Abstract
Superconducting qubits are among the most advanced candidates for achieving faulttolerant quantum computing. Despite recent significant advancements in the qubit lifetimes, the origin of the loss mechanism for stateoftheart qubits is still subject to investigation. Furthermore, the successful implementation of quantum error correction requires negligible correlated errors between qubits. Here, we realize longlived superconducting transmon qubits that exhibit fluctuating lifetimes, averaging 0.2 ms and exceeding 0.4 ms – corresponding to quality factors above 5 million and 10 million, respectively. We then investigate their dominant error mechanism. By introducing novel timeresolved error measurements that are synchronized with the operation of the pulse tube cooler in a dilution refrigerator, we find that mechanical vibrations from the pulse tube induce nonequilibrium dynamics in highly coherent qubits, leading to their correlated bitflip errors. Our findings not only deepen our understanding of the qubit error mechanisms but also provide valuable insights into potential errormitigation strategies for achieving fault tolerance by decoupling superconducting qubits from their mechanical environments.
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Introduction
Superconducting qubits have become a viable platform both for scientific and technological applications, ranging from fundamental quantum optical experiments^{1}, hybrid quantum systems^{2} to quantum information science^{3}. In particular, they have attracted much attention in faulttolerant quantum computing, achieving important milestones, including the realization of highfidelity quantum gate and readout on a multiplequbit system^{4,5}, the reports of quantum supremacy^{6,7}, and the demonstrations of surface codes^{8,9,10}. Despite such encouraging progress, realizing largescale superconducting quantum computing is still an outstanding challenge^{11}. Although quantum error correction promises reliable and scalable quantum computing, it strictly requires the physical errors of a large number of qubits to be sufficiently smaller than a certain threshold, and, more importantly, to be uncorrelated^{12,13}.
Since the physical errors in superconducting qubits are dominantly limited by their coherence^{4,5,14}, considerable efforts towards improvements in the qubit lifetimes have been made using insights from diverse fields^{15}, ranging from classical and quantum circuit engineering^{16,17,18,19,20} to material science^{21,22,23,24,25}. However, it still remains unclear whether the qubit lifetimes can be enhanced steadily. In addition, more coherent superconducting qubits are more sensitive to small changes in their environments^{25}, imposing a challenge on their scalability. For instance, when a superconducting qubit is dominantly coupled to a few twolevel systems (TLSs), its relaxation time is largely fluctuating^{26,27}. More recently, it has been reported that the absorption of ionizing radiation generates highenergy phonons in a qubit substrate, which leads to nonequilibrium quasiparticles, causing correlated chargeparity switching^{28} and energy relaxations^{29} of superconducting qubits. To verify fault tolerance, it is, therefore, more and more important to characterize a highly coherent multiplequbit system more carefully, i.e., not only reporting their averaged coherence times but also studying the time and frequency dependence, as well as confirming the absence of correlated errors. Indeed, such characterizations have revealed dominant loss mechanisms and sources of fluctuations in superconducting qubits, such as surface dielectric loss^{30}, TLSs^{26,31,32,33}, nonequilibrium quasiparticles^{34,35}, and ionizing radiation^{36,37,38}.
Here, we realize longlived superconducting transmon qubits based on niobium capacitor electrodes with the average and the longest lifetimes exceeding 0.2 ms and 0.4 ms, respectively, and report a new source of correlated bitflip errors, caused by mechanical bursts generated by the pulse tube cooler of a dry dilution refrigerator^{39}. This is revealed by a novel timeresolved analysis of the residual excitedstate probabilities and quantum jumps of multiple qubits, which are synchronized with the operation of the pulse tube cooler. Although the origin of the mechanical sensitivity of longlived superconducting qubits could not be determined unambiguously in this work, our observations are consistent with quasiparticle and TLSmediated qubit decays to phononic baths^{40,41}. Moreover, these findings suggest future strategies for faulttolerant superconducting quantum computing, including the development of acoustically shielded superconducting devices^{42,43,44}, mechanical shockresilient sample packaging^{45,46}, and a vibrationfree dilution refrigerator^{47,48,49}.
Results
Longlived superconducting transmon qubits
We develop superconducting transmon qubits, each formed by a single Al/AlO_{x}/Al Josephson junction shunted by a Nb capacitor, fabricated on a highresistivity silicon substrate (see the fabrication details in Supplementary Note 1). Figure 1a, c show an optical micrograph of a fabricated multiple superconducting qubit device and its equivalent circuit model, respectively, including four frequencyfixed transmon qubits (Q_{0}–Q_{3}) with resonance frequencies ranging from 4.8 GHz to 6.2 GHz and anharmonicities of −0.26 GHz on average. As the metalsubstrate interface of the Al film fabricated by a liftoff process can not be as clean as that of the Nb film directly sputtered on the silicon substrate, we minimize the area of the Al electrodes and bandage patches^{50} to reduce the energy participation ratio in the interface (see Fig. 1b). To realize multiplexed dispersive readout, all the qubits are individually coupled to λ/4 readout resonators with different resonance frequencies around 7 GHz, sharing a λ/2 Purcell filter^{51}. The filter is connected to a feed line, along which frequencymultiplexed control and readout signals are sent. The filter is designed to have a 7 GHz resonance frequency and a 300 MHz bandwidth, suppressing the qubit radiative decay rates to a level of \({{{{{{{\mathcal{O}}}}}}}}(10\,{{{{{{{\rm{Hz}}}}}}}})\). The statedependent dispersive shifts and the readout resonator bandwidths are designed to be \({{{{{{{\mathcal{O}}}}}}}}(1\,{{{{{{{\rm{MHz}}}}}}}})\). Note that the different dispersive shifts of the readout resonators for the first and second excited states enable us to distinguish between the readout signals corresponding to the first three states (G, E, and F) in a single shot. See Table 1 in Method for the full characterization of the system parameters.
