Abstract
Superconducting qubits are a leading candidate for quantum computing but display temporal fluctuations in their energy relaxation times T_{1}. This introduces instabilities in multiqubit device performance. Furthermore, autocorrelation in these time fluctuations introduces challenges for obtaining representative measures of T_{1} for process optimization and device screening. These T_{1} fluctuations are often attributed to time varying coupling of the qubit to defects, putative two level systems (TLSs). In this work, we develop a technique to probe the spectral and temporal dynamics of T_{1} in single junction transmons by repeated T_{1} measurements in the frequency vicinity of the bare qubit transition, via the ACStark effect. Across 10 qubits, we observe strong correlations between the mean T_{1} averaged over approximately nine months and a snapshot of an equally weighted T_{1} average over the Stark shifted frequency range. These observations are suggestive of an ergodiclike spectral diffusion of TLSs dominating T_{1}, and offer a promising path to more rapid T_{1} characterization for device screening and process optimization.
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Introduction
Superconducting qubits are a leading platform for quantum computing^{1,2}. This has been driven, in part, by improvements in coherence times over five orders of magnitude since the realization of coherent dynamics in a Cooperpair box^{3}. However, further improving coherence times remains crucial for enhancing the scope of noisy superconducting quantum processors as well as the longterm challenge of building a fault tolerant quantum computer. Recent advances^{4,5,6,7} in twoqubit gate control have placed their fidelities at the cusp of their coherence limit, implying that improvements in coherence could directly drive gate fidelities past the fault tolerant threshold. In this context, coherence stability and its impact on multiqubit device performance is also an important theme, since superconducting qubits have been shown to display large and correlated temporal fluctuations (i.e., 1/f^{α}) in their energy relaxation times T_{1}^{8,9,10,11,12,13,14}. This places additional challenges for benchmarking the coherence properties of these devices^{13}, and also for error mitigation strategies such as zero noise extrapolation^{12}.
The fluctuations of qubit T_{1} are often attributed to resonant couplings with twolevel systems (TLSs) that have been historically studied in the context of amorphous solids^{15,16} and their low temperature properties. More recently, TLSs have attracted renewed interest due to their effect on the coherence properties of superconducting quantum circuits^{11,13,14,17,18,19,20}, and are attributed to defects in amorphous materials at surfaces, interfaces, and the Josephson junction tunnel barrier. Frequency resolved measurements of T_{1} in flux and stress tunable devices^{11,19,20,21} have also displayed fluctuations, suggesting an environment of TLSs with varying coupling strengths around the qubit frequency. The variability of T_{1} over time is explained^{11,16}, at least in part, by temporal fluctuations in this frequency environment, associated with the spectral diffusion of the TLSs^{15,22}.
Furthermore, twoqubit gates that involve frequency excursions^{5,11,23} can also interact with TLSs near the qubit frequency leading to additional incoherent error. The fluctuations in TLS peak positions, therefore, can also introduce fluctuations in twoqubit fidelity. Spectroscopy of defect TLS is, therefore, central to understanding the short and long time T_{1} and gate fidelity of qubits.
Single Josephson junction transmons with fixed frequency couplings represent a successful device architecture achieving networks of over 60 qubits^{1} with all microwave control and state of the art device coherence. The single junction configuration offers advantages such as reduced sensitivity to flux noise, while preserving the transmon charge insensitivity and also reducing system complexity with fewer control inputs. However, there is little TLS spectroscopy of single junction transmons because of the limited tunability, despite the central importance of understanding the TLS environment both for device and process characterization.
In this work, we introduce an allmicrowave technique for the fast spectroscopy of TLSs in single junction transmon qubits that requires no additional hardware resources. In contrast to fluxbased approaches to TLS spectroscopy, we employ offresonant microwave tones to drive ACStark shifts of the fundamental qubit transition and spectrally resolve qubit relaxation times. Dips in relaxation times serve as a probe of the frequency location of a strongly coupled TLS. We use repeated frequency sweeps to probe the time dynamics of the relaxation probabilities including tracking the spectral diffusion of strongly coupled TLS. Across 10 qubits, we observe strong correlations between the long time mean, averaged over several months \({\langle {T}_{1}\rangle }_{T}\), and the short time mean, averaged around the local qubit frequency \({\langle {T}_{1}\rangle }_{\omega ,t}\).
