Investigating the Individual Performances of Coupled Superconducting Transmon Qubits

Abstract

The strong requirement for high-performing quantum computing led to intensive research on novel quantum platforms in the last decades. The circuital nature of Josephson-based quantum superconducting systems powerfully supports massive circuital freedom, which allowed for the implementation of a wide range of qubit designs, and an easy interface with the quantum processing unit. However, this unavoidably introduces a coupling with the environment, and thus to extra decoherence sources. Moreover, at the time of writing, control and readout protocols mainly use analogue microwave electronics, which limit the otherwise reasonable scalability in superconducting quantum circuits. Within the future perspective to improve scalability by integrating novel control energy-efficient superconducting electronics at the quantum stage in a multi-chip module, we report on an all-microwave characterization of a planar two-transmon qubits device, which involves state-of-the-art control pulses optimization. We demonstrate that the single-qubit average gate fidelity is mainly limited by the gate pulse duration and the quality of the optimization, and thus does not preclude the integration in novel hybrid quantum-classical superconducting devices.

1. Introduction

Superconducting Quantum processors are one of the most advanced platforms in the frame of both NISQ (Noisy-Intermediate-Scale-Quantum) [1] or Fault-Tolerant (FT) Quantum computing [2,3,4,5,6,7,8,9,10]. In order to efficiently solve application-specific problems for cryptography, pharmaceutical, quantum information and processing, both the paradigms require a large number of superconducting qubits [11,12], combined with long coherence times. This is not only meant for the implementation of several quantum gates, but most importantly, to get suitable and large single- and two-qubit gates fidelities [13].

Coherence times above hundreds of microseconds have been demonstrated for specific circuital designs, such as for transmon [14,15,16,17,18,19,20,21] or fluxonium qubits [22,23,24,25]. Within typical coherence times measured in these devices, it is in principle possible to reach the fault-tolerant error gate threshold $p_{th}={10}^{-2}$ [2], corresponding to single qubit-gate fidelities of ~$99\%$, or even reach the gold standard of the three-nines, i.e., of the order of $99.9\%$ [2,26].

In order to solve complex quantum algorithms, we not only need high-performance single-qubit systems, but most importantly, the number of physical qubits in a superconducting Quantum Processing Unit (QPU) should increase by a few hundred, i.e., the state-of-the-art so far [27], to over ${10}^5$ [26]. The possibility to engineer a QPU with such a number of physical qubits is the most challenging problem for quantum computing, also known as Input/Output (I/O) challenge.

Quantum controllers use analogue microwave-based control techniques to activate and readout the states of the QPU, which may overwhelm the cooling capacity of even the most powerful cryostats when increasing the number of qubits above ~400 [27,28]. Possible solutions to tackle the I/O problem are to employ analogue multiplexing [29,30], to minimize the heat load of wiring [28], or to introduce compact cryogenic circuitry [31,32,33,34,35,36,37]. These techniques are meant to optimize both the number and the volume of the microwave coaxial cabling at room temperature, as well as in the cryostat. Another approach is to search for novel hybrid Josephson devices to be integrated into superconducting quantum circuits in order to provide novel ways to scale superconducting QPU [38,39,40,41,42,43,44,45,46,47,48]. Finally, it has been proposed to integrate at the millikelvin stage both superconducting quantum processors and energy-efficient classical digital circuits [49,50,51,52], such as Single-Flux Quantum (SFQ) controllers [53,54,55] and Josephson-based readout circuits [56,57,58,59,60,61], into a quantum multi-chip module (qMCM) [62,63,64,65]. The request to separate the SFQ control electronics and the QPU in two different sub-modules comes from the need to mitigate quasiparticle dissipation arising in SFQ circuits [66]. Particularly relevant for SFQ controllers integration in the qMCM, the final goal is to replace analogue microwave control and readout signals supplied from room-temperature electronics via dedicated coaxial cables, with SFQ digital pulse patterns [67,68,69,70]. The expected reduced thermal load related to the replacement of RF-coaxial cabling when using an SFQ controller promise to unlock scaling to over 1000 qubits, but most importantly, it allows for low latency controller feedback [67,68,69,70], which is a fundamental prerequisite for FT quantum computing.

The realisation of qMCMs faces several engineering and physical challenges. As a matter of fact, the circuital parameters of the module hosting the quantum processor (quantum chip) and the superconducting electronics module (carrier chip) are strictly dependent on each other [65]. On the other hand, 3D-integration approaches allow fabrication and diagnosis of the qubits and the control modules separately [65]. In this work, we specifically focus on a preliminary all-microwave experimental characterization of a planar QPU subcircuit, which includes a pair of coupled transmon qubits. The testing protocol aims at the measurements of the circuital parameters, the relaxation and coherence times and Randomized Benchmarking single-qubit gate fidelities of the individual qubits in the QPU. We demonstrate that special care on the optimization of readout and control protocols is required in order to reach the theoretical best values for coherence and single-qubit fidelity in our device. We also demonstrate that we have successfully established a coupling between two qubits, which is a fundamental request for scalability and the implementation of two-qubit gates. By establishing and diagnosing experimentally the circuital parameters of the planar quantum chip, including the readout-qubit and qubit-qubit coupling factors, it is possible to simulate and predict the qMCM stack module performances. Therefore, the results reported in this work play a fundamental role in the future multi-qubit QPU integration in a qMCM stack module.

