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A Modified Depolarization Approach for Efficient Quantum Machine Learning

Bikram Khanal, Pablo Rivas

TL;DR

The paper tackles the computational burden of simulating depolarizing noise on NISQ devices by introducing a reduced two-Kraus depolarization channel that uses only Pauli $X$ and $Z$ operations. The authors derive the equivalent representation $\\rho_{m}'=(1-\\frac{2p}{3})\\rho+\\frac{2p}{3}Z((\\rho X)^T X)Z$ with Kraus $K_0=\\sqrt{1-\\frac{2p}{3}}I$ and $K_1=i\\sqrt{\\frac{2p}{3}} ZX$, proving equivalence to the standard model and deriving a first-order expansion for repeated applications. They validate the approach experimentally on a Quantum Machine Learning task with the Iris dataset, showing consistency with the standard channel for small $p$ and demonstrating that a Rotational encoding with variational layers maintains accuracy up to intermediate circuit depths while mitigating noise-related degradation. The results indicate substantial computational savings—fewer multiplications and no $Y$-gate usage—facilitating scalable depolarization simulations and robust QML training in the NISQ era. Overall, the work provides a practical, resource-efficient noise model that preserves essential dynamics and supports more scalable quantum learning on noisy hardware.

Abstract

Quantum Computing in the Noisy Intermediate-Scale Quantum (NISQ) era has shown promising applications in machine learning, optimization, and cryptography. Despite the progress, challenges persist due to system noise, errors, and decoherence that complicate the simulation of quantum systems. The depolarization channel is a standard tool for simulating a quantum system's noise. However, modeling such noise for practical applications is computationally expensive when we have limited hardware resources, as is the case in the NISQ era. We propose a modified representation for a single-qubit depolarization channel with two Kraus operators based only on X and Z Pauli matrices. Our approach reduces the computational complexity from six to four matrix multiplications per execution of a channel. Experiments on a Quantum Machine Learning (QML) model on the Iris dataset across various circuit depths and depolarization rates validate that our approach maintains the model's accuracy while improving efficiency. This simplified noise model enables more scalable simulations of quantum circuits under depolarization, advancing capabilities in the NISQ era.

A Modified Depolarization Approach for Efficient Quantum Machine Learning

TL;DR

The paper tackles the computational burden of simulating depolarizing noise on NISQ devices by introducing a reduced two-Kraus depolarization channel that uses only Pauli and operations. The authors derive the equivalent representation with Kraus and , proving equivalence to the standard model and deriving a first-order expansion for repeated applications. They validate the approach experimentally on a Quantum Machine Learning task with the Iris dataset, showing consistency with the standard channel for small and demonstrating that a Rotational encoding with variational layers maintains accuracy up to intermediate circuit depths while mitigating noise-related degradation. The results indicate substantial computational savings—fewer multiplications and no -gate usage—facilitating scalable depolarization simulations and robust QML training in the NISQ era. Overall, the work provides a practical, resource-efficient noise model that preserves essential dynamics and supports more scalable quantum learning on noisy hardware.

Abstract

Quantum Computing in the Noisy Intermediate-Scale Quantum (NISQ) era has shown promising applications in machine learning, optimization, and cryptography. Despite the progress, challenges persist due to system noise, errors, and decoherence that complicate the simulation of quantum systems. The depolarization channel is a standard tool for simulating a quantum system's noise. However, modeling such noise for practical applications is computationally expensive when we have limited hardware resources, as is the case in the NISQ era. We propose a modified representation for a single-qubit depolarization channel with two Kraus operators based only on X and Z Pauli matrices. Our approach reduces the computational complexity from six to four matrix multiplications per execution of a channel. Experiments on a Quantum Machine Learning (QML) model on the Iris dataset across various circuit depths and depolarization rates validate that our approach maintains the model's accuracy while improving efficiency. This simplified noise model enables more scalable simulations of quantum circuits under depolarization, advancing capabilities in the NISQ era.
Paper Structure (11 sections, 4 theorems, 40 equations, 3 figures)

This paper contains 11 sections, 4 theorems, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Derivation
  3. Standard Expression of the Depolarizing Channel
  4. Alternative Expression of the Depolarizing Channel
  5. Experiment
  6. Quantum Circuit behavior analysis under Depolarization channel up to m times
  7. Data Encoding
  8. Variational Layers
  9. Training
  10. Discussions
  11. Conclusion

Key Result

Theorem 1

Eq. (eq:generalDepolarize) and (eq:modifiedDepolarize) are equivalent.

Figures (3)

  • Figure 1: 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) and plot (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 2: 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.
  • Figure 3: 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 that highlight the impact of noise rate and circuit depth on learning efficacy.

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1