As schematically shown in Fig. 1d, the fabricated device is mounted at the mixing chamber stage (~10 mK) of a dry dilution refrigerator, enclosed in a multilayer shielding: copper radiation shields and magnetic shields of aluminum and mumetal. The transmon qubits are characterized using a nearly quantumlimited broadband Josephson traveling wave parametric amplifier (JTWPA)^{52}, allowing us to perform the simultaneous singleshot readout of the qubits by frequencymultiplexing^{5}. The readout errors of the G states, primarily determined by separation errors, yield ≲0.08% for Q_{0} and ≲0.04% for Q_{1}, while those of the E (F) states, primarily influenced by stateflip errors, yield ≲2% (5%) for Q_{0} and ≲5% (8%) with additional influence from separation errors for Q_{1} (see more details in Supplementary Note 3). To suppress thermal and backward amplifier noises, the input and output lines are heavily attenuated and isolated, respectively, while both are equipped with lowpass filters and eccosorb filters (see more details in “Methods” section).
Figure 1e, f show the time traces of the excitedstate probability of the transmon qubit with a resonance frequency of 4.8 GHz (Q_{0}), showing the longest relaxation times (T_{1} = 0.45 ± 0.04 ms) and the longest Ramsey and Hahnecho dephasing times (T_{2*} = 0.18 ± 0.02 ms and T_{2e} = 0.37 ± 0.08 ms), respectively. Our observations confirm that coherence improvements are still possible with widely employed superconducting material systems involving Al/AlO_{x}/Al Josephson junctions and Nb electrodes fabricated on a silicon substrate^{20}.
As shown in Fig. 1g, we measure the longterm stability of the relaxation times of the four qubits, showing significantly large fluctuations, especially for the longerlived qubits (Q_{0} and Q_{1}) with average T_{1} of approximately 0.2 ms and relative standard deviations of 30%, while Q_{2} and Q_{3} with average T_{1} of 0.04–0.08 ms exhibit smaller relative deviations of 10–20%. The Allan deviation analysis of the fluctuations implies that the relaxation times of our longlived transmon qubits are mainly limited by TLSs^{27} (see Supplementary Note 2).
Effect of pulse tube cooler on qubit excitations
We perform the singleshot readout for the longest lifetime transmon qubit (Q_{0}). As shown in Fig. 2a, we apply 2.5 μs long readout pulses repeatedly with an interval of 1 ms (≈5 times larger than the average T_{1}), which is sufficiently long to prepare the qubit in the equilibrium. In addition to the conventional qubit measurement setup, we anchor an accelerometer on the top plate of the dilution refrigerator, converting the pulse tube vibrational noise to a voltage signal. The converted signal is acquired by an oscilloscope that is operated synchronously with the qubit readout sequence via a trigger signal generated by the qubit measurement setup, enabling us to simultaneously record both the vibrational noise and the singleshot readout outcomes.
Figure 2b, d show the synchronized time trace records of the vibrational noise and the qubit singleshot readout quadrature amplitudes, respectively. When the pulse tube cooler is on (orange), the qubit is more frequently excited to the first excited state (E), even to the second excited state (F), while with the cooler off (blue), it mostly remains in the ground state (G). Here, we obtain the data with the pulse tube cooler off by deactivating it temporarily without affecting the base temperature (≈5 min). Note that the readout signals for the E and F states of Q_{0} can be distinguished well in one quadrature projected in an optimal phase. Importantly, when the pulse tube cooler is on, the qubit becomes excited periodically in time, synchronized with the periodic vibrational noise. Figure 2c, e show the amplitude spectral densities of the vibrational noise and the qubit readout quadrature, respectively, showing both have harmonics with exactly the same fundamental frequency of approximately 1.4 Hz when the pulse tube cooler is on. Note that we show the spectral density of the absolute value of the raw vibrational data to capture the pulse tube repetition frequency (see Supplementary Note 7).
Figure 2f shows the histograms of the readout quadratures (the number of each data ≈3 × 10^{5}) with the pulse tube cooler on and off. To mitigate the readout separation errors, we obtain the occupation probability in each qubit state by fitting the histogram to a mixture of multiple Gaussian distributions. When the pulse tube cooler is switched off, we achieve a high initialization fidelity of 99.88% (P_{E} = 0.12%) by passive cooling, where the corresponding effective temperature is T_{eff} = 34 mK, although the base temperature is approximately 10 mK. Importantly note that the excitedstate probability due to the stateflip error induced by the readout backaction is ≲ 0.02%, not dominantly limiting the residual qubit excitation (see Supplementary Note 3). In contrast, when the pulse tube is on, the qubit is excited to the E state with a higher probability (P_{E} = 1.25%), and even the occupation probability in the F state is not negligible (P_{F} = 0.15%). More interestingly, the occupation probability distribution is not in the thermal equilibrium, i.e., the effective temperature is T_{eff} = 53 mK for the GE transition, while T_{eff} = 102 mK for the EF transition. This implies that the qubit is not simply excited by a local thermal heating of the environment, but it is excited by nonequilibrium dynamics of the mechanical vibrations generated by the pulse tube cooler.
