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Characterizing the Reproducibility of Noisy Quantum Circuits^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### Example

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NISQ | noisy intermediate scale quantum |

SPAM | state preparation and measurement |

## Appendix A. Samples Required for Reliable Estimation of n-Dimensional Distribution

## Appendix B

**Lemma**

**A1.**

**Proof.**

**Figure A1.**Schematic of the ibmq_toronto device, produced by IBM. Circles represent register elements, while edges denote the connectivity for performing two qubit operations.

**Figure A2.**Fidelity distributions for computational basis states of ibmq_toronto for qubits 9–17. Raw data for ibmq_toronto, collected on 8 April 2021, between 8:00–10:00 p.m. (UTC-05:00).

**Figure A3.**Fidelity distributions for computational basis states of ibmq_toronto for qubits 18–26. Raw data for ibmq_toronto, collected on 8 April 2021, between 8:00–10:00 p.m. (UTC-05:00).

**Figure A4.**Register variation of the experimentally-obtained Hellinger distance. Each of the 27 register elements was used to verify Equation (36) for $n=1$. The dotted red line denotes the register mean for Hellinger distance (averaged over all qubits). Qubit 19 is the closest to ideal, while qubit 24 is the farthest. The error bars show the standard deviation of the population mean across $L=203$ experiments. A consistent register color scheme has been used for all the figures.

**Figure A5.**Each of the 27 register elements was used to verify Equation (36) for $n=1$. The dotted red line denotes the register mean for readout asymmetry (averaged over all qubits). Qubit 0 has the best performance for this parameter, while qubit 24 was the worst. The error bars show the standard deviation of the population mean across L = 203 experiments. A consistent register color scheme has been used for all the figures.

**Figure A6.**Register variation of the derived parameter ${\gamma}_{D}\left(\mathcal{C}\right)$. A high ${\gamma}_{D}\left(\mathcal{C}\right)$ adversely impacts the computational reproducibility. However, low ${\gamma}_{D}\left(\mathcal{C}\right)$ does not necessarily mean that the device is close to perfect because the terms in Equation (30) can cancel each other out and lead to an improvement in the output distance. For an unstable device, ${\gamma}_{D}\left(\mathcal{C}\right)$ will vary with time and must be re-estimated. The dotted red line denotes the register mean averaged over all qubits. Qubit 16 has the best performance for this composite parameter, while qubit 24 has the worst. A consistent register color scheme has been used for all the figures.

Register # | ${\mathit{\gamma}}_{\mathbf{max}}$ | ${\mathit{\gamma}}_{\mathit{D}}\left(\mathit{\tau}\right)$ | Register # | ${\mathit{\gamma}}_{\mathbf{max}}$ | ${\mathit{\gamma}}_{\mathit{D}}\left(\mathit{\tau}\right)$ |
---|---|---|---|---|---|

0 | 1.4590 | 1.3040 | 14 | 11.9104 | 11.9038 |

1 | 1.1365 | 0.6755 | 15 | 2.0713 | 2.0228 |

2 | 2.7284 | 2.7118 | 16 | 1.3392 | 0.2359 |

3 | 6.9946 | 6.9931 | 17 | 4.8557 | 4.8553 |

4 | 4.3229 | 4.3226 | 18 | 1.5986 | 1.4980 |

5 | 5.8171 | 5.8157 | 19 | 1.0322 | 0.4378 |

6 | 4.5425 | 4.5325 | 20 | 9.0893 | 9.0886 |

7 | 2.6946 | 2.6649 | 21 | 1.2259 | 1.0620 |

8 | 5.5066 | 5.4724 | 22 | 10.9146 | 10.9136 |

9 | 8.9672 | 8.9666 | 23 | 3.0018 | 3.0017 |

10 | 2.7272 | 2.7231 | 24 | 14.1254 | 14.1241 |

11 | 11.5502 | 11.5486 | 25 | 1.4325 | 1.2624 |

12 | 3.2212 | 3.0797 | 26 | 1.3103 | 0.9567 |

13 | 1.7818 | 0.6460 |

**Figure A7.**Plot of readout asymmetry vs. observed Hellinger distance for $L=203$ experiments. Each register element is shaded by a different color.

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**Figure 1.**Circuit used for our experiment. In this figure, H represents the Hadamard gate. The meter symbol denotes measurement gate.

**Figure 2.**The quantum channel maps $|0\rangle $ and $|1\rangle $ to their expected outcomes, with probabilities ${f}_{0}$ and ${f}_{1}$, respectively.

**Figure 3.**Fidelity distributions for computational basis states of ibmq_toronto for qubits 0–8. Raw data for ibmq_toronto, collected on 8 April 2021, between 8:00–10:00 p.m. (UTC-05:00). The remaining register elements are shown in the Appendix B in Figure A2 and Figure A3.

**Figure 4.**Register variation of the experimentally-obtained Hadamard gate error (in degrees). Each of the 27 register elements was used to verify Equation (36) for $n=1$. The dotted red line denotes the register mean for the gate error (averaged over all qubits). Qubit 21 is the closest to ideal, while qubit 24 is the farthest. A consistent register color scheme has been used for all the figures.

**Figure 5.**Characterizing circuit reproducibility on ibmq_toronto. Plot of ${\gamma}_{\mathrm{max}}$ (dashed line) and ${\gamma}_{D}$ (blue dots) for ibmq_toronto on 8 April 2021, when $\delta $ is set to be the observed Hellinger distance $\left(d\right)$. Equation (8) should hold for all $\delta $ and hence it must hold for the corner case when $\delta $ is set to be the experimentally observed Hellinger distance. The blue dots are experimentally-observed data plotted using the characterization data versus the actual observed Hellinger distance $\left(d\right)$ for each register element. Only a subset of qubits are shown. See Table A1 for all 27 qubits.

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Dasgupta, S.; Humble, T.S.
Characterizing the Reproducibility of Noisy Quantum Circuits. *Entropy* **2022**, *24*, 244.
https://doi.org/10.3390/e24020244

**AMA Style**

Dasgupta S, Humble TS.
Characterizing the Reproducibility of Noisy Quantum Circuits. *Entropy*. 2022; 24(2):244.
https://doi.org/10.3390/e24020244

**Chicago/Turabian Style**

Dasgupta, Samudra, and Travis S. Humble.
2022. "Characterizing the Reproducibility of Noisy Quantum Circuits" *Entropy* 24, no. 2: 244.
https://doi.org/10.3390/e24020244