2. Results

In this work, we characterize two flux-tunable symmetric split-transmon qubits ($Q_a$ and $Q_b$) integrated into a matrix of five transmon qubits, coupled by means of a high-frequency NbTiN bus. The nominal resonance frequency of bus coupling is 25–30 GHz.

Transmon qubits, proposed for the first time in Ref. [16], are modified Cooper-pair Boxes (CPBs), in which a shunt capacitor is added in parallel with the Josephson element to decrease the charging energy $E_c=e^2/\left(2C\right)$ of the device, and therefore increase the ratio $E_J/E_c$ [16]. In our samples, the Josephson element is a superconducting symmetric DC-SQUID loop with two Al/AlO${}_x$/Al Josephson junctions. By increasing the $E_J/E_c$, the sensitivity to charge-noise can be suppressed, thus making the transmon qubit protected against charge noise fluctuations compared to the CPB design [14,15,16,17,18,19,20,21].

Additional information on the sample circuital schematics and materials employed can be found in Section 3.

2.1. Read-Out Resonator and Qubit Spectroscopy

One-tone spectroscopy protocol as a function of the power of the resonators’ excitation signal provides the possibility to inspect the bare and the dressed (or low-photon) frequencies of the readout resonator. Additional information on the experimental setup can be found in Section 3. The transmission through the feedline shows the usual Lorentzian-like dip associated with the adsorption of photons in the readout resonator, which occurs on resonance with the readout frequency [71], as shown in Figure 1a,b for the readout resonators of $Q_a$ and $Q_b$, respectively. At high-input power signals (red data points, specifically at $-70$ dBm on the device-under-test, corresponding to around 6–8 × 10${}^6$ photons), the resonance frequency corresponds to the resonator bare state, and a positive shift ${\chi }^{lp}$ of the frequency occurs in the low-photon regime (blue data points, specifically at $-130$ dBm on the device-under-test, corresponding to around tens of photons), demonstrating the coupling between the readout resonator and the qubit [72,73]. The low-photon shift at zero flux-field for the readout cavity of $Q_a$ is approximately ${\chi }_a^{lp}=(2.14\pm 0.05)$ MHz, while for $Q_b$ is ${\chi }_b^{lp}=(4.47\pm 0.05)$ MHz, where $50$ kHz is a maximum error.

Dashed yellow and green curves in Figure 1 are the resonator fits in the high- and low-power regimes, from which we extract the intrinsic quality factor of the resonator $Q_i$ due to internal losses in the resonator, and the coupling quality factor $Q_c$, which identifies the quality of the coupling with the environment and the external circuitry [71]. The total (loaded) quality factor $Q_l$ is given by $Q_l^{-1}=Q_c^{-1}+Q_i^{-1}$, and is $7\times {10}^3$ for $Q_a$ and $Q_l=9\times {10}^3$ for $Q_b$. Moreover, as reported in

Table 1. Summary of the Quantum Processing Unit (QPU) with the two coupled split-transmon qubits, $Q_a$ and $Q_b$: the $\left|0\right.〉\to \left|1\right.〉$ qubit frequency transition ${\nu }_{01}$ at the Sweet-Spot (SS), the qubit anharmonicity $\alpha$, calculated as twice the frequency separation between ${\nu }_{01}$ and the two-photon transition ${\nu }_{02}/2$ in Figure 1d, the readout resonator frequency in the bare-state ${\nu }_r$, the resonator coupling and intrinsic quality factors $Q_c$ and $Q_i$, the readout-qubit detuning $\Delta$, the relaxation time $T_1$, the Hahn-echo time $T_2$, the Ramsey time $T_2^{*}$ and the qubit-qubit coupling $2J$. The error on the qubit frequency, estimated as the centre of a Ramsey fringes experiment, is a maximum error related to the acquisition step as a function of the qubit-drive frequency. The error on the resonator frequency is a maximum error related to the readout frequency acquisition step. The error on the quality factors is given a maximum error of $10\%$ of the fitting routine. The errors on $T_2^{*}$ and the qubit-qubit coupling strength are the fit errors, while for $T_1$, $T_2^{Echo}$ they are given as an estimation from statistical measurements.