Mechanically induced correlated excitations
We study the existence of correlated excitations for two of the longlived qubits, Q_{0} and Q_{1} with the average T_{1} ≈ 0.2 ms, by simultaneously reading out both states with a frequencymultiplexed readout pulse. Figure 3a shows the 2D histograms of the simultaneous readout outcomes of approximately 3 × 10^{5} and 10^{7} measurements when the pulse tube cooler is off and on, respectively. Note that the different numbers of data points are due to the limited measurement time when the pulse tube is switched off. As shown in Fig. 3a, we can obtain the excitedstate probability for both qubits, or the probability when the state is found in the E or F states (\(\bar{G}\)), by using only the real part of the readout complex amplitude. To quantitatively study the correlated excitations, we use mutual information (MI) in the unit of bit, which is defined as
Here, \(P({{{{{{{\mathcal{X}}}}}}}},{{{{{{{\mathcal{Y}}}}}}}})\) is the joint probability of qubit Q_{0} in the \({{{{{{{\mathcal{X}}}}}}}}\) state and qubit Q_{1} in the \({{{{{{{\mathcal{Y}}}}}}}}\) state, while \(P({{{{{{{\mathcal{X}}}}}}}})\) and \(P({{{{{{{\mathcal{Y}}}}}}}})\) are the marginal probabilities of Q_{0} in \({{{{{{{\mathcal{X}}}}}}}}\) and Q_{1} in \({{{{{{{\mathcal{Y}}}}}}}}\), respectively, where \({{{{{{{\mathcal{X}}}}}}}}\) and \({{{{{{{\mathcal{Y}}}}}}}}\) can be G and \(\bar{G}=E\) or F. This quantifies how much information about the excitation of one qubit we obtain from the other qubit (0 ≤ I ≤ 1 bit). We correct the systematic bias and obtain the standard error on the estimated MI with the method described in ref. ^{53}. For the case when the pulse tube cooler is off, the MI is I < 10^{−6} bit, while that with the pulse tube on is found to be I = 0.00278 bit (orange dashed line in Fig. 3e), showing there is a significant correlation in their excitations that are induced by the pulse tube mechanical vibrations.
To investigate the origin of the correlated excitations, we develop a timeresolved analysis of the qubit excitedstate probabilities, synchronized with the periodic vibrational noise generated by the pulse tube cooler. Using the protocol shown in Fig. 2a, we repeat a sequence consisting of approximately 4000 multiplexed readout pulses (≈4 s in total) while simultaneously recording the vibrational noise. Since the starting time of each sequence is not synchronized with the phase of the periodic vibrational noise, we need to timealign every time trace of the singleshot data with respect to the periodic vibrational noise. To this end, we first specify one period of the vibrational noise as a reference, as shown in Fig. 3b. Then, we timealign every trace of the singleshot outcomes by maximizing the crosscorrelation of the simultaneously recorded vibrational noise with the referential one. Consequently, we can accumulate a sufficient number of the singleshot outcomes at an arbitrary time of interest within the vibrational period to obtain the timeresolved excitedstate probabilities of the qubits. Figure 3d shows the results of the timeresolved measurements of the E and Fstate probabilities for both qubits, obtained from the sequences repeated approximately 3000 times. Although the two operational phases of the pulse tube cooler exhibit similar vibrational noises, the qubits are frequently excited only during the gasflowin phase, significantly deviating from thermal distributions. Figure 3c shows the result of the timefrequency analysis of the vibrational noise, by taking a shorttime Fourier transform with a 5 ms Hann window, revealing there is a difference between the gasflowin and out phases. However, there is no clear correlation between the timeresolved excitedstate probabilities and the result of the timefrequency analysis up to 50 kHz. This could be because superconducting qubits can be excited by gigahertz or higher frequency vibrational noise that cannot be measured by the accelerometer as long as a multiphonon excitation process and local heating do not play a dominant role. Moreover, we find that the E and Fstate probabilities of both qubits are increased synchronously from their lowest values (P_{E} ≈ 0.2% and P_{F} ≈ 0.005%) by a factor of approximately one hundred and one thousand, respectively. This implies that the two qubits are dominantly excited by the common mechanical vibrations via their phononic baths.
Figure 3e shows the timeresolved MI as a function of time within the one vibrational period, which is obtained from the timeresolved multiplexed singleshot readout outcomes for the two qubits using Eq. (1). The timeresolved MI (blue solid line) consistently exhibits smaller values compared to the nontimeresolved counterpart (orange dashed line). This implies that individual qubit excitation events are not strongly correlated, while the excitedstate probabilities vary in time synchronously between the qubits due to the global effect of the pulse tube mechanical vibrations, explaining the existence of the correlation observed in the nontimeresolved data.
