# A Modified Depolarization Approach for Efficient Quantum Machine Learning

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## Abstract

**:**

## 1. Introduction

#### 1.1. Contribution

- Depolarization Channel Representation: We propose a modified representation of the depolarization channel for single-qubit quantum states.
- Kraus Operators Configuration: The proposed method contains only two Kraus operators.
- Pauli Matrices Utilization: Unlike the standard approach that uses three Pauli matrices, our channel only uses two, X and Z, Pauli matrices.
- Computational Efficiency: The proposed representation reduces the computational complexity from six to four matrix multiplications for each channel execution.
- Theoretical Verification: We rigorously prove the validity of our proposed Kraus operators and the modified channel.
- Experimental Validation: We empirically tested the proposed channel representation using a QML model on the Iris dataset. We evaluated the model performance across various circuit depths and depolarization rates.

#### 1.2. Organization

## 2. Preliminaries

#### 2.1. Quantum State and Quantum Gates

#### 2.2. Depolarization Channel Representation

#### 2.3. Quantum Machine Learning

## 3. Derivation

#### Alternative Expression of the Depolarizing Channel

**Proof.**

**Theorem**

**2.**

**Proof.**

**Proof.**

**Lemma**

**1.**

## 4. Experiment

#### 4.1. Quantum Circuit Behavior Analysis under Depolarization Channel up to m Times

#### 4.2. Data Encoding

#### 4.3. Variational Layers

#### 4.4. Training

## 5. Discussions

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**An arbitrary single qubit quantum circuit starting at $|0\rangle $, applying a Hadamard gate, followed by a sequence of unspecified single quantum gates, then a Pauli-X gate, and finally measurement.

**Figure 2.**Scatter plots present the difference between the standard channel and modified depolarization channel expectation value. Each channel was applied to a quantum circuit with single qubit gates of 3, 8, and 15, respectively. The result for 3 single qubit gates is presented in plot (

**a**), while plot (

**b**,

**c**) represent the results for 8 and 15 gates circuits, respectively. The x-axis of each plot represents the number of times the noisy channel was applied and is given by m, while the y-axis gives the varying depolarization rates.

**Figure 3.**Feature Mapping of the Iris dataset using Amplitude Encoding and Rotational encoding method. The Rotational encoding scheme, a combination of $RX$ and $RY$, provides better mapping results for the classification problem. The red color represents Class 1, the blue color represents Class 2, and the green color represents Class 3. (

**a**) Bloch Sphere representation of the quantum states obtained by Amplitude encoding of the features vectors. (

**b**) Bloch Sphere representation of the quantum states obtained by Angle encoding of the features vectors.

**Figure 4.**Various encoding schemes for single qubits using the rotational encoding. The combination of $RZ$ and $RX$ gates provides the best mapping for binary classification. The red color represents Class 1, the blue color represents Class 2, and the green color represents Class 3.

**Figure 5.**Experimental results for decision boundary evolution presented in the right column and training dynamics in the left column for a QML model on the Iris dataset, with varied noise levels (p) and depolarization channel applied up to (m) times. The decision boundaries are plotted for depths of $1,3,5,10,$ and 15, at noise levels ranging from $0.0$ to $0.5.$ The results across rows are presented in chronological order in circuit depth. Accuracy and loss graphs display the model’s performance over 30 epochs, highlighting the impact of noise rate and circuit depth on learning efficacy.

Gate Count | Mean Difference | Standard Deviation | % of Differences above 0.001 |
---|---|---|---|

3 | 0 | 0 | 0% |

8 | 0.00023 | 0.00038 | 6% |

15 | 0.00065 | 0.00106 | 22% |

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**MDPI and ACS Style**

Khanal, B.; Rivas, P.
A Modified Depolarization Approach for Efficient Quantum Machine Learning. *Mathematics* **2024**, *12*, 1385.
https://doi.org/10.3390/math12091385

**AMA Style**

Khanal B, Rivas P.
A Modified Depolarization Approach for Efficient Quantum Machine Learning. *Mathematics*. 2024; 12(9):1385.
https://doi.org/10.3390/math12091385

**Chicago/Turabian Style**

Khanal, Bikram, and Pablo Rivas.
2024. "A Modified Depolarization Approach for Efficient Quantum Machine Learning" *Mathematics* 12, no. 9: 1385.
https://doi.org/10.3390/math12091385