signal is normalized to the maximum visibility of the qubit |0 and |1 states.Then for qubit relaxation data, a fit to the following function [8,43] is first attempted: P e (t) = exp(n qp (exp(−t/T 1qp ) − 1)) exp(−t/T 1 ) (1) where P e (t) is the probability of the qubit being excited, n qp is an average number of quasiparticles near the qubit junction, T 1qp is the relaxation time associated with a single quasiparticle and T 1 is the remaining relaxation time associated with any other process.The intention of fitting to this form was to examine whether n qp is zero, meaning that the decay is single exponential, or whether n qp is finite leading to a double-exponential decay shape.Instead we find that the confidence intervals of the parameters n qp and T qp are non-physically large.This leads us to conclude that the model in Eq. ( 1) has too many parameters to describe the features of our data.So instead we fit the data with a a single exponential decay shape.While this could be an indication of our measurement environment leading to lower quasiparticle densities, a more likely explanation is that n qp has some scaling with the number of Josephson junctions (or more precisely islands separated by Josephson junctions).Therefore, it is less frequent that a quasiparticle is close to our single-junction transmon, compared to the 3-junction flux qubit [8] or 95-junction fluxonium [43].
For the Ramsey measurements, the initial processing is as described above.However, the Ramsey frequency (f R ) 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 data-sets examined, the FFT reveals only one oscillation frequency.

D. 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 Welch-method FFT (S T1 (f )) and an overlapping Allan deviation (σ T1 (τ )).From here, beginning with the Allan deviation, we consider the standard power-law model [44] 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 power-law noise processes create a well-like shape in the Allan analysis, where, with the terms listed above, the walls have slopes of ∝ τ − 1 2 and ∝ τ 1 2 .If more terms are included in the powerlaw noise model, the available slope gradients increase, but the well-like shape remains.When applied to the T 1 fluctuations (Fig. 3 in the main paper), this model is not able to describe the most striking feature: the hill-like 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 [45]: 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 Welch-FFT and Allan deviation.From the main text, the T 1 fluctuation is then parameterised by h 0 = 3×10 −9 .For Lorentzian 1, A = 5.4×10 −6 and τ 0 = 6.3×10 3 seconds.For Lorentzian 2, A = 3.2×10 −6 and τ 0 = 1.5×10 4 seconds.Therefore, the dominant switching rates (1/τ 0 ) are 158.7 µHz and 83.3 µHz.For the rest of the data-sets, we tabulate the Lorentzian parameters and white noise level in Table II.

E. TLS discussion
Within the main text, there are data sets which show revival features in time within measurements of the qubit relaxation.These features arise due to the coherent coupling between the qubit and a single TLS, described by the Hamiltonian [46], Due to this coupling, the qubit excited state can bridize form two, almost degenerate, states.The coupling strength g TLS can be extracted from measuring the energy relaxation decay of the qubit fitting it to (see Fig. 2c), + a e −Γ ↓,2 t + osc cos(2πf osc t)e −Γosct (8) where σ z is the expectation value of the Pauli matrix for the qubit, σ z ∞ is the zero-temperature equilibrium value, a ↓,k and Γ ↓,k are the amplitude and decay rate from the two excited states k (k = 1, 2) to the ground state, and a osc , f osc and Γ osc describes the amplitude, frequency, and decay rate of an oscillation in σ z .These parameters can be rewritten in terms of coupling g TLS and detuning δf = f 01 − f TLS f osc = g 2 TLS + δf 2 (9) From this model we find the mean coupling rate to be 4.8 kHz.Additionally, we find the TLS to possess a relaxation time of approximately 100 µs.Such a lifetime is considerable larger than those found within the tunnel barrier of phase qubits [47].However, the lifetime is strongly dependent on the coupling strength to the qubit.In absence of coupling [16], the phonon-limited relaxation time of a TLS is approximately 1 ms.
The main text also finds TLS switching rates as low as 75 µHz.These low switching rates are important in the general context of understanding TLS dynamics.Generally, measurements of charge noise [34] are used to determine the switching rates of TLS.Those measurements found a minimum switching rate of γ min = 100 Hz and a maximum switching rate of γ max = 25 kHz.Combining these leads to a TLS switching ratio (P γ = 1/ln(γ max /γ min )) of 0.18, a value in agreement with some experiments [48,49], although other studies have found lower values [19].A lower value of P γ can be obtained if γ min is smaller.From T 1 data we find switching rates ranging from 75 µHz to 2 mHz.Therefore, even the fastest rate is slower than those found in charge noise studies.This demonstrates that superconducting qubits are excellent probes of the TLS and highlights the need for further study of TLS dynamics.