Mechanically induced correlated quantum jumps
Next, we study the mechanical effect on quantum jumps of the transmon qubits, i.e., the pulse tube effect on their transition rates among the G, E, and F states. As shown in Fig. 4a, we repeat a sequence to continuously monitor the states of Q_{0} and Q_{1} about 3000 times^{54}, while simultaneously recording the vibrational noise from the pulse tube cooler. Each sequence consists of approximately 5 × 10^{5} successive 2.5 μs long multiplexed readout pulses with an interval of 3 μs, corresponding to a length of ≈1.5 s. Here, we will first focus on the timeresolved measurement of the transition rates of qubit Q_{0} (Fig. 4), and then study the existence of a correlation in quantum jumps between Q_{0} and Q_{1} (Fig. 5).
In a similar manner to the previous timeresolved measurements, we can timealign every continuous monitoring trace with respect to the referential periodic vibrational noise, displayed in Fig. 3b. As shown in Fig. 4b, every timealigned trace contains several quantum jump events, from which we sample the event time (vertical dashed line) and the dwell time (a doublesided horizontal arrow) for the G, E, and F states, individually. Figure 4c–e show the 2D histograms of the event time and the dwell time for the three states, respectively, where each histogram contains >10^{6} dwell events.
To determine the transition rates among the G, E, and F states in a timeresolved fashion, we calculate the distribution of the dwell time in each state at a time of interest within the vibrational period. Namely, the timeresolved dwelltime distributions are obtained by using the dwell events within a bin centered at the chosen time. In addition, the bin width for all three states is set to be proportional to the average G dwell time around the chosen time. This is because the time resolution of the measurement of the transition rate for the G state is limited by the inverse of the rate, approximately corresponding to the average of the G dwell time. Furthermore, the number of the E (F) dwell events within a bin width of the average E (F) dwell time (fundamental time resolution) is not sufficient, requiring a wider bin width to accumulate more data points, and eventually limiting the time resolution.
Figure 4f, g show examples of the timeresolved dwelltime distributions in the three states at the times specified with the dashed lines in Fig. 4h, respectively. As exemplified by the G dwelltime distribution in Fig. 4g, some dwelltime distributions display two distinct characteristic timescales, for which an unambiguous reason is not identified in this study. However, similar phenomena are observed in various superconducting qubit systems, potentially linked to nonequilibrium quasiparticle dynamics^{55,56,57}. To include this effect and accurately obtain the transition rates, we fit each dwelltime distribution to two distinct models: an exponential distribution and a mixture of two exponential distributions, and choose the fitting model with the lower Bayesian information criterion value (see more details in Supplementary Note 3). As shown in Fig. 4h, this analysis results in the timeresolved total transition rates from the G, E, and F states to the other two (Γ_{G}, Γ_{E}, and Γ_{F}), respectively, as a function of time within one period of the vibrational noise.
To extract the individual transition rate from the \({{{{{{{\mathcal{X}}}}}}}}\) state to the \({{{{{{{\mathcal{Y}}}}}}}}\) state (\({{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}\to {{{{{{{\mathcal{Y}}}}}}}}}\)) based on the inferred total transition rate (\({{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}}={{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}\to {{{{{{{\mathcal{Y}}}}}}}}}+{{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}\to {{{{{{{\mathcal{Z}}}}}}}}}\)), we utilize the probability of eventually transitioning to \({{{{{{{\mathcal{Y}}}}}}}}\) conditioned on dwelling in \({{{{{{{\mathcal{X}}}}}}}}\), where \({{{{{{{\mathcal{X}}}}}}}}\), \({{{{{{{\mathcal{Y}}}}}}}}\) and \({{{{{{{\mathcal{Z}}}}}}}}\) traverse the G, E, and F states. Since the conditional probability of the event yields \({{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}\to {{{{{{{\mathcal{Y}}}}}}}}}/{{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}}\), which can be experimentally inferred as the number of the \({{{{{{{\mathcal{X}}}}}}}}\) dwell events resulting in transitions to \({{{{{{{\mathcal{Y}}}}}}}}\) divided by the total number of the \({{{{{{{\mathcal{X}}}}}}}}\) dwell events, we can determine \({{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}\to {{{{{{{\mathcal{Y}}}}}}}}}\) as the product of the total transition rate \({{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}}\) and the conditional probability \({{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}\to {{{{{{{\mathcal{Y}}}}}}}}}/{{{\Gamma }}}_{{{{{{{{\mathcal{X}}}}}}}}}\). Note that we correct a nontrivial effect of the readout separation errors on the inferred transition rates. See more details in Supplementary Note 4.
Figure 4i shows the timeresolved transition rates among the G, E, and F states. The transition rates exhibit a dynamic pattern consistent with the previously described behavior observed in the excitedstate populations (Fig. 3d). In detail, they show an interval where the pulse tube mechanical vibrations induce significant increases in all the transition rates, including the direct transition processes between the G and F states. Interestingly, a nonnegligible direct transition process between the G and F states is observed even in the “quiet” period, when the pulse tube effect is minimal. This can be explained by readoutinduced higherorder transitions^{58} or auxiliary modeassisted twophoton transitions^{59}. Finally, in the “quiet” period, the decay rates of the E and F states are larger compared to those in the free evolution (the inverse of the respective relaxation times). We attribute such increases to the readoutinduced state flip^{60}.