This strong correlation suggests a quasiergodic behavior of the TLS spectral diffusion in the nearby frequency neighborhood of the qubit. In contrast, there is lower correlation between \({\langle {T}_{1}\rangle }_{T}\) and T_{1} measured over a single day. The \({\langle {T}_{1}\rangle }_{\omega ,t}\) can provide, therefore, a more rapid estimate of longtime behavior.
Results
Device and spectroscopy technique
The experiments reported in this letter were performed on ibmq_almaden, a 20 qubit processor based off single junction transmons and fixed couplings. The device topology is shown in Fig. 1a, and qubit frequencies are around ~5 GHz. Figure 1b depicts the characteristic spread of the qubit T_{1}s and their mean, from ~250 measurements over 9 months. The base plate (to which the device was mounted) temperature of the dilution refrigerator was typically ~13 mK excepting several temperature excursions to ~1 K, which were not observed to have any significant effects on the long time T_{1} time series or distributions of T_{1} values discussed in this work, discussed later. Several qubits on the device display mean T_{1}s exceeding 100 μs. However, the large spread in individual qubit T_{1}s highlights the challenge for rapid benchmarking of device coherence, since any single T_{1} measurement can disagree substantially from its longtime mean.
We study the spectral dynamics of these T_{1} times by employing offresonant microwave tones^{24} to induce an effective frequency shift Δω_{q} in single junction transmons by the AC Stark effect. This has been employed previously for coherent state transfer between coupled qubits that are Stark shifted into resonance^{25}. In this work, shifting the qubit frequency into resonance with a defect TLS induces a faster relaxation time, which in turn is used to detect the frequency location of the TLS^{26}, as depicted in Fig. 2a. The Stark shift can be described analytically by a Duffing oscillator model^{27,28}
where δ_{q} is the qubit anharmonicity, Ω_{s} is the drive amplitude and Δ_{qs} = ω_{q} − ω_{s} is the detuning between the qubit frequency and the Stark tone.
As seen from the expression above, the magnitude and sign of the Stark shift can be manipulated by the detuning and the drive amplitude of the Stark tone, Fig. 2c. Very large frequency shifts can be obtained by driving close to the transmon transitions, but this typically leads to undesired excitations/leakage out the twostate manifold. In this work, we obtain Stark shifts of 10’s of MHz, with modest drive amplitudes and a fixed detuning Δ_{qs} of ±50 MHz. The frequency shifts are experimentally measured using a modified Ramsey sequence^{29}, schematically shown in Fig. 2b, and display good agreement with the quadratic dependence of the perturbative model in the lowdrive limit. A representative case is shown in Fig. 2d.
We focus on the spectrally resolved T_{1} measurements in Fig. 3 that we use as a probe of defect TLS transition frequencies. However, instead of measuring the entire T_{1} decay, we use the excited state probability, P_{1}, after a fixed delay time as a measure of T_{1}. This speeds up the spectral scans significantly. Our experiments are performed at a repetition rate of 1 kHz, but our scheme can be further accelerated with reset techniques^{30}, which can be crucial for probing faster TLS dynamics. For an effective frequency sweep, we run an amplitude sweep with offresonant pulses at fixed detuning (±50 MHz) and duration (delay time of 50 μs), after exciting the qubit with an initial π pulse. The pulsed Stark sequence enables faster spectroscopy by circumventing the need to recalibrate the π, π/2 pulses at every frequency. The offresonant pulses have Gaussiansquare envelopes with a 2σ risefall profile, where σ = 10 ns. This pulse sequence is shown in Fig. 2b. The amplitude points in the sweep are then related to Stark shifts by Ramsey sequences. Figure 3 shows representative data of such a sweep on qubit 19 (Q_{19}) with distinctive dips in P_{1} that we attribute to strongly coupled TLS at their transition frequencies. T_{1} measurements at Stark tone amplitudes corresponding to high/low P_{1} points, as seen in the bottom panel of Fig. 3, explicitly show the substantial variation in T_{1} as a function of frequency and the consistent tracking of T_{1} with P_{1}.
