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Noise Resilience and Robust Convergence Guarantees for the Variational Quantum Eigensolver

Mirko Legnini, Julian Berberich

TL;DR

This work analyzes how both coherent and incoherent noise affect the optimal parameters and convergence of the Variational Quantum Eigensolver. By recasting noise effects as perturbations to the observable (perturbed observable framework), it derives bounds on the distance between noisy and nominal optima using Łojasiewicz-type arguments, showing polynomial dependence on the perturbation level and, under local surjectivity, a linear bound: dist(˜θ^*_ε, Θ^*) ≤ C $\epsilon^{(1-\alpha)/\alpha}$, with α = 1/2 yielding linear scaling. The study further demonstrates that coherent noise can be reduced to a first-order observable perturbation, while incoherent noise maps to a scaled observable perturbation; depolarization leaves optimal parameters invariant but may induce barren plateaus. Numerical simulations on a 5-qubit circuit with Pennylane validate the theoretical predictions, confirming robust convergence to a neighborhood of the nominal optimum under small noise. The results inform robust VQE design and noise-aware optimization for near-term quantum devices, and point to extensions to broader VQA classes and non-compact parameter spaces.

Abstract

Variational Quantum Algorithms (VQAs) are a class of hybrid quantum-classical algorithms that leverage on classical optimization tools to find the optimal parameters for a parameterized quantum circuit. One relevant application of VQAs is the Variational Quantum Eigensolver (VQE), which aims at steering the output of the quantum circuit to the ground state of a certain Hamiltonian. Recent works have provided global convergence guarantees for VQEs under suitable local surjectivity and smoothness hypotheses, but little has been done in characterizing convergence of these algorithms when the underlying quantum circuit is affected by noise. In this work, we characterize the effect of different coherent and incoherent noise processes on the optimal parameters and the optimal cost of the VQE, and we study their influence on the convergence guarantees of the algorithm. Our work provides novel theoretical insight into the behavior of parameterized quantum circuits. Furthermore, we accompany our results with numerical simulations implemented via Pennylane.

Noise Resilience and Robust Convergence Guarantees for the Variational Quantum Eigensolver

TL;DR

This work analyzes how both coherent and incoherent noise affect the optimal parameters and convergence of the Variational Quantum Eigensolver. By recasting noise effects as perturbations to the observable (perturbed observable framework), it derives bounds on the distance between noisy and nominal optima using Łojasiewicz-type arguments, showing polynomial dependence on the perturbation level and, under local surjectivity, a linear bound: dist(˜θ^*_ε, Θ^*) ≤ C , with α = 1/2 yielding linear scaling. The study further demonstrates that coherent noise can be reduced to a first-order observable perturbation, while incoherent noise maps to a scaled observable perturbation; depolarization leaves optimal parameters invariant but may induce barren plateaus. Numerical simulations on a 5-qubit circuit with Pennylane validate the theoretical predictions, confirming robust convergence to a neighborhood of the nominal optimum under small noise. The results inform robust VQE design and noise-aware optimization for near-term quantum devices, and point to extensions to broader VQA classes and non-compact parameter spaces.

Abstract

Variational Quantum Algorithms (VQAs) are a class of hybrid quantum-classical algorithms that leverage on classical optimization tools to find the optimal parameters for a parameterized quantum circuit. One relevant application of VQAs is the Variational Quantum Eigensolver (VQE), which aims at steering the output of the quantum circuit to the ground state of a certain Hamiltonian. Recent works have provided global convergence guarantees for VQEs under suitable local surjectivity and smoothness hypotheses, but little has been done in characterizing convergence of these algorithms when the underlying quantum circuit is affected by noise. In this work, we characterize the effect of different coherent and incoherent noise processes on the optimal parameters and the optimal cost of the VQE, and we study their influence on the convergence guarantees of the algorithm. Our work provides novel theoretical insight into the behavior of parameterized quantum circuits. Furthermore, we accompany our results with numerical simulations implemented via Pennylane.
Paper Structure (13 sections, 8 theorems, 46 equations, 3 figures)