	${\mathit{Q}}_{\mathit{a}}$		${\mathit{Q}}_{\mathit{b}}$
${\nu }_{01}$ at SS (GHz)	$4.8475\pm 0.0001$		$5.5167\pm 0.0001$
$\alpha$ at SS (MHz)	-		$321.2\pm 1$
${\nu }_r$ (bare state) (GHz)	$7.5008\pm 0.0001$		$7.6611\pm 0.0001$
$Q_c$	$(9\pm 1)\times {10}^3$		$(10\pm 1)\times {10}^3$
$Q_i$	$(2.7\pm 0.3)\times {10}^4$		$(1.5\pm 0.3)\times {10}^5$
$\Delta$ (GHz)	$2.6534\pm 0.0002$		$2.1444\pm 0.0002$
g (MHz)	$112\pm 1$		$88\pm 1$
$T_1(\mathsf{\mu }$s)	$23\pm 2$		$14\pm 1$
$T_2^{Echo}(\mathsf{\mu }$s)	$44\pm 3$		$23\pm 2$
$T_2^{*}(\mathsf{\mu }$s)	$5.8\pm 0.5$ at $\delta \nu =580\pm 2$ kHz		$9.1\pm 0.6$ at $\delta \nu =554\pm 1$ kHz
$2J$ (MHz)		$29\pm 7$

, $Q_c$ is roughly one order of magnitudes smaller than $Q_i$, thus suggesting that the resonators fall in the overcoupled regime.

Two-tone spectroscopy measurements provide a first estimation of the qubit frequency [72]. In Figure 1c,d, we report an example of the two-tone pulsed spectroscopy for $Q_a$ and $Q_b$, respectively. For the former, we show the dependence of the qubit resonance frequency ${\nu }_{01}^a$ as a function of the power of the qubit-drive pulse, which is here attenuated by means of a PXIe variable attenuator card. Additional information on the room-temperature experimental setup can be found in Section 3. The results in Figure 1c are acquired at the sweet spot (SS) of the qubit. For $Q_b$, we report the dependence of the qubit resonance frequency ${\nu }_{01}^b$ as a function of the external flux. Qubit frequencies are ${\nu }_{01}^a=4.8475$ GHz and ${\nu }_{01}^b=5.5167$ GHz at the SS. Therefore, given the bare resonance frequencies of the readout resonators in Figure 1a,b, the qubit-readout frequency detuning ${\Delta }_i={\nu }_r^i-{\nu }_{01}^i$ is, for $Q_a$, ${\Delta }_a=2.65335$ GHz and, for $Q_b$, ${\Delta }_b=2.14443$ GHz. The dispersive shit $\chi$ ranges from $(1.35\pm 0.05)$ MHz to $(-0.91\pm 0.05)$ MHz for $Q_a$ and $Q_b$, respectively, thus giving readout-qubit coupling factors of $(112\pm 1)$ MHz and $(88\pm 1)$ MHz [16,72]. All these parameters are also collected in Table 1.

2.2. Relaxation and Coherence Properties

In order to investigate the performances of superconducting transmon qubits, the experimental protocol requires an estimation of the relaxation time $T_1$, and the coherence time $T_2$ and $T_2^{*}$ of the qubits. Here, the ${}^{*}$ symbol conventionally indicates the $T_2$ value sensitive to quasi-static, low-frequency fluctuations that contribute to dephasing [72].

In the relaxation measurement, we first prepare the qubit in the excited state $\left|1\right.〉$, and we measure in time the free-evolution of the qubit state. In order to prepare the qubit state in $\left|1\right.〉$, we send a qubit-drive pulse on-resonance with the qubit frequency, and with the same duration as a $\pi$-pulse [73]. This parameter is experimentally accessed through Rabi oscillations measurements [73]. As an example, we show the experimental results obtained for qubit $Q_a$ and $Q_b$ in Figure 2, in which half the period of Rabi oscillations is the $\pi$-pulse.

For what concerns the study of decoherence processes in a superconducting transmon qubit, we use Hahn-Echo and Ramsey interferometry protocols. In the Hahn-Echo sequence, we prepare the qubit on the equator of the qubit Bloch sphere by sending a $\pi /2$ qubit drive pulse, i.e., a pulse with half the amplitude of a $\pi$ pulse. Then, we apply a refocus $\pi$-pulse, in order to perform a rotation around the z-axis. After that, we send another $\pi /2$-pulse, thus driving the qubit towards the $\left|1\right.〉$ state. Finally, we measure the output signal after a waiting time that corresponds to the total Hahn-Echo sequence duration, which is gradually increased. Ramsey interferometry, instead, uses a $\pi /2$ qubit drive pulse to prepare the qubit state on the equator of the Bloch sphere [72]. After a certain evolution time, in which dephasing effects can freely move the qubit on the equator, another $\pi /2$-pulse is sent, and then the qubit state is measured within the usual pulsed readout technique [73]. The experiment is repeated by increasing the total Ramsey sequence duration [73].