Variations in P_{1} can potentially be caused by sources other than TLS. In our experiments, P_{1} is spectrally resolved to ~± 25 MHz around the individual qubit frequencies. The narrow frequency range combined with measuring nonneighbor sets of qubits simultaneously avoids strong P_{1} suppression from resonances with neighboring qubits, the coupling bus or common lowQ parasitic microwave modes. Control experiments show that time insensitive features in the P_{1} fingerprint are robust to choice of the Stark tone detuning, ruling out a power dependence for the power range used in this work. Finally, while a recent report^{31} modeled their broadband T_{1} scatter as arising from quasiparticle fluctuations, this is not sufficient to explain the sharp frequencydependent P_{1} features depicted, for instance in Fig. 3. Furthermore, recent experiments on our qubits suggest a quasiparticle limit to T_{1} that exceeds several milliseconds^{32}.
TLS dynamics and correlations of P _{1}(ω, t) and 〈T _{1}〉_{T}
We repeat the line traces of Fig. 3 for both positive and negative 50 MHz detuning, approximately once every 3–4 h, extended over hundreds of hours for all the qubits. A representative example of the cumulative scans is shown in Fig. 4 for Q_{15}. Spectroscopy of the other qubits is shown in the supplemental information S1. The TLS dynamics around the qubit frequency are qualitatively similar to previous TLS spectroscopy using flux or stress tunable devices^{16}.
In the case of Q_{15}, Fig. 4, there are prominent dips in relaxation probability around positive 1 MHz, negative 5–10 MHz, and negative 15–20 MHz. The spectral diffusion of the positions of the T_{1} dips can vary between order of 1 to 10 MHz over the 272 h of measurement providing a qualitative measure of linewidths. A more quantitative discussion of linewidths can be found in supplemental information S2. The background is covered by an ensemble of smaller dips of relaxation, Fig. 3, that also dynamically evolve, with features that are larger than the sampling noise in the measurement.
As discussed previously, T_{1} fluctuations introduce uncertainty in the coherence benchmarking, stability of multiqubit circuit performance and process optimization of superconducting qubit devices. In this context of better estimator, we examine if the longtime averages (T ~ 9 months) \({\langle {T}_{1}\rangle }_{T}\) and \({\langle {P}_{1}\rangle }_{T}\) are correlated with the frequency neighborhood of the qubit \({\langle {T}_{1}\rangle }_{\omega ,t}\) and \({\langle {P}_{1}\rangle }_{\omega ,t}\), respectively. The averaged relaxation probabilities and T_{1}s are defined as
where definitions of variables can be found in Table 1.
We compare \({\langle {P}_{1}\rangle }_{\omega ,t}\) to \({\langle {P}_{1}\rangle }_{T}\) from the daily T_{1} measurements over \({T}_{\max } \sim\) 9 months evaluated at τ = 53 μs, shown in Fig. 1. The \({\langle {P}_{1}\rangle }_{\omega ,t}\) are calculated for a T_{1} delay time of τ = 50 μs for 10 qubits in the device for the first time slice and a cutoff frequency Δω/2π = 5 MHz. A qualitatively close agreement for all 10 qubits is observed, see Fig. 5a.
A \({\langle {T}_{1}\rangle }_{\omega ,t}\) can also be estimated for each \({\langle {P}_{1}\rangle }_{\omega ,t}\) at τ = 50 μs by assuming an exponential decay. The approximate equivalence of \({\langle {T}_{1}\rangle }_{\omega ,t}\) and \({\langle {T}_{1}\rangle }_{T}\) is seen in the scatter plot of Fig. 5a inset. A near 1:1 relationship is observed when this approach is applied more broadly across many IBM devices, see supplemental information S3. Furthermore, the poorer correlation between \({\langle {T}_{1}\rangle }_{T}\) and a single instance of T_{1} measurements, is also shown by larger scatter, as seen in Fig. 5a inset.
