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Learning Robust Observable to Address Noise in Quantum Machine Learning

Bikram Khanal, Pablo Rivas

TL;DR

The paper addresses the instability of quantum machine learning models in the presence of noise on NISQ devices by proposing a framework to learn observables that remain invariant under noisy channels. It combines a theoretical invariant condition expressed via Kraus operators with a practical machine-learning approach to optimize observables across multiple circuits and noise models. Key contributions include a toy Bell-state example demonstrating a robust observable, a learning protocol with a parameter-shift gradient across six circuits and five channels, and empirical statistics showing widespread robustness of learned observables. The work suggests that noise-robust observables can enhance the reliability of QML measurements without extra quantum resources, offering a practical pathway to more stable QML in realistic noisy environments.

Abstract

Quantum Machine Learning (QML) has emerged as a promising field that combines the power of quantum computing with the principles of machine learning. One of the significant challenges in QML is dealing with noise in quantum systems, especially in the Noisy Intermediate-Scale Quantum (NISQ) era. Noise in quantum systems can introduce errors in quantum computations and degrade the performance of quantum algorithms. In this paper, we propose a framework for learning observables that are robust against noisy channels in quantum systems. We demonstrate that it is possible to learn observables that remain invariant under the effects of noise and show that this can be achieved through a machine-learning approach. We present a toy example using a Bell state under a depolarization channel to illustrate the concept of robust observables. We then describe a machine-learning framework for learning such observables across six two-qubit quantum circuits and five noisy channels. Our results show that it is possible to learn observables that are more robust to noise than conventional observables. We discuss the implications of this finding for quantum machine learning, including potential applications in enhancing the stability of QML models in noisy environments. By developing techniques for learning robust observables, we can improve the performance and reliability of quantum machine learning models in the presence of noise, contributing to the advancement of practical QML applications in the NISQ era.

Learning Robust Observable to Address Noise in Quantum Machine Learning

TL;DR

The paper addresses the instability of quantum machine learning models in the presence of noise on NISQ devices by proposing a framework to learn observables that remain invariant under noisy channels. It combines a theoretical invariant condition expressed via Kraus operators with a practical machine-learning approach to optimize observables across multiple circuits and noise models. Key contributions include a toy Bell-state example demonstrating a robust observable, a learning protocol with a parameter-shift gradient across six circuits and five channels, and empirical statistics showing widespread robustness of learned observables. The work suggests that noise-robust observables can enhance the reliability of QML measurements without extra quantum resources, offering a practical pathway to more stable QML in realistic noisy environments.

Abstract

Quantum Machine Learning (QML) has emerged as a promising field that combines the power of quantum computing with the principles of machine learning. One of the significant challenges in QML is dealing with noise in quantum systems, especially in the Noisy Intermediate-Scale Quantum (NISQ) era. Noise in quantum systems can introduce errors in quantum computations and degrade the performance of quantum algorithms. In this paper, we propose a framework for learning observables that are robust against noisy channels in quantum systems. We demonstrate that it is possible to learn observables that remain invariant under the effects of noise and show that this can be achieved through a machine-learning approach. We present a toy example using a Bell state under a depolarization channel to illustrate the concept of robust observables. We then describe a machine-learning framework for learning such observables across six two-qubit quantum circuits and five noisy channels. Our results show that it is possible to learn observables that are more robust to noise than conventional observables. We discuss the implications of this finding for quantum machine learning, including potential applications in enhancing the stability of QML models in noisy environments. By developing techniques for learning robust observables, we can improve the performance and reliability of quantum machine learning models in the presence of noise, contributing to the advancement of practical QML applications in the NISQ era.
Paper Structure (17 sections, 1 theorem, 22 equations, 3 figures)

This paper contains 17 sections, 1 theorem, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Depolarizing Channel
  4. Amplitude Damping Channel
  5. Phase Damping Channel
  6. Phase Flip Channel
  7. Bit Flip Channel
  8. Quantum Observables
  9. Problem Definition
  10. Result
  11. Problem Example
  12. Learning Robust Observables
  13. Statistics of the Learned Observables
  14. Discussion
  15. Conclusion
  16. ...and 2 more sections

Key Result

theorem thmcountertheorem

The expectation value $\expval{O}$ of an observable $O$ on a quantum state $\rho$ remains invariant under a noise model $\mathcal{E}$, represented by Kraus operators $\{K_i\}$, if and only if each $K_i$ commutes with $O$, i.e., $K_i^\dag O K_i = O$ for all $i$.

Figures (3)

  • Figure 1: Expectation value of different observables on the depolarized Bell state as a function of the depolarization rate $p$. $Z$ is the Pauli-Z matrix, $X$ is the Pauli-X matrix, $H$ is the Hadamard gate, and $O_{optimized}$ is an arbitrary single qubit Hermitian measurement operator. The expectation value of the observable $O_{optimized}$ remains constant as the depolarization rate $p$ increases.
  • Figure 2: Results of the learning process
  • Figure 3: Count of the standard deviation of the expectation value of the observables for all the circuit-channel combinations.

Theorems & Definitions (2)

  • theorem thmcountertheorem
  • proof