# Correcting Coherent Errors by Random Operation on Actual Quantum Hardware

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## Abstract

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## 1. Introduction

## 2. CPT Maps on the Bloch Sphere

## 3. Single-Qubit Quantum Process Tomography

#### 3.1. CNOT Noisy Channel

#### 3.2. Random Unitaries

## 4. Error Correction by Randomization

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Cartan’s KAK Decomposition of the Unitary Group

**Figure A1.**A quantum circuit implementing a two-qubit unitary gate using the KAK parametrization of $SU\left(4\right)$.

## Appendix B. CNOT Noisy Channel Ccoherent Errors Correction

**Figure A2.**A circuit implementing the coherent error correction for the CNOT noisy channel. Two echo steps are shown.

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**Figure 1.**Fidelities of the 4 states used to reconstruct the CPT map as a function of the number of steps (one step of the CNOT noisy channel corresponds to two CNOT gates). (

**Left**) simulations with the noise parameters of ibm_lagos, calibration of 2 November 2022. (

**Right**) actual results obtained with ibm_lagos on 2 November 2022.

**Figure 2.**Evolution of the single qubit Bloch ball as a function of the number of CNOT map steps. From left to right, the number of steps increases in correspondence with the data shown in Figure 1. (

**a**) Qiskit Simulations with the noise parameters of ibm_lagos. The segment highlighted in red shows the major axis of the ellipsoid; the gray sphere is the unit-radius Bloch ball. (

**b**) Results obtained with ibm_lagos. The segment highlighted in green shows the major axis of the ellipsoid; the gray sphere is unit-radius Bloch ball. Data from the quantum processor, taken on 2 November 2022, with the corresponding calibration parameters used for Qiskit simulations.

**Figure 3.**As in Figure 1, but for random unitaries. Qiskit (

**left**) and actual hardware (

**right**) data were obtained with ibm_lagos on 2 November 2022.

**Figure 5.**Fidelities of the 4 basis states as a function of the number of steps (one step in the CNOT noisy channel corresponds to two CNOT gates) after the correction procedure described above. On the left, the computational basis is rotated with random unitaries; that on the right undergoes single-axis rotations. The purple curve represents the average fidelities obtained without correction. Results obtained with $ibm\_lagos$ on 20 November 2022.

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**MDPI and ACS Style**

Cenedese, G.; Benenti, G.; Bondani, M.
Correcting Coherent Errors by Random Operation on Actual Quantum Hardware. *Entropy* **2023**, *25*, 324.
https://doi.org/10.3390/e25020324

**AMA Style**

Cenedese G, Benenti G, Bondani M.
Correcting Coherent Errors by Random Operation on Actual Quantum Hardware. *Entropy*. 2023; 25(2):324.
https://doi.org/10.3390/e25020324

**Chicago/Turabian Style**

Cenedese, Gabriele, Giuliano Benenti, and Maria Bondani.
2023. "Correcting Coherent Errors by Random Operation on Actual Quantum Hardware" *Entropy* 25, no. 2: 324.
https://doi.org/10.3390/e25020324