Using the same continuous monitoring data, we further study whether there is a correlation in quantum jumps between qubits Q_{0} and Q_{1}. When the qubits are continuously monitored for a certain time interval, an error, corresponding to a state transition, will occur with a finite probability. For simplicity, we do not distinguish between the E and F states in the correlation analysis (\(\bar{G}=E\) or F). We chunk the continuous monitoring data and classify them into 16 possible events: the 2 possible initial states \(\left\{G,\bar{G}\right\}\) and the error occurrence within the time interval {,×} for each of the 2 qubits. Here, “" stands for the occurrence of no bitflip event, and “×" stands for the occurrence of at least one bitflip event. As shown with an example in Fig. 5a, we can, therefore, calculate a 4 × 4 matrix of the error probabilities at a specific time interval.
To quantitatively study the existence of correlated bitflip errors during continuous monitoring, we use mutual information (MI), defined in Eq. (1), using the 4 × 4 error probability matrix in this case. The MI corresponds to a quantitative value of the amount of the correlation in the quantum jumps, i.e., how much information about the occurrence of a bitflip error in one qubit we can obtain from the other one (0 ≤ I ≤ 2 bit). Figure 5b presents the MI of the bitflip error probabilities between Q_{0} and Q_{1} as a function of the time interval, showing the maximum of I = 0.1328 bit at 2.5 ms. It clearly shows that there is a correlation in the quantum jump events between the two qubits, which corresponds to the existence of a correlated error in their gates.
Effect of a controlled mechanical shock
To rule out that the qubit excitations are due to possible electrical noise produced by the pulse tube cooler, we mount on the top plate of the dilution refrigerator an electric hammer based on an electromagnet (see Fig. 6a). In this manner, we generate a purely mechanical shock by a pulsed DC current in the electromagnet, synchronized with the qubit readout sequence via a trigger signal. We simultaneously record both the vibrational noise and the qubit singleshot readout outcomes while the pulse tube cooler is deactivated.
Figure 6b, c present the time traces of the acceleration generated by the controlled mechanical shock and the result of the timefrequency analysis, respectively, showing a broadband impulse shock, followed by a damped oscillation at eigen frequencies of the refrigerator. Figure 6d shows the timeresolved E and Fstate probabilities of qubits Q_{0} and Q_{1}, measured synchronously with the mechanical shock. The excitedstate probabilities are obtained from the singleshot outcomes of approximately 10^{4} measurements for each time of interest with a 10 ms time resolution. Note that we accumulate 1 ms timeresolved data sets for 10 ms to obtain a sufficiently large number of data points to characterize the small Fstate probabilities. We observe that both qubits are similarly excited purely by the mechanical shock, while its effect on the microwave bath is negligible (see Supplementary Note 6). This implies that possible electrical noise generated from the pulse tube cooler does not dominantly contribute to the qubit excitations. Interestingly, the qubits are not excited more frequently by the electric hammer when compared to the case of the pulse tube mechanical shock although the hammer mechanical shock is larger even when the mechanical shocks for the two cases are compared at the qubit device at room temperature (see Supplementary Note 7). We believe that this is because gigahertz or higher frequency vibrational noise, which can not be characterized by the accelerometer and could be different between the two cases, needs to be responsible for the qubit excitations.
Timeresolved measurement of microwave bath
One possible process to excite the qubits mechanically is that a side effect of a mechanical shock would change the property of the microwave bath, leading to an increase in the microwave background noise. For example, a change in the transmission through the filters and attenuators may alter the amount of thermal noise coming from the higher temperature stages of the refrigerator. In addition, we use an attenuator and a terminator made of crystalline quartz in order to acoustically thermalize the feed line to the base temperature (see the detailed experimental setup in “Methods” section). Such microwave components would convert a mechanical shock to microwave noise, eventually exciting the qubits electrically. Moreover, local heating and a triboelectric effect could also increase the effective temperature of the microwave bath. These concerns motivate us to perform the timeresolved analysis of the microwave transmission coefficient (∣S_{21}∣) and background noise of the measurement chain, synchronized with the measurement of the periodic vibrational noise generated by the pulse tube cooler.
As shown in Fig. 7a, we apply coherent pulses at around 7 GHz, which are offresonant with the readout resonators and qubits. By measuring the average and variance of the coherent amplitude of the pulses in a timeresolved manner (similar to the experiment for Fig. 3), we can characterize the microwave transmission coefficient and background noise at a time of interest within the period of the vibrational noise (Fig. 7b). Figure 7c, d show the timeresolved microwave transmission coefficient (∣S_{21}∣) and background photon noise, respectively, confirming that there is no significant time variation.
Here, we can estimate at least how much background noise increase is required to excite the qubits to the extent observed in Fig. 3d. First, we consider the scenario that the qubits are excited by mechanically induced microwave noise in the feedline. From numerical simulations based on the finiteelement method, the external coupling rates of qubits Q_{0} and Q_{1} to the feed line (microwave bath) are of the order of Γ_{ex}/2π ≈ 10 Hz, showing the strong suppression of the radiation losses by the Purcell filter. In contrast, the experimentally measured relaxation rates are of the order of 1 kHz, confirming that the qubits are strongly undercoupled to the feed line (Γ_{in}/2π ≈ 1 kHz, where Γ_{in} is the qubit internal loss rate). Assuming the intrinsic qubit bath temperature is negligible, we can estimate the microwave background photon noise (n_{bg}) required to have the residual excitedstate probability of P_{E} ≈ 0.2 by using
This results in n_{bg} ≈ 30.
