# Investigating the Individual Performances of Coupled Superconducting Transmon Qubits

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Read-Out Resonator and Qubit Spectroscopy

#### 2.2. Relaxation and Coherence Properties

#### 2.3. Control-Pulses Optimization

#### 2.4. Single-Qubit Gate Fidelities

#### 2.5. Evidence of Two-Qubit Coupling

## 3. Materials and Methods

## 4. Discussion and Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**Readout resonator and qubit spectroscopy for superconducting coupled transmon qubits ${Q}_{a}$–${Q}_{b}$. In (

**a**), readout cavity of ${Q}_{a}$ and low-photon shift; in (

**b**) for ${Q}_{b}$. Scatter-line plots represent the readout response to large ($-70$ dBm on the device-under-test, corresponding to around 6–8 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{6}$ photons, in red) input power signal, corresponding to the resonators bare-state, and in the low-power regime ($-130$ dBm on the device-under-test, corresponding to tens of photons, in blue). Yellow dashed and green dash-dotted curves correspond to the resonator fit performed with an open-source traceable Python fit routine [71] for the estimation of quality factors in the bare and low-photon states (Table 1), respectively. In (

**c**), ${Q}_{a}$ pulsed spectroscopy as a function of the qubit-drive attenuation, which adds to roughly $-70$ dB drive input power on the device-under-test and the qubit-drive frequency. In (

**d**) ${Q}_{b}$ two-tone pulsed two-tone spectroscopy as a function of the flux and the qubit-drive frequency at fixed qubit drive power on the device-under-test of roughly $-80$ dB. Please note that the readout resonator frequency has been kept fixed during the measurement, thus causing lower resolution of the qubit frequency peaks far from the flux sweet spot.

**Figure 2.**In (

**a**,

**b**), Rabi oscillations measured for ${Q}_{a}$ and ${Q}_{b}$, respectively. Data points correspond to the population of the qubit ground state as a function of the duration of the qubit-drive pulse. The red curve represents the Rabi fit, which gives the $\pi $-pulse duration reported in the legends.

**Figure 3.**Relaxation, Hahn-Echo and Ramsey fitting for qubit ${Q}_{a}$ and ${Q}_{b}$. In (

**a**,

**b**), black (blue) scatter points correspond to the qubit state population as a function of the ${T}_{1}$ (${T}_{2}^{Echo}$) sequence duration. Straight orange (red) curves are the fit functions used for the estimation of ${T}_{1}$ (${T}_{2}^{Echo}$) time, reported also in the legend. In (

**c**,

**d**), Ramsey oscillations fit for the estimation of ${T}_{2}^{*}$ for ${Q}_{a}$ and ${Q}_{b}$, respectively. Black scatter points correspond to the excited state population as a function of the Ramsey sequence duration, while red lines correspond to the fit. Please note that the ${\chi}^{2}$-minimization of the fitting procedure compensates for the higher ground state population for long sequence durations in the Ramsey oscillations, due to unoptimized $\pi $ and $\pi /2$ control pulses, with a non-physical ground state probability >1.

**Figure 4.**Example of optimization protocol for Randomized Benchmarking on qubit ${Q}_{b}$. In (

**a**), Ramsey fringes for qubit ${Q}_{b}$ from interferometry experiment, fundamental for the optimization of qubit drive frequency signal. In (

**b**), an example of a Derivative-Reductive-Adiabatic-Gate calibration with the All-XY technique on ${Q}_{b}$, fundamental for the optimization of qubit drive pulses shape. In (

**c**), All-XY protocol scheme. In (

**d**), experimental demodulated voltage as a function of the DRAG scaling parameter $\alpha $ within the All-XY technique.

**Figure 5.**Randomized Benchmarking measurements for single-qubit average gate fidelity F estimation for qubit ${Q}_{a}$ and ${Q}_{b}$. Black scatter data corresponds to a real part of the demodulated readout voltage signal as a function of the number of Clifford gates in the randomized benchmarking sequence, averaged over a number of random seeds ${n}_{seed}$: in (

**a**), ${n}_{seed}=25$ and in (

**b**), ${n}_{seed}=26$, respectively. The error bars represent the standard deviation of the measured data among different repetitions. Straight red curves are the fit functions used for the estimation of the single-qubit randomized benchmarking average gate fidelity F, which is $99.43\%$ for ${Q}_{a}$ and $99.34\%$ for ${Q}_{b}$.