The results are reported in Figure 3 for qubit $Q_a$ and for $Q_b$. Blue dots in Figure 3a,b correspond to the typical relaxation exponential decay in superconducting transmon qubits. The red curve is the exponential fit performed on the experimental data, which gives a relaxation time of $T_1^a=23\pm 2\mathsf{\mu }$s for $Q_a$ and $T_1^b=14\pm 1\mathsf{\mu }$s for $Q_b$. The order of magnitude of the relaxation times is compatible with those found in the literature for state-of-the-art transmon qubit devices [72,74]. We here stress that the Hahn-Echo measurements (black dots in Figure 3a,b) and Ramsey oscillations in Figure 3c,d have been acquired before the optimization of $\pi$ and $\pi /2$ control pulses, thus leading to: (i) a revival of the ground state population at low sequence durations; (ii) a final ground state population deviating from the expected value of $0.5$. Even though dephasing effects due to unoptimized pulses in Echo and Ramsey sequences cannot be neglected in single-qubit gate fidelity measurements, at this stage of the characterization the main goal is to provide the relaxation and coherence times in order to have a first estimation of the theoretical maximum average single qubit gate fidelities.

In the limit of no detrimental dephasing of the qubit state, the coherence time should approach the theoretical limit $T_2\sim 2T_1$ [72]. The Hahn-Echo sequence allows us to mitigate low-frequency dephasing noise to a certain extent, as reported in Refs. [72,73]. Therefore, in the ideal-case scenario of complete noise-mitigation, the values of $T_{2,Echo}$ should be consistent with the theoretical limit of $2T_1$ [72]. The exponential decay fit performed on the experimental Hahn-Echo data (orange curves in Figure 3a,b provides a Hahn-Echo decay time of $T_{2,Echo}^a=44\pm 3\mathsf{\mu }$s for $Q_a$ and $T_{2,Echo}^b=23\pm 2\mathsf{\mu }$s for $Q_b$, which is fairly close to the theoretical $2T_1$ limit. Moreover, these values are larger than $T_2^{*}$ obtained by the study of Ramsey oscillations, confirming that the Hahn-Echo sequence efficiently suppressed some low-frequency noise components, which cannot be neglected in the Ramsey experiment.

Finally, the coherence time of a superconducting transmon qubit contributes to the intrinsic limitation of the maximum single-qubit gate fidelity. Following Ref. [13], the single-qubit gate fidelity affected by uncorrelated energy relaxation with rate $T_1^{-1}$, and pure dephasing with rate $T_{\varphi }^{-1}={T_2^{*}}^{-1}-{\left(2T_1\right)}^{-1}$, reads as:

$$F=1-\frac{1}{3}\tau \left(T_1^{-1}-T_{\varphi }^{-1}\right),$$

(1)

where $\tau$ is the mean gate sequence duration. For $\tau$ of the order of a few tens of nanoseconds, the single-qubit gate fidelity expected within the experimental $T_1$ and $T_2^{*}$ are ~$99.88\%$ for $Q_a$, and of ~$99.81\%$ for $Q_b$, i.e., they are close to the three-nines $99.9\%$ golden standard. These values, in fact, promise one error in thousands of operations on the qubit, and are in line with average gate fidelity measured on state-of-the-art superconducting transmon qubits [6,74,75,76,77,78]. However, in order to reach the theoretical limit, an intensive optimization of the control pulses is required.

2.3. Control-Pulses Optimization

The experimental measurement of single-qubit gate fidelity is related to the error rate that occurs when multiple operations are performed on the qubit [13,79]. We here quantify the single-qubit gate fidelity by means of Randomized Benchmarking (RB) technique [76]. The RB protocol provides information on the single-qubit gate fidelity averaged over randomly chosen single-qubit Clifford gates [80]:

$$\mathcal{C}=\{I,X_{\pm \pi },Y_{\pm \pi },Z_{\pm \pi },X_{\pm \pi /2},Y_{\pm \pi /2},Z_{\pm \pi /2}\}.$$

(2)

Here, $X,Y,Z$ identify the qubit rotation around a particular axis of the Bloch sphere, which is obtained by means of Pauli rotations, while the subscript specifies the rotation angle. Within this technique, we are able to quantify the quality of the qubit initial state preparation, and how well we are able to control and measure the qubit state after the application of a gate. Therefore, RB intrinsically includes the possibility to disentangle the effect of State Preparation And Measurement (SPAM) errors, which are typically introduced by the user if control pulses are not properly optimized from intrinsic depolarization errors [76].

The optimization of control pulses in superconducting qubits is of fundamental importance in order to evaluate the performances of single- and multi-qubit gates. As a matter of fact, high fidelities can be achieved by reducing SPAM errors, by carefully generating the correct drive pulse, in terms of their amplitude, shape, duration and frequency [76,81,82,83].