In the scenario of an abrupt thermal heating due to the pulse tube mechanical vibrations at the mixing chamber plate of the dilution refrigerator, the microwave bath would undergo a heating process analogous to that observed in the qubits. Consequently, the background microwave photon noise would be elevated to approximately n_{bg} ≈ 0.2 during the transient period when the qubits reach the maximal excitation due to the pulse tube operation, as shown in Fig. 3d.
When it is characterized by a microwave detector with a quantum efficiency of η, the background microwave photon noise is measured to be ηn_{bg} + 1/2 in the unit of quanta (see more details in “Methods” section). By using the relationship between the signaltonoise ratio in singleshot qubit readout and the readoutinduced qubit dephasing, we confirm that our microwave measurement achieves a quantum efficiency of η > 0.2 (see Supplementary Note 5). This is facilitated by the JTWPA, operated as a nearly quantumlimited phaseinsensitive amplifier. If the microwave background noise dominantly heats up the qubits, the measured photon noise on top of the vacuum noise (1/2), therefore, needs to be increased to be of the order of ηn_{bg} ≳ 6 and 0.04 for the two aforementioned scenarios, respectively. However, there is no increase observed at such a level in the background noise, as shown in Fig. 7d. This confirms that possible nonequilibrium dynamics of the microwave bath do not play a dominant role in the mechanically induced excitations of the qubits.
Discussion
While we clearly observe that the transmon qubits suffer from correlated bitflip errors due to the global nature of mechanical vibrations, the physical origin still remains uncertain. Here, we propose and discuss possible explanations for the mechanically induced excitations and quantum jumps of the qubits.
First, we can rule out the possibility that the qubits are excited by an abrupt thermal heating of the mixing chamber plate caused by a mechanical shock. In fact, this is not consistent with the nonthermal probability distribution of the qubit produced by the pulse tube mechanical vibrations (see Fig. 2f). Next, we verify that the mechanical vibrations do not affect the microwave environment around the qubit frequencies, although they could cause a triboelectric effect, generating lowfrequency electrical noise in cables and degrading the coherence of a spin qubit^{47}. Indeed, we record no fluctuation in the transmission coefficient and background noise of the feed line while the mechanical bursts excite the qubits (see Fig. 7). Furthermore, the absence of variation in the microwave background noise also supports the lack of heating of the mixing chamber plate (thermally coupled to the microwave environment via the attenuators). In addition, this is also consistent with the qubit excitations caused by a pure mechanical shock, as shown in Fig. 6, where possible electrical noise generated by the pulse tube cooler does not play an important role in the qubit excitations. Nevertheless, our characterization does not cover the frequency range for infrared photons, which are more sensitive to a small mechanical displacement in microwave components and can induce a qubit decay by breaking Cooper pairs^{61}.
We associate the mechanical sensitivity of the qubits with quasiparticle and TLSmediated interactions to their phononic baths. On one hand, highenergy phonons break Cooper pairs, resulting in nonequilibrium quasiparticles in the qubit electrodes^{41,62}. The quasiparticles are quickly cooled down to the lowest energy level above the superconducting energy gap, remaining there for several tens of milliseconds before recombining into Cooper pairs^{29,34}. This nonequilibrium quasiparticle bath could be heated by scattering with the phonons produced by the pulse tube mechanical vibrations, resulting in excitations or relaxations of the qubits. On the other hand, TLSs couple to a qubit via their electric dipoledipole interactions while coupling to a phononic bath via their strain potential^{40}. Therefore, they can mediate between the qubit and the phononic bath, resulting in the mechanical sensitivity of the qubit. Finally, both a change in the strain^{63} and the saturation of the TLS bath^{64} can alter the coupling between the qubit and the TLSs, leading to a fluctuation in the qubit decay rate to the phononic bath. Additionally, the TLS model also explains the longterm fluctuations that are observed in the lifetimes of our qubits^{26,27}, as shown in Fig. 1g.
In summary, in this work, we presented a novel timeresolved measurement technique to study the mechanical sensitivity of longlived transmon qubits, synchronized with the operation of the pulse tube cooler of a dilution refrigerator. Our results demonstrated that the mechanical vibrations generated by the pulse tube cooler induce dominant bitflip errors in the qubits. Moreover, the global nature of the mechanical bursts on the multiqubit device causes correlated errors among the qubits, which are detrimental to realizing largescale quantum computing based on quantum error correction. While the origin of mechanical sensitivity of the qubits could not be established unequivocally, our observations are consistent with quasiparticle and TLSinduced qubit decay models^{40,41}, and provide valuable insights into the loss mechanisms that limit the stateoftheart qubit coherence.