**Figure 6.**Avoided Level Crossing between ${Q}_{a}$ and ${Q}_{b}$. The colour bar represents the normalized demodulated voltage readout signal $(V-\u2329V\u232a)/{\sigma}_{V}$, where $\u2329V\u232a$ is the average magnitude voltage and ${\sigma}_{V}$ is the standard deviation, obtained for a two-tone pulsed spectroscopy measurement of ${Q}_{b}$ on resonance with ${Q}_{a}$. The qubit ${Q}_{b}$ is put on resonance with ${Q}_{a}$ by using an external flux coupled to the DC-SQUID in the split-transmon. The red curve represents the Avoided Level Crossing fit reported in Refs. [91,92,94], with which we extract a qubit-qubit coupling energy $2J=29\pm 7$ MHz.

**Figure 7.**In (

**a**), circuit schematics of the coupled superconducting transmon qubits ${Q}_{a}$ (in blue) and ${Q}_{b}$ (in black). The two qubits are symmetric split-transmons, coupled by means of a high-frequency resonator bus coupler (in red). Read-out is performed through read-out resonators, capacitively coupled to a common feedline for multiplexing (dark blue). External flux is applied through inductively coupled superconducting flux lines on the chip, while control is done by capacitively coupled dedicated drive lines (not shown). In (

**b**), an example of an optical microscope picture of a similar sample with four qubits coupled through high-frequency bus resonators. Sample geometry and chip dimensions are the same for both devices.

**Figure 8.**Schematics of the room-temperature and cryogenic electronics for two-qubit characterization. In blue (red), ${Q}_{a}$ (${Q}_{b}$) setup drive lines. Dashed blue (red) lines are related to flux-bias circuitry. In green, input and output lines schematics. The down-conversion is obtained through a single-sideband three-port mixer, while the up-conversion for both drive and readout input signals is provided by an I-Q mixer.

**Figure 9.**Simulations of the total relaxation time due to Purcell and radiative decay in the drive lines ${T}_{1,P}$ and the dielectric losses induced relaxation time ${T}_{1,d}$ for ${Q}_{b}$ in case of a flip chip configuration. The thickness of the lines represents the error on the estimated values of the order of $30\%$. The data are reported as a function of the qubit frequency, and the distance between the QPU and the carrier in the flip-chip qMCM module.

**Table 1.**Summary of the Quantum Processing Unit (QPU) with the two coupled split-transmon qubits, ${Q}_{a}$ and ${Q}_{b}$: the $\left|0\right.\u232a\to \left|1\right.\u232a$ 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) | $7.5008\pm 0.0001$ | $7.6611\pm 0.0001$ | |

${Q}_{c}$ | $(9\pm 1)\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | $(10\pm 1)\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{3}$ | |

${Q}_{i}$ | $(2.7\pm 0.3)\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ | $(1.5\pm 0.3)\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{5}$ | |

$\Delta $ (GHz) | $2.6534\pm 0.0002$ | $2.1444\pm 0.0002$ | |

g (MHz) | $112\pm 1$ | $88\pm 1$ | |

${T}_{1}\phantom{\rule{0.277778em}{0ex}}(\mathsf{\mu}$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) | $29\pm 7$ |

**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 |

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Ahmad, H.G.; Jordan, C.; van den Boogaart, R.; Waardenburg, D.; Zachariadis, C.; Mastrovito, P.; Georgiev, A.L.; Montemurro, D.; Pepe, G.P.; Arthers, M.;
et al. Investigating the Individual Performances of Coupled Superconducting Transmon Qubits. *Condens. Matter* **2023**, *8*, 29.
https://doi.org/10.3390/condmat8010029

**AMA Style**

Ahmad HG, Jordan C, van den Boogaart R, Waardenburg D, Zachariadis C, Mastrovito P, Georgiev AL, Montemurro D, Pepe GP, Arthers M,
et al. Investigating the Individual Performances of Coupled Superconducting Transmon Qubits. *Condensed Matter*. 2023; 8(1):29.
https://doi.org/10.3390/condmat8010029

**Chicago/Turabian Style**

Ahmad, Halima Giovanna, Caleb Jordan, Roald van den Boogaart, Daan Waardenburg, Christos Zachariadis, Pasquale Mastrovito, Asen Lyubenov Georgiev, Domenico Montemurro, Giovanni Piero Pepe, Marten Arthers,
and et al. 2023. "Investigating the Individual Performances of Coupled Superconducting Transmon Qubits" *Condensed Matter* 8, no. 1: 29.
https://doi.org/10.3390/condmat8010029