To quantify with a single value the correlation between \({\langle {T}_{1}\rangle }_{T}\) or \({\langle {P}_{1}\rangle }_{T}\) and their estimators for many qubits, we use a Pearson R measure across the ten oddlabeled qubits,
where d is the number of qubits in the device or analysis, 10 in this case, and X is the observable P_{1} or T_{1}. The Pearson correlation is a normalized covariance between two variables reflecting a linear correlation from 1 to −1, where R = 1 (−1) represents a 100% positive (negative) correlation and R = 0 indicates no correlation. Strong R correlation can therefore signal the existence of a potential linear mapping between the estimator and \({\langle {T}_{1}\rangle }_{T}\), in particular, possibly one that is 1:1 or a scaling factor that will reliably estimate \({\langle {T}_{1}\rangle }_{T}\).
For a single frequency sweep that takes ~20 min, we obtain 0.76 < R(t_{i}) < 0.84 correlation between \({\langle {T}_{1}\rangle }_{T}\) and \({\langle {T}_{1}\rangle }_{\omega ,t}\) for 0.5 MHz < Δω < 5 MHz. Using the P_{1} values without assuming an exponential dependence leads to stronger correlations of 0.87 < R(t_{i}) < 0.91. Both of these are substantially stronger than the correlation found between the representative instance of T_{1} and \({\langle {T}_{1}\rangle }_{T}\), which was R = 0.29. We note this instance of R can have a large spread, as seen by simulations of Gaussiandistributed fluctuations in supplemental information S4.
A better estimate of the \({\langle {T}_{1}\rangle }_{T}\) for each qubit, Q_{k}, in the device can be obtained from a moving average of multiple, N, measurements. We show the evolution of \({\langle R\rangle }_{{T}_{0\to N}}\) using a moving average of the T_{1}(T_{i}) measurements, \({\langle {T}_{1}\rangle }_{{T}_{0\to N}}\), for each qubit, Fig. 5b. The \({\langle R\rangle }_{{T}_{0\to N}}\) exceeds R ~ 0.8 (i.e., strong correlation) after ~10 measurements, corresponding to a time exceeding 100 h. Approximately 10 independent measurements is sufficient for fluctuations with magnitude \(\sim 0.2{\langle {T}_{1}\rangle }_{T}\) to obtain a strong correlation, R ~ 0.8, between an estimator (e.g., \({\langle {T}_{1}\rangle }_{{T}_{0}\to N}\)) and \({\langle {T}_{1}\rangle }_{T}\). The details of R dependence on fluctuation magnitude and number of measurements in the moving average are discussed more completely in supplemental information S4.
Autocorrelation between T_{1}(T_{i}) and T_{1}(T_{i−1}) measurements is an underlying challenge to fast estimation of \({\langle {T}_{1}\rangle }_{T}\). Evidence of autocorrelation can be seen for example in longterm drifts in the average and shortterm correlations between T_{1}, inset of Fig. 5b. On shorter time scales, our experimental data shows evidence of stronger autocorrelation frustrating faster accurate estimation of \({\langle {T}_{1}\rangle }_{T}\) and that the fastest R ~ 0.8 can be obtained on order of 1–2 days, see supplemental information S5 and S6. We conclude that \({\langle {T}_{1}\rangle }_{\omega ,t}\) shows promise as a method for faster estimation of \({\langle {T}_{1}\rangle }_{T}\) than repeated T_{1}(ω = ω_{q}) measurements at only the qubit frequency. Extending the \({\langle {T}_{1}\rangle }_{\omega ,t}\) estimator to a set of many qubits, {Q_{k}}, in a device result in larger R, in the same time, compared to relying only on T_{1}(ω_{q}) measurements for each qubit. The R value simply being a quantitative single value expression of the high correlation between each \({\langle {T}_{1}\rangle }_{\omega ,t}\) and \({\langle {T}_{1}\rangle }_{T}\) across the entire set of qubits.