Our findings suggest several strategies to mitigate the mechanical sensitivity of superconducting qubits, including the use of a suspended qubit substrate with phononic crystal structures^{42,43,44} and sample packages that employ both mechanical isolation and thermal conductivity^{45,46}, in order to isolate superconducting qubits from mechanical vibrations. In addition, our results would emphasize the importance of vibrationfree dilution refrigerators^{47,48,49} in achieving long and stable coherence times in superconducting devices. Moreover, our measurement scheme revealed that superconducting qubits become out of equilibrium within a specific time window during the periodic pulse tube operation. This observation suggests a straightforward mitigation strategy: synchronizing qubit experiments with the pulse tube operation and acquiring data only when the pulse tube effect is minimal. We believe that these insights will be valuable for the development of nextgeneration superconductingqubit technologies that pave the way for realizing faulttolerant quantum computing.
Methods
System parameters
The system parameters of the multiqubit device are summarized in Table 1.
The qubit frequencies (ω_{q}), relaxation times (T_{1}), and dephasing times (T_{2*} and T_{2e}) are obtained as the averages of the longterm stability measurement data collected over 400 h, respectively, shown in Fig. S3 of SI, while the error bars are calculated as the standard deviations. The longest relaxation and dephasing times, shown in Fig. 1e, f, are observed in the first cooling down for the multiqubit device, while the longstability measurement is conducted in the following cooling down, where the relaxation and dephasing times are slightly degraded possibly due to additional oxidation of the Nb and Si surfaces. The anharmonicities (α) are characterized by the EF control in the time domain (Q_{0} and Q_{1}) and the twophoton transition in the qubit excitation spectra (Q_{2} and Q_{3}). We perform the Fstate relaxation measurement for qubits Q_{0} and Q_{1} and obtain the Fstate probability using the averaged readout complex amplitude for different delay times. The exponential fitting to the time trace resulted in the T_{1} of the F state.
The frequencies, external coupling rates, and internal loss rates of the readout resonators are characterized from the reflection spectra with the qubits in the G states in continuouswave (CW) measurement. Since the qubits are well cooled down to their ground states in our experimental setup, the effective intrinsic losses due to the qubit excitations are negligible^{65}. The dispersive shifts for Q_{0} and Q_{1} are determined by the resonance frequency difference of the qubitstatedependent reflection spectra of the resonators in the timedomain protocol (see Fig. S4a in SI), while those for Q_{2} and Q_{3} are obtained from the photonnumber resolved qubit excitation spectra.
Cryogenic wiring and room temperature measurement setup
The experimental setup is shown in Fig. 8. The chip is fixed on a sample table made of goldplated Cu, wirebonded to a PCB based on coplanar waveguides sandwiched by double ground planes, and covered by an Al lid (see Fig. 9b). There is an air gap designed under the chip on the sample table to increase the frequency of spurious chip modes well above our working frequencies^{66}. We thermally and mechanically anchor the device under test (DUT) to the mixing chamber stage (≈10 mK) of a dilution refrigerator (BlueFors BFLD250). We isolate the DUT from environment fluctuations with multiple shielding: Al and Amumetal4 K shields are used for reducing magnetic noise, while a Cu shield is used for thermalizing the qubit radiation field to the base temperature (see Fig. 9a). In addition to the standard equipment defined by BlueFors (300 K vacuum can and 50 K, 3 K, and 0.8 K radiation shields, not shown in Fig. 8), an outer Amumetal shield is equipped and thermalized to the vacuum can (300 K) while a Cu radiation shield is thermalized to the 10 mK stage, covering the entire 10 mK setup. The input line to the DUT is equipped with a series of cryogenic attenuators (−72 dB in total) to suppress thermal noise from the higher temperature stages, while the output line is equipped with several isolators to prevent back heating from amplifiers. The readout signals are amplified by a Josephson traveling wave parametric amplifier (JTWPA) and a high electron mobility transistor (HEMT) amplifier in the dilution refrigerator, allowing us to realize a nearly quantumlimited microwave measurement. The continuous pump signal to operate the JTWPA is added to the readout chain after the DUT via a directional coupler. We optimize the pump power and frequency to maximize the signaltonoise ratio around the readout frequencies, resulting in about a 20 dB gain. In addition, all the input, output, and pump lines are equipped with lowpass filters (LPFs) and eccosorb filters (Eccos) to reduce the contamination of highfrequency photons.
For multiplexed control and readout of the transmon qubits, we employ an arbitrary waveform generator (Zurich Instruments HDAWG) and a quantum analyzer (Zurich Instruments UHFQA) to generate, acquire, and analyze intermediate frequency (IF) pulse sequence. We up and downconvert frequencymultiplexed IF signals using IQ mixers operated with continuous waves generated from a multichannel microwave source (AnaPico APMS12G). The upconverted control signal is amplified, filtered to cut the output amplifier noise around the readout frequencies, and combined with the readout signal via a directional coupler. The frequencymultiplexed readout microwave signals are downconverted, amplified by room temperature amplifiers (RTA), digitized, and, demodulated by the UHFQA, leading to the I and Q quadratures for each readout frequency. The operation of the UHFQA, generating and digitizing the readout signals, is synchronized with the HDAWG via a trigger signal.
