# Characterization of a Transmon Qubit in a 3D Cavity for Quantum Machine Learning and Photon Counting

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Transmon Fabrication

^{2}was used for the capacitors, while the junctions were written with a smaller 15 $\mathsf{\mu}$m aperture and a dose of 112.5 $\mathsf{\mu}$C/cm

^{2}. Post exposure, the samples were developed using Allresist AR 600-546 and Kayaku diluted 101A for the top and bottom layers, respectively.

^{−8}mbar of pressure. The evaporation was performed at a 55° tilt angle using the Manhattan approach. In total, 35 nm of Al were deposited at a rate of 0.6 nm/s, followed by a static oxidation at 0.625 mbar for 25 min, and a second Al evaporation of 55 nm.

^{2}and the two antenna pads separated by 20 $\mathsf{\mu}$m are 556 $\mathsf{\mu}$m long and 144 $\mathsf{\mu}$m wide.

#### 2.2. Experimental Setup for Transmon Characterization

## 3. Results

#### 3.1. Transmon Spectroscopic Characterization

#### 3.2. Time Domain Transmon Characterization

## 4. Simulation

#### 4.1. Capacitance

^{M}which is obtained from the ANSYS-Q3D extraction tool. A schematic of the capacitance network of the device is shown in Figure 8. The components of the 2 × 2 matrix are given by:

#### 4.2. Dipole Coupling

#### 4.3. Relaxation Time ${T}_{1}$

## 5. Fit to the $\mathit{u}$-Quark Parton Distribution Function of the Proton with a Superconducting Transmon Qubit in a 3D Cavity

`Qibo`framework, with its modular structure, is exploited to develop and test pure quantum full-stack algorithms [42,43,44,45].

`Qibo`[46,47] is a full stack open-source middleware framework for quantum computing. The

`Qibo`suite includes full-state vector simulators, which have been shown to be compatible with the state-of-the-art [48,49] and several tools dedicated to quantum control and quantum calibration [50,51]. Quantum control is implemented through a dedicated backend,

`Qibolab`, able to provide control over a different set of electronics including Radio Frequency Systems on Chip (RFSoC) [52].

`Qibolab`[53] also provides primitives to compile and transpile quantum circuits. The calibration and characterization of QPUs is delegated to

`Qibocal`[54], a

`Qibo`module which includes several pre-coded experiments necessary to fine-tune calibration parameters, reporting tools, and methods to automatically update QPU parameters. In this particular setup, the qubit is controlled with a RFSoC 4 × 2 through

`Qibolab`and the calibration was performed using

`Qibocal`.

## 6. Measurement Protocol for a Low Dark-Count Photon Detector with Two Qubits

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Average Photon Number inside the Cavity

**Figure A1.**Qubit absorption spectra peaks intensity distribution extracted from the data reported in Figure 4 as a function of the Fock state number. Left, ${\mathrm{P}}_{probe}$ = −103 dbm, right, ${\mathrm{P}}_{probe}$ = −98 dbm. The black dots are the experimental data, the broken line is a fit performed using a Poisson distribution. The y axis is in logarithmic scale.

## Appendix B. Quantum Treatment of LC+Transmission Line

## Appendix C. Capacitance Matrix and Total Capacitance

## References

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**Figure 1.**(

**Left**): Al cavity hosting the transmon chip. (

**Right**): optical image of the transmon shunt capacitance pads acquired with a 100× magnification. The JJ is not observable since it is roughly $200\times 200$ nm

^{2}, but it is located between the pads, in proximity of the two observable metal extensions.

**Figure 3.**Cavity power scan. The high power feature (B) corresponds to the bare cavity transmission peak. The low power feature (A) is the transmission peak of the dressed cavity-qubit system with the qubit in the ground state.

**Figure 4.**Qubit spectroscopy of individually resolved photon numbers inside the cavity, for an average photon population of $\overline{N}$ = 1.8 (

**left**, ${P}_{probe}=-102$ dbm) and $\overline{N}$ = 4.1 (

**right**, ${P}_{probe}=-98$ dbm). Each peak is separated by 2$\chi /2\pi =-6.82\pm 0.16$ MHz.

**Figure 5.**(

**a**) measurement scheme for the Rabi spectroscopy. (

**b**) Chevron plot, acquired with excitation power P = $-93$ dbm. The y-axis is given as detuning from the resonance frequency of 6.4194 GHz. (

**c**) Rabi oscillation dependence from the excitation tone power. Excitation frequency = 6.4194 GHz. (

**d**) Linear dependence of the Rabi frequency from the square root of the average photon number. The data have been fit with a straight line angular coefficient $6.6\times {10}^{-3}$ MHz

^{−1}and intercept $2.6\times {10}^{-4}$ MHz

^{−1}.