It is important to note that our calculations of \({\langle {T}_{1}\rangle }_{\omega ,t}\) employ an equal weighting of P_{1} associated with every frequency bin and the same choice of Δω for every qubit. However, it is not a priori clear that equal weighting is a representative choice over the Δω range. For example, how evenly does the spectral diffusion of each TLS contribute to the T_{1} of the qubit? The strong correlation of \({\langle {T}_{1}\rangle }_{\omega ,t}\) with \({\langle {T}_{1}\rangle }_{T}\) with equal weighting suggests that an ergodiclike sampling of the TLSs near the qubit frequency is a reasonable first approximation. The ergodic behavior of the T_{1} estimators is examined more completely in supplemental information S7 and supplemental information S8. Central to the question of assigning a T_{1} estimate to any qubit, we observe that \({\langle {T}_{1}\rangle }_{T}\) behaves ergodically for all the qubits despite shortterm 1/f^{α} correlated behavior (i.e., a constant mean \({\langle {T}_{1}\rangle }_{T}\) can be identified). Assignment of any T_{1} estimate could alternatively be made impossible in the presence of drift, which is not observed in these qubits, see supplemental information S9 and supplemental information S7 for further details about weak stationarity and ergodicity. Furthermore, the strong correlation of \({\langle {T}_{1}\rangle }_{T}\) to \({\langle {T}_{1}\rangle }_{\omega ,t}\) using only the P_{1}(ω, τ, t) spectrum around the qubit is consistent with a leading hypothesis that the \({\langle {T}_{1}\rangle }_{T}\) is dominated by TLS behavior rather than other stochastic or static contributions.
Correlation dependence on frequency and measurement time
A natural question about the estimator \({\langle {T}_{1}\rangle }_{\omega ,t}\) is, what are the optimal parameter choices for frequency range Δω, n autocorrelated samples and the spacing in time, Δt = t_{i} − t_{i−1}, to obtain sufficiently weakly autocorrelated measurements and a fast, accurate measure of \({\langle {T}_{1}\rangle }_{T}\). Since the optimum choices are presently not known a priori, we evaluate and plot \({\langle R\rangle }_{{t}_{0\to n}}\) versus Δω and t_{i} in Fig. 5c to guide future application of this approach. Equal frequency bin weighting of P_{1} is used. While this order of magnitude choice of Δω produces a reasonably good first approximation for correlation across the entire range, the plot displays several unexplained features (e.g., nonmonotonic dependence on Δω) indicating the unsurprising insufficiency of these two globally applied parameters (i.e., Δω and t) alone to weight the frequency contribution of all the qubits and approach R ~ 1. Additional sensitivity analysis in supplemental information S8 also examines correlation between frequencies and highlights that individual qubits have different sensitivity to the range sampled, Δω. We see that a wide span of Δω produces high \({\langle R\rangle }_{{t}_{0\to n}}\), comparable or better than R(T_{i}) from a single T_{1}(ω_{q}) measurement. We further show that not only is there a strong R correlation (e.g., linear dependence) but that \({\langle {T}_{1}\rangle }_{\omega ,t}\) approaches 1:1 quantitative agreement with \({\langle {T}_{1}\rangle }_{T}\). The degree to which a T_{1} estimator, from sampling the nearby frequency space, is quasiergodic and would converge to 1:1 agreement is addressed in much more detail in supplemental information S8 and supplemental information S3.
Discussion
Implications for process characterization
The strong correlation between \({\langle {T}_{1}\rangle }_{\omega ,t}\) and \({\langle {T}_{1}\rangle }_{T}\) suggests that longtime T_{1} averages might be estimated relatively rapidly using spectroscopy. This is in contrast to overcoming correlation times in T_{1} at a single ω_{q} to obtain a representative \({\langle {T}_{1}\rangle }_{T}\) for the qubit.
Identification of better choices of Δω and n in this study were made with preknowledge of what \({\langle {T}_{1}\rangle }_{T}\) was. These parameters will have to be chosen without this precharacterization for future implementation of this method. Encouragingly, the R dependence on both these parameters appears to be relatively weak suggesting that a heuristic choice for a single Δω and n might be sufficient to obtain useful estimates (i.e., R > 0.8) of \({\langle {T}_{1}\rangle }_{T}\) for new processes when using this simple equal weighting approach until improved choices can be formulated (i.e., different frequency spans for each qubit and or weighted averaging over frequency).