First of all, the RF I-Q mixers typically used for upconversion mechanisms may introduce an imbalance between the I and Q signal amplitudes [28,72]. Therefore, the calibration of $\pi$- and $\pi /2$-pulses is a fundamental step prior to fidelity measurements. This is achieved by: (i) preparing the qubit in the ground state; (ii) applying $2\pi$-pulses ($4\pi /2$-pulses), thus driving the qubit again to the ground state, and then measuring the qubit state. The experiment is repeated for an increasing number of operations, and as a function of the relative amplitude of the I and Q signals. The optimal relative amplitude is obtained as the one for which the qubit returns to the ground state with the maximum amount of operations.

The same protocol can be used for the optimization of the $\pi$-pulse duration, while fine optimization of the qubit drive frequency is typically performed by Ramsey interferometry (Figure 4a). This measurement is very sensitive to the detuning between the qubit resonance and drive frequency. As a matter of fact, for drive frequencies ${\nu }_d\ne {\nu }_q$ the typical Ramsey fringes are observed, while for ${\nu }_d={\nu }_{01}$ a pure exponential decay of the qubit state is measured, with a decay rate related to qubit dephasing $T_2^{*}$ [72]. Therefore, a useful way to identify the on-resonance qubit frequency is to set the qubit drive frequency to the centre of Ramsey fringes. An example in terms of the qubit population is shown in Figure 4a. For our device, qubit frequencies reported in Table 1 are consistent with the fine qubit drive frequency optimization by using the Ramsey protocol.

Finally, a fundamental role is also played by the control pulse shape [74,84]. As a matter of fact, spurious higher-order harmonics in the upconversion setup and overshoot (undershoot) in the control pulses may move the qubit outside the computational space (leakage errors), or introduce under- or over-rotations on the Bloch sphere (phase errors) [74,84]. In this work, we used the Derivative Reduction by Adiabatic Gate (DRAG) technique, proposed in Ref. [85], and largely employed in literature for the reduction of SPAM errors [75,76,78,86,87,88,89].

The DRAG technique corrects the pulse shape $\Omega \left(t\right)$ by introducing a term depending on the derivative of the control pulse $d\Omega \left(t\right)/dt$ [74],

$${\Omega }^{\prime }\left(t\right)=\Omega \left(t\right)-i\frac{\alpha }{\Delta }\frac{d}{dt}\Omega \left(t\right),$$

(3)

where $\alpha$ is a scaling factor, and $\Delta$ is the difference between the transitions ${\nu }_{12}$ and ${\nu }_{01}$, and can be further corrected by adding a detuning parameter $\delta f$ as [74]:

$$\Delta ={\nu }_{12}-{\nu }_{01}-2\pi \delta f.$$

(4)

It has been demonstrated that the optimization of the scaling parameter $\alpha$ allows us to define a sweet spot for phase errors, while at the same time minimizing leakage errors [74]. Therefore, we focused on the optimization of the DRAG scaling parameter, keeping $\delta f=0$.

We used the All-XY technique [84] to optimize the scaling factor $\alpha$, which consists in sending to the qubit a sequence of X and Y single-qubit gates. A complete list of possible XY sequences is reported in Ref. [84]. For un-optimized control pulses, each XY sequence is characterized by a specific error syndrome sign, which can assume a positive or negative value. This corresponds to the overshoot or undershoot of the control pulses, respectively. Therefore, as proposed in Ref. [84], we have chosen to compare the readout voltage magnitude as a function of the DRAG scaling parameters for two XY sequences with opposite syndrome error signs, like $X_{\pi }-Y_{\pi /2}$ and $Y_{\pi }-X_{\pi /2}$. For the former sequence, the dependence on the error accumulated within un-optimized control pulses causes an undershooting on the final arrival state, quantified in terms of an error $-ϵ$, as depicted in Figure 4b. At the same time, the $Y_{\pi }-X_{\pi /2}$ sequence leads to an overshooting ($+ϵ$) on the final arrival state. A schematic representation of the experimental All-XY protocol in terms of drive and readout pulses is reported in Figure 4c. Intuitively, the readout voltage signal as a function of $\alpha$ has a positive slope for positive error sign, i.e., for overshoot signals, while it has a negative slope for negative error sign, i.e., for undershoot (Figure 4d). The optimal scaling factor is then found at the crossing between the two experimental curves [84].

2.4. Single-Qubit Gate Fidelities

The RB test provides an estimate of the average error rate of a qubit when applying sequences of random gates [82,90]. In the RB protocol, the qubit is first prepared in the ground state. Then, a sequence of Clifford gates, randomly chosen among the Clifford gate-set in Equation (2), followed by its inverse (or return gate) is applied. Finally, the qubit state is measured as a function of the number of Clifford gates.