**Figure 6.**(

**Left**): Ground state population (P

_{g}) as a function of the delay time between excitation pulse and qubit readout. ${\nu}_{excitation}=6.4194$ GHZ, ${P}_{excitation}=-93$ dbm. (

**Right**): Ramsey spectroscopy, 600 KHz detuned. ${\nu}_{excitation}=6.42$ GHZ, ${P}_{excitation}=-93$ dbm.

**Figure 7.**(

**Left**): Model in Ansys HFSS of the resonance cavity with the transmon qubit at its center. The TE110 cavity mode, which is influenced by the transmon, is also shown. (

**Right**): Zoom in on the transmon with its charge distribution and the local electric field.

**Figure 8.**Scheme of the capacitances and the charge density distribution of the transmon qubit. The two rectangles represent the upper pad (

**up**) and the bottom pad (

**down**) of the qubit. The charge of the pads is ${q}_{up}=-{q}_{down}=q$ for the symmetry of the system, with a charge distribution $\rho \left(\overrightarrow{r}\right)={\rho}_{up}\left(\overrightarrow{r}\right)+{\rho}_{down}\left(\overrightarrow{r}\right)$. The green rectangle represents an infinite ground plane. The capacitances between the pads and this plane are ${C}_{\mathrm{up}}$ and ${C}_{\mathrm{down}}$, while the capacitance between the two pads is ${C}_{\mathrm{pads}}$.

**Figure 9.**(

**Left**), fit of the u-quark Parton Distribution Function values performed using a Variational Quantum Circuit trained with gradient descent on a self-hosted single-qubit superconducting device. The estimation of the PDF value for each point is calculated as the average of ${N}_{\mathrm{runs}}=50$ predictions computed using the trained qubit. The uncertainty intervals are computed instead using one (dark orange) and two (light orange) standard deviations from the mean considering the same ${N}_{\mathrm{runs}}$ predicted values. (

**Right**), Mean Squared Error (MSE) values as function of the optimization iterations.

**Figure 10.**Phase shift for the reflected photon as a function of the state of the qubits. To build this graph, we assumed a resonator frequency of 5 GHz.

Variables | Values |
---|---|

$\chi /2\pi $ [MHz] | −3.41± 0.08 |

${\chi}_{01}/2\pi $ [MHz] | −10.2 ± 0.2 |

${\chi}_{12}/2\pi $ [MHz] | −13.6 ± 0.3 |

$\alpha $ [MHz] | 421 ± 84 |

${g}_{01}/2\pi $ [MHz] | 92.5 ± 1; 75 ± 12 |

C [fF] | 46 ± 5 |

${T}_{1}$ [$\mathsf{\mu}$s] | 8.68 ± 0.72 |

${T}_{2}$ [$\mathsf{\mu}$s] | 2.30 ± 0.11 |

${T}_{\varphi}$ [$\mathsf{\mu}$s] | 2.65 ± 0.15 |

${L}_{J}$ [nH] | 13 ± 2 |

${I}_{C}$ [nA] | 24.7 ± 1.3 |

Variables | Values |
---|---|

${P}_{tot}$ | 4.4 $\times {10}^{-4}$ |

${T}_{purcell}$ [$\mathsf{\mu}$s] | 156 |

${T}_{int}$ [$\mathsf{\mu}$s] | 57 |

${T}_{1}$ [$\mathsf{\mu}$s] | 42 |

${C}_{tot}$ [fF] | 56 |

${g}_{01}/2\pi $ [MHz] | 97 |

Parameter | ${N}_{\mathrm{train}}$ | ${N}_{\mathrm{params}}$ | Optimizer | ${N}_{\mathrm{shots}}$ | ${\mathrm{MSE}}_{\mathrm{final}}$ | Inst. | ${T}_{\mathrm{exe}}$ |

Value | 30 | 14 | Adam | 250 | $3.6\times {10}^{-3}$ | ZCU111 | ${78}^{\prime}$ |

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D’Elia, A.; Alfakes, B.; Alkhazaleh, A.; Banchi, L.; Beretta, M.; Carrazza, S.; Chiarello, F.; Di Gioacchino, D.; Giachero, A.; Henrich, F.;
et al. Characterization of a Transmon Qubit in a 3D Cavity for Quantum Machine Learning and Photon Counting. *Appl. Sci.* **2024**, *14*, 1478.
https://doi.org/10.3390/app14041478

**AMA Style**

D’Elia A, Alfakes B, Alkhazaleh A, Banchi L, Beretta M, Carrazza S, Chiarello F, Di Gioacchino D, Giachero A, Henrich F,
et al. Characterization of a Transmon Qubit in a 3D Cavity for Quantum Machine Learning and Photon Counting. *Applied Sciences*. 2024; 14(4):1478.
https://doi.org/10.3390/app14041478

**Chicago/Turabian Style**

D’Elia, Alessandro, Boulos Alfakes, Anas Alkhazaleh, Leonardo Banchi, Matteo Beretta, Stefano Carrazza, Fabio Chiarello, Daniele Di Gioacchino, Andrea Giachero, Felix Henrich,
and et al. 2024. "Characterization of a Transmon Qubit in a 3D Cavity for Quantum Machine Learning and Photon Counting" *Applied Sciences* 14, no. 4: 1478.
https://doi.org/10.3390/app14041478