More specifically we observe that \({{{\mathcal{O}}}}\)(10) independent measurements is sufficient to obtain an R ~ 0.8 or higher, see supplemental information S4. We conjecture that one can obtain 10 approximately independent samples, S, in a single scan by sampling at frequency spacings, χ, that are greater than the autocorrelation frequency width (i.e., a frequency spacing where correlation drops below ~ 0.2). In this work, we found the correlation to become weak for \({{{\mathcal{O}}}}\)(1 MHz), see supplemental information S8. Then by this heuristic, a single spectroscopy scan would require a Δω = \(\frac{(S1)}{2}\chi\), where S = 10 for the target of R ~ 0.8. We assume one of the measurements is done at the qubit frequency, T_{1}(ω_{q}), so for a χ ~ 1 MHz, a scan from ± 4.5 MHz would be suggested by such a heuristic. Extra n measurements can be obtained by waiting longer than the autocorrelation time. The autocorrelation width, furthermore, can be evaluated in the same scan as that used for the \({\langle {T}_{1}\rangle }_{T}\) estimate as long as a sufficiently wide range is sampled. Alternatively, a second scan can be taken if the initial Δω guess was too small.
Empirically we see diminishing gains in using ever larger Δω. Further research is needed to guide better limits on Δω beyond the operational observation that \(S \sim {{{\mathcal{O}}}}(10)\) produces a quasiergodic result for qubits with \({\langle {T}_{1}\rangle }_{T}\) in the range of 10–200 μs, see supplemental information S8 for more details on quasiergodicity. Since we do find ~ 1:1 agreement using a relatively small Δω ~ 10 MHz for the ~9 month time series and we observe that the distribution of T_{1}(ω_{q}, T_{i}) produces a constant standard deviation, see supplemental information S9, rather than growing (e.g., proportional to a random walk \(\propto \sqrt{t}\)), we speculate that optimal Δω is bounded rather than growing indefinitely from spectral diffusion processes. Notably, Klauder et al. calculate that dipolecoupled ensembles that are proposed for TLS spectral diffusion^{22}, will produce a truncated linewidth^{33}.
Remarks on technique, correlations, and ergodicity
In this work, we probe the temporal and spectral dynamics of superconducting qubit relaxation times. We study these dynamics in high coherence, singlejunction transmons by developing a technique for energy relaxation spectroscopy of defect TLSs via the AC Stark effect. Our technique requires no additional hardware resources and can be easily sped up further by integration with reset schemes. Autocorrelation of T_{1} frustrates rapid characterization of the longtime average \({\langle {T}_{1}\rangle }_{T}\) and therefore accurate characterization of devices. Our analysis of the dynamics identifies a strong correlation between \({\langle {T}_{1}\rangle }_{T}\) and its short time average over the local frequency span, \({\langle {T}_{1}\rangle }_{\omega ,t}\). The strong correlation of \({\langle {T}_{1}\rangle }_{T}\) with \({\langle {T}_{1}\rangle }_{\omega ,t}\) is also consistent with a TLS dominated T_{1} that quasiergodically samples the qubit local frequency neighborhood in contrast to static or uncorrelated stochastic processes. This work opens up several new promising directions for rapid process characterization and evaluation of device stability.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
We acknowledge technical support on the ibmq_almaden device from the IBM Quantum deployment team. Additional insightful discussions, suggestions and assistance came from Nick Bronn, Andrew Cross, Oliver Dial, Doug McClure, Easwar Magesan, Hasan Nayfeh, James Raferty, Martin Sandberg, Srikanth Srinivasan, Neereja Sundaresan, Jerry Tersoff, Ben Fearon, Karthik Balakrishnan, James Hannon, and Jerry Chow. M.C. also acknowledges support from Princeton Plasma Physics Laboratory through the Department of Energy Laboratory Directed Research and Development program and contract number DEAC0209CH11466 to complete parts of the analysis and manuscript.
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S.R. and A.K. developed the technique with contributions from M.C., I.L., and P.J. M.C. and S.R. performed the experiments. M.C., S.R., and A.K. analyzed the data. M.C., S.R., and A.K. wrote the manuscript with feedback from the other authors.
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Carroll, M., Rosenblatt, S., Jurcevic, P. et al. Dynamics of superconducting qubit relaxation times. npj Quantum Inf 8, 132 (2022). https://doi.org/10.1038/s4153402200643y
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DOI: https://doi.org/10.1038/s4153402200643y
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