# Error Mitigation-Aided Optimization of Parameterized Quantum Circuits: Convergence Analysis

arXiv: https://arxiv.org/abs/2209.11514 · HTML: https://mlsanity.com/paper/2209.11514/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 46 references that are in the map.
- First posted on arXiv 23 Sep 2022.

## Builds on (most cited first, 8 of 8)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [On the Convergence of SGD with Biased Gradients](https://mlsanity.com/paper/2008.00051/) (2020)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [An Introduction to Quantum Machine Learning for Engineers](https://mlsanity.com/paper/2205.09510/) (2022)
- [Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?](https://mlsanity.com/paper/2109.01051/) (2021)
- [Noise-Resilient Variational Hybrid Quantum-Classical Optimization](https://mlsanity.com/paper/1912.06744/) (2019)

## Built on by (most cited first, 6 of 6)

- [Quantum Conformal Prediction for Reliable Uncertainty Quantification in Quantum Machine Learning](https://mlsanity.com/paper/2304.03398/) (2023)
- [Sporadic Federated Learning Approach in Quantum Environment to Tackle Quantum Noise](https://mlsanity.com/paper/2507.12492/) (2025)
- [Differentially Private Federated Quantum Learning via Quantum Noise](https://mlsanity.com/paper/2508.20310/) (2025)
- [Towards Heterogeneous Quantum Federated Learning: Challenges and Solutions](https://mlsanity.com/paper/2511.22148/) (2025)
- [Communication-Efficient Quantum Federated Learning over Large-Scale Wireless Networks](https://mlsanity.com/paper/2603.01222/) (2026)
- [Distributed Quantum Learning over Near-term Devices: Convergence Analysis and Security Design](https://mlsanity.com/paper/2606.20606/) (2026)

## Abstract (arXiv)

Variational quantum algorithms (VQAs) offer the most promising path to obtaining quantum advantages via noisy intermediate-scale quantum (NISQ) processors. Such systems leverage classical optimization to tune the parameters of a parameterized quantum circuit (PQC). The goal is minimizing a cost function that depends on measurement outputs obtained from the PQC. Optimization is typically implemented via stochastic gradient descent (SGD). On NISQ computers, gate noise due to imperfections and decoherence affects the stochastic gradient estimates by introducing a bias. Quantum error mitigation (QEM) techniques can reduce the estimation bias without requiring any increase in the number of qubits, but they in turn cause an increase in the variance of the gradient estimates. This work studies the impact of quantum gate noise on the convergence of SGD for the variational eigensolver (VQE), a fundamental instance of VQAs. The main goal is ascertaining conditions under which QEM can enhance the performance of SGD for VQEs. It is shown that quantum gate noise induces a non-zero error-floor on the convergence error of SGD (evaluated with respect to a reference noiseless PQC), which depends on the number of noisy gates, the strength of the noise, as well as the eigenspectrum of the observable being measured and minimized. In contrast, with QEM, any arbitrarily small error can be obtained. Furthermore, for error levels attainable with or without QEM, QEM can reduce the number of required iterations, but only as long as the quantum noise level is sufficiently small, and a sufficiently large number of measurements is allowed at each SGD iteration. Numerical examples for a max-cut problem corroborate the main theoretical findings.