For monitoring the vibrations of the top plate of the dilution refrigerator, we use an oscilloscope (Keysight 1000X), operated synchronously with the HDAWG via a trigger signal. The acceleration of the top plate is continuously converted to a voltage signal by an accelerometer (KEMET VSBV203B) mounted on it (see Fig. 9c). The accelerometer can be activated with a 5 V DC voltage bias (not shown in Fig. 8). Upon a trigger signal from the HDAWG, the oscilloscope starts to record the converted voltage signal. The recorded voltage signals are expressed in the unit of acceleration by using the sensitivity of 20 mV/m/s^{2}.
For artificially generating a pure mechanical shock on the top plate of the refrigerator, we use an electric hammer based on a circuit consisting of an electromagnet, a 15 V DC voltage bias, and an electrical switch (see Fig. 9c). When the electrical switch is on, a current flows in the electromagnet, accelerating the magnet core and resulting in a mechanical shock on the top plate. When the electrical switch is off, the magnet core is detached from the top plate by the elastic force of a spring. We use an AWG (Tektronix AFG3252C), operated synchronously with the HDAWG, to control the electrical switch by a pulsed voltage signal. In the experiment for Fig. 6, the electric hammer is activated periodically with 50 ms pulsed signals with a period of 1.5 s.
Quantum efficiency of microwave measurement
To exclude possible responsibility of electrical noise or local heating for the qubit nonequilibrium dynamics, we study the effect of the pulse tube on the background noise of the microwave bath. To achieve nearly quantumlimited microwave measurement, we amplify microwave background noise interacting with the qubit device by the JTWPA and the HEMT amplifier in the dilution refrigerator, followed by another amplification, demodulation, and digital sampling at room temperature. The measurement outcome of an input noise (n_{in}) can be simply described as
where C_{m} and n_{m} are the total scaling factor and the effective inputreferred noise, respectively, which are uniquely determined by the combined contributions of propagation losses, amplifier gains, and amplifieradded noises in the measurement chain. Note that the expression is described in the unit of quanta.
For convenience, the outcome n_{out} is normalized by the output noise for the vacuum input (n_{in} = 1/2) and rescaled to 1/2, resulting in
where a quantum efficiency is defined as
Thus, the full microwave measurement chain can be effectively considered as a single quadrature detector with a quantum efficiency of η, or an insertion of a beam splitter with a transmittance η in front of an ideal quadrature detector^{67}. Since our measurement chain adopts a phaseinsensitive amplification enabled by the JTWPA as a preamplifier, the inputreferred noise is limited by n_{m} ≥ 1/2, resulting in η ≤ 1/2. In our analysis, we normalized the outcome by the nontimeresolved (averaged) noise, corresponding to the possible lowest noise since there is no significant increase in the timeresolved noise data. This is valid as long as the measurement bath is close to being in the vacuum at least when the pulse tube effect is minimal. A finite thermal photon noise would affect the estimated quantum efficiency^{68}, which we can safely assume is quite small given the heavily attenuated, filtered, and isolated input and output lines.
Within this formalism, the output for the input of thermal background noise (n_{in} = 1/2 + n_{bg}) is described as
which are shown in Fig. 7d in the main text.
We calibrate the attenuation in the dilution refrigerator based on photonnumber resolved qubit spectra. With this, we estimate the readout photon flux incoming to the readout cavity, resulting in the readoutinduced qubit dephasing rate. By taking the ratio of the signaltonoise ratio rate of the singleshot readout to the estimated dephasing rate, we characterize the quantum efficiency of our measurement chain as η > 0.2 (see more details in Supplementary Note 5).
Data availability
The data used to produce the plots within this paper are available on Zenodo (https://doi.org/10.5281/zenodo.11034817). All other data are available from the corresponding authors upon request.
Code availability
The code used to produce the plots within this paper will be available on Zenodo (https://doi.org/10.5281/zenodo.11034817).
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Acknowledgements
We thank Sebastian Cozma and Pasquale Scarlino for helping with designing the sample package and Yang Xu for developing the Nb deposition and etching, respectively. Moreover, we thank MIT Lincoln Laboratory and William D. Oliver for providing the JTWPA. This work was supported by the European Research Council (ERC) grant No. 835329 (ExCOMcCEO), as well as the Swiss National Science Foundation (SNSF) under grant No. 204927 and the NCCR QSIT grant No. 51NF40185902. S.K. acknowledges support from the EU H2020 research and innovation program under the Marie SklodowskaCurie grant agreement No. 101033361 (QuPhon). M.S. acknowledges support from the EPFL Center for Quantum Science and Engineering postdoctoral fellowship. All devices were fabricated in the Center of MicroNanoTechnology (CMi) at EPFL.
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S.K. conceived the experiment. S.K. and M.C. designed the device and developed the fabrication process with the assistance of X.W. S.K. and J.P. established the measurement setup and performed the measurements. S.K. and J.P. analyzed the data with the assistance of M.C. S.K., J.P., M.C., M.S., and T.J.K. wrote the manuscript with feedback from all the coauthors. T.J.K. and S.K. supervised the project.
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Kono, S., Pan, J., Chegnizadeh, M. et al. Mechanically induced correlated errors on superconducting qubits with relaxation times exceeding 0.4 ms. Nat Commun 15, 3950 (2024). https://doi.org/10.1038/s41467024482303
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DOI: https://doi.org/10.1038/s41467024482303
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