F. Local vs non-local origins
Within the main text, a simultaneous measurement of both qubits is performed to examine whether the observed fluctuations in T 1 are local to each qubit.To as- sess this, we calculate the magnitude-squared coherence of the two data sets (shown in Fig. 14).This examines how much the T 1 of qubit A corresponds to the T 1 of qubit B. The value of the magnitude-squared coherence is normalized to between 0 and 1, where 1 relates to completely correlated (at that frequency).In Fig. 14 there are also dashed lines representing the threshold lev-els for significant correlation.These thresholds are calculated by statistical bootstrapping (repeatedly examining the magnitude-squared coherence of randomly resampled sets of one of the data set vs. the other original data set).The data in Fig. 14 is clearly far below these thresholds, as would be expected for uncorrelated noise that is local to each qubit.

FIG. 1 .
FIG. 1.(a) Multiple T1 measurements performed simultaneously on qubits A (black) and B (green).The data consists of 2000 consecutive T1 measurements that lasted a total duration of 2.36 × 10 5 s (approximately 65 hours).(b) Histograms of the T1 values in (a).The histograms have been fit (solid line) to Gaussian distributions with the parameters shown.

FIG. 2 .
FIG. 2. (a) Consecutive T1 measurements, spanning 4.6 × 10 5 s (approximately 128 hours), of qubit A. (b) A data set showing a change in T1 within a single iteration.These jumps are found to occur in approximately 3% of all measurements.(c) A data set showing a decaying sinusoidal (rather than a purely exponential) decay profile.The appearance of revivals are due to resonant exchange with a TLS.These profiles are found to occur in approximately 5% of all iterations.

FIG. 3 .
FIG. 3. Statistical analysis of 2001 sequential T1 measurements of qubit A spanning a total measurement duration of 2.36 × 10 5 s.(a) Overlapping Allan deviation of T1 fluctuations.(b) Welch-method spectral density of T1 fluctuations.In both plots there are fits to the total noise (red line) which is formed of white noise (green lines) and two different Lorentzians (blue lines).The amplitudes and time constants of all noise processes are the same for both types of analysis.

FIG. 4 .
FIG. 4.An interleaved series of 1000 T1 relaxation and T * 2 Ramsey measurements of qubit A. (a) Qubit frequency (f01) shift relative to its mean (f01) determined from the Ramsey experiments.(b) Extracted T1 (black), T * 2 (blue), and T φ (red).(c) Histogram of T * 2 from the data in (b).(d) Histogram of the data in (a).The frequency fluctuations from (a) are analyzed by overlapping Allan deviation (e) and by Welch-method spectrum (f ).The solid and dashed lines represent the modeled noise, where the noise amplitudes are the same for both types of analysis.(g) Histogram of T φ from the data in (b).The solid line indicates the T φ limit calculated from integrating the frequency noise from (e).(h) Histogram of T * 2 /T1 from the data in (b).We find 1.4 < T * 2 /T1 < 2.2 in 81.7 % of the counts.
FIG. 5. (a) Simplified schematic of the experimental setup.The main features are the various shielding layers.(b) Optical image of the qubit sample.It shows a common microwave transmission line, a λ/4 resonator and a transmon qubit with a coplanar capacitor (Xmon-geometry).

FIG. 13 .
FIG.13.A series of histograms from N measurements spanning a measurement duration (D) for qubit A (in black) and qubit B (in green) across several separate cooldowns.Setup 2 represents the full schematic demonstrated in Fig.5a, while setup 1 does not include the absorber layers from Fig.5a

FIG. 14 .
FIG. 14.(a) Multiple T1 measurements performed simultaneously on qubits A (black) and B (green).The data consists of 2000 consecutive T1 measurements that lasted a total duration of 2.36 × 10 5 s (approximately 65 hours).(b) Magnitudesquared coherence of the data in (a); the dashed lines represent the significance levels obtained from statistical bootstrapping.

TABLE I .
Summary of parameters for both devices.fR is the frequency of the readout resonator and f01 that of the qubit's 01 transition.f12 − f01 is the frequency difference between the qubit's 12 transition and 01 transition.EJ is the qubit's Josephson energy, Ec its charging energy, its charge dispersion, and χ its dispersive coupling rate.

TABLE II .
Summary of the noise parameters for modeling T1 fluctuations.The data is labeled as Q (qubit A or B) C (cooldown number).The Lor 1/2 corresponds to the Lorentzian being parameterised.