In Figure 5a,b, we show the experimental results achieved on qubit $Q_a$ and $Q_b$, respectively. The black dots are obtained by averaging 25 different repetitions, characterized by a different seed number for the pseudo-random generation of Clifford gates applied. The readout signal is averaged over 1000 demodulated digitized traces. The ground state population in RB measurements typically decays from maximum probability at zero Clifford gates applied to $0.5$. In our device, while for $Q_a$ this is consistent with the experimental outcomes, for $Q_b$ the arrival state deviates from the expected value (Figure 5). This is suggesting that for $Q_b$ further optimization of the $\left|0\right.〉$–$\left|1\right.〉$ state calibration is required. Regardless of this behaviour, we can estimate the single-qubit gate fidelity, by using the fitting equation (red curve in Figure 5) [76]:

$$\mathcal{F}\left(N\right)=A+B{p}^N,$$

(5)

where the fitting parameter A depends on the dimensionality of the system, the parameter B on the SPAM errors and p is also known as depolarization error. The depolarization error p for a single-qubit system depends on the average error committed per sequence r as:

$$r=\frac{1-p}{2}.$$

(6)

Therefore, we can calculate the average gate fidelity as $F=1-r$, recovering for $Q_a$ an average single-qubit gate fidelity of $F_a=99.42\pm 0.01\%$, and for qubit $Q_b$$F_b=99.34\pm 0.02\%$.

2.5. Evidence of Two-Qubit Coupling

Within the future final goal of investigating the performances of coupled superconducting transmon qubits, and as a perspective to characterize multi-qubits gates performances on a scalable quantum processor, we here show preliminary results on the evidence of two-qubits coupling. In Figure 6, we report an experimental measurement of an Avoided Level Crossing (ALC) between $Q_a$ and $Q_b$, which use a high-frequency resonator bus as a coupling element [91,92,93]. The colour scale identifies the normalized demodulated readout voltage phase signal measured within a two-tone spectroscopy technique on $Q_b$, the highest frequency qubit. We then apply a flux bias to the SQUID in order to move the $Q_b$ frequency to be on resonance with $Q_a$ frequency. Close to the crossing point between the two frequencies, the appearance of a gap indicates the coupling between two-level system, with a strength J related to the amplitude of the gap [91,92,93,94]. Following the work in Refs. [91,92,94], we estimate a coupling energy through the symmetric and antisymmetric branches fit in Figure 6, and an energy gap $2J=29\pm 7$ MHz. This value promises two-qubit i-SWAP gate time of $t_{i-SWAP}⩾1/\left(2\pi \left(2J\right)\right)\sim 5$ ns [86,93,95], and therefore two-qubit gate fidelities of the order of >$90\%$ [96,97].

3. Materials and Methods

A circuital schematic of our sample is reported in Figure 7a. The optical micrograph of a similar sample with four coupled transmon qubits, rather than five, is reported in Figure 7b. Chip dimension and general layout are the same as in the device analyzed in this work. Each qubit is equipped with a NbTiN readout superconducting CoPlanar-Waveguide (CPW) resonator, capacitively coupled to a common chip feedline in a notch-type geometry for simultaneous read-out of multiple qubits. The specifics on the NbTiN material are reported in

Table 2. Design and materials parameters of the Quantum Processing Unit (QPU) with the two coupled split-transmon qubits, $Q_a$ and $Q_b$: the critical temperature $T_c$ of the NbTiN readout and coupling resonators, the width of the resonator w and the length gap $L_{gap}$ and the phase-velocity $v_{ph}$.

$T_c$	$14.6$ K
w	12 $\mathsf{\mu }$m
$L_{gap}$	4 $\mathsf{\mu }$m
$v_{ph}$	112 × 10${}^6$ m/s

. In order to guarantee optimal control of the qubits, each qubit is capacitively coupled to a dedicated drive line. Frequency tunability is guaranteed by a mutually inductively coupled dedicated flux line. Aluminium air bridges have been integrated in order to prevent the propagation of parasitic slot line modes in CPWs.

The sample is thermally and mechanically anchored at the mixing chamber (MXC) plate of a dry dilution refrigerator, and it is protected by infrared radiation and external magnetic fields by means of copper-plated cryo-perm screens (Figure 8). The temperature of the MXC plate is about $10$ mK. Cryogenic lines for the input feedline and the drive lines are equipped with cryogenic attenuators, low-pass and eccosorb RF filters with $10$ GHz, and >$12$ GHz cutoff frequencies, respectively. The detailed attenuation scheme for drive and input lines is reported in Figure 8, and highlighted by straight blue or red lines for qubit $Q_a$ and $Q_b$, respectively. The input line is represented by the green straight line. Cryogenic flux-bias lines are equipped with low-pass filters with $1$ GHz cutoff frequency and $30$ dB attenuation instead of $50$ dB, to efficiently flux-tune the qubit frequency (see dashed blue (red) lines in Figure 8, for qubit $Q_a$ and $Q_b$, respectively). The total attenuation scheme for the two lines share a 10–20 dB attenuator at the 4K-plate in order to reduce the current noise below the room-temperature Johnson-Nyquist noise [28]. The remaining attenuation is provided at the cold plate (CP) and the other plates in order to distribute the attenuators’ thermal load [28]. Finally, the cryogenic output line is equipped with a standard High-Electron Mobility Transistor (HEMT) amplifier at $4$ K, a couple of cryogenic isolators (with a total isolation level of $40$ dB) at the MXC plate and room-temperature RF-amplifiers.

The room-temperature electronics is a compact PXIe chassis able to both generate and measure Continuous Wave (CW) and Pulsed Wave (PW) RF signals. The former measurements use standard RF-signals generators and a Vector Network Analyzer (VNA), while the latter exploits the Heterodyne detection technique by means of Arbitrary Waveform Generators (AWGs) and I-Q mixers. The down-converted signal is measured by a PXIe digitizer with a sampling rate of $500$ MSa/s. Such a signal is processed into real and imaginary quadratures, which can be also expressed in terms of amplitude and phase. The flux tuning is provided by a dedicated DC-offset voltage source.

4. Discussion and Concluding Remarks

The characterization and the study of a planar QPU with two split-transmon qubits coupled through a high-frequency bus resonator provided fundamental knowledge on the single-qubit gate fidelities and coherence times. Despite the relaxation and coherence times measured should allow for single-qubit gate fidelities close to $99.9\%$, the measured ones deviate from the theoretical predictions of about $0.5\%$. One possible explanation may be that the optimization protocols for the control pulses need to be improved. We must note that the control pulse optimization procedures hereby used must be performed iteratively in order to find the optimal control pulse parameters [82]. In this work, we have limited the iterative tune-up to a few iterations. However, automatic calibration/optimization of control pulses becomes fundamental for a fine optimization of the average gate fidelity, thus resulting in the possibility to achieve higher performance results. Moreover, we stress here that we may need to extend the optimization procedure to other parameters, such as the DRAG detuning parameter $\delta f$, or the shape of the drive pulses. For these results, we used cosine-shaped drive pulses, but as a future perspective, it may be worth investigating the effect of the drive pulse shape on single-qubit gate fidelities [98,99,100]. Finally, we also point out that RB measurements reported have been performed by using $\pi$-pulses of the order of ~$20$ ns. We can not exclude that by increasing the speed of the implemented all-microwave gates we may achieve more favourable results.

As a future perspective, we aim in replacing analogue microwave-based control of the qubits with SFQ control electronics [54] by integrating the QPU subcircuit in a qMCM. SFQ voltage pulses with an amplitude of roughly $1$ mV and duration of the order of $2$ ps induce rotation on the Bloch sphere of an angle $\delta \theta$ [54],

$$\delta \theta =C_c{\varphi }_0\sqrt{\frac{4{\omega }_{01}{E}_c}{\hslash e^2}},$$

(7)

where $C_c$ is the SFQ controller-qubit coupling capacitance, and $E_c$ is the charging energy.

The results reported in this work represent a fundamental starting point in order to predict possible modifications on the relaxation and coherence times, and single-qubit gate fidelities, arising because of the coupling with an SFQ controller. As an example, we here discuss about the possibility to implement a flip-chip configuration [65].

In a flip-chip MCM configuration, signal lines, such as the feedline for multiplexed readout, the control and the flux lines, are located in the carrier chip, and separated from the quantum elements of the QPU. The distance between the carrier and the QPU induces variation in fundamental parameters, including the readout resonator and qubit frequencies, and therefore the dispersive detuning, the readout-qubit coupling, the qubit anharmonicity, and the control and readout quality factors. Specifically speaking, the charging energy $E_c$ in Equation (7) decreases compared with a planar architecture, as well as the qubit frequency, the readout and drive quality factors, as reported in Ref. [65].

Taking as an example the charging energy for qubit $Q_b$, which can be inspected from the qubit spectroscopy measurement in Figure 1d, we observe that the two-photons assisted $\left|0\right.〉\to \left|2\right.〉$ transition occurs at $5.35$ GHz at the SS, thus giving $E_c=327\pm 1$ MHz. In a flip-chip configuration, we will require to keep $E_c$ consistent with this value, given the fundamental request to maintain $E_J/E_c$ suitable for a transmon configuration, i.e., of the order of 50–100 [16].

Other parameters that may be affected by the integration in a flip-chip qMCM module are the readout resonator frequency, the readout-qubit detuning, and the readout-qubit coupling, which typically increase when decreasing the distance between the carrier and the QPU, if compared to the single chip limit [65]. All these parameters play a fundamental role in the relaxation and coherence times.

Relaxation in superconducting transmon qubits is most likely related to spontaneous radiative decay due to both the Purcell effect in the readout resonators and the feedline, and the radiative decay in the coplanar waveguide lines for drive and control of the qubit [16]. From the readout-qubit frequency detuning $\Delta$, the readout-qubit coupling g and the readout resonators quality factors $Q_l$, the Purcell-limited relaxation time for the readout is

$$T_{1,P}^{RO}=\frac{Q_l}{{\nu }_r}\frac{{\Delta }^2}{g^2}.$$

(8)

The radiative losses in the drive lines induce a relaxation time that depends, instead, on the drive line-qubit coupling quality factor $Q_c^d$ as

$$T_1^D=\frac{{\nu }_{01}}{Q_c^d},$$

(9)

where $Q_c^d$ is of the order of ${10}^6$–${10}^7$ [65]. Therefore, the total relaxation time is $T_{1,P}=({T_1^D}^{-1}+{T_{1,P}^{RO}}^{-1})$.

Another possible noise source that may limit relaxation times is due to dielectric losses in the materials composing the device, which gives a relaxation time $T_1^{die}$ [101]:

$$T_1^{die}=\frac{1}{{\nu }_{01}{\sum }_i{p}_i/Q_i+{\Gamma }_0},$$

(10)

where $Q_i=1/tan\delta i$ is the quality factor of the i-th material, characterized by the loss-tangent $tan\delta i$, and the participation ratio $p_i$, defined as the fraction of electric field energy stored within the volume of this material, and ${\mathsf{\Gamma }}_0$ is the relaxation rate induced by non-dielectric channels [101].

For what concerns the device analyzed in this work, we estimated the relaxation time due to Purcell and radiative losses by using the parameters experimentally measured (Table 1), and the relaxation time due to dielectric losses with the Python package scqubits [102,103], where we set the capacitive quality factor to $0.5\times {10}^6$ and the temperature to the nominal MXC temperature of $10$mK. We stress that the capacitive quality factor used represents a rough estimation of dielectric losses induced decay, given by the relation $Q={\nu }_{01}{T}_1$. With these parameters, the estimated total relaxation time due to Purcell and radiative decay through the drive lines is of the order of $700\mathsf{\mu }$s, which is one order of magnitude larger compared to the experimental ones. The relaxation time due to dielectric losses, instead, is consistent with measured $T_1$, thus possibly being the most important limitation to $T_1$ in our devices.

Let us now consider fixing the circuital design parameters of the QPU in a multi-chip module to the experimental ones measured in the planar single-chip version. Compared to the single-chip case, corresponding to an infinite separation between the two modules, both Purcell and radiative losses through the drive lines, and dielectric relaxation times are expected to depend on the distance between the carrier and the QPU d [65]. This is due to the fact that in a flip-chip configuration, all the capacitive elements in the device will depend on d [65]. Specifically, as reported in the Supplementary material of Ref. [65], the qubit self-capacitance, the drive coupling capacitance and the resonator capacitance increase when decreasing d. Therefore, the qubit frequency will effectively decrease with d, thus decreasing the radiative relaxation time for the drive lines $T_1^D$ and increasing the relaxation time due to dielectric losses $T_{1,die}$ [65]. However, even though the resonator capacitance increases when decreasing the inter-chip distance, in a flip-chip configuration the most important effect on the resonator frequency is given by the resonator inductance, which instead tends to decrease as a function of d [65]. This effectively increases the readout resonator frequency by decreasing d. As a consequence, the behaviour of the total Purcell relaxation time as a function of d is not trivial and takes into account the interplay between all these fundamental parameters [65]. In order to perform a comparison between the Purcell and radiative losses through the drive lines, and dielectric losses in a flip-chip qMCM, in Figure 9 we report a simulation of the expected $T_1$ for the two noise sources as a function of the qubit frequency, which is changed according to its dependence on the distance between the QPU and the carrier. As a reference, we have used $Q_b$ parameters (Table 1). In this scenario, while the relaxation time due to radiative losses in the drive lines and Purcell effect will decrease in the qMCM regime, the dielectric losses will likely still dominate the relaxation (Figure 9).

This suggests that we do not expect to worsen the relaxation time when integrating the QPU in a flip-chip qMCM configuration.

Additionally to the possibility to implement a flip-chip MCM configuration, it is not excluded to search for novel modular configurations [104]. DC to microwave low-losses connectivity can be provided by vertical superconducting vias through the QPU substrate. This ensures a separation between the SFQ-controller chip and the QPU, while at the same time preserving the circuit parameters, which are instead dependent on the MCM gap in the flip-chip configuration [104].

In conclusion, we have reported on an experimental investigation of single-qubit coherence and fidelity in a superconducting transmon planar QPU architecture for future integration in an MCM configuration. The reported results provide insightful predictions on the possibility to replace analogue microwave control pulses with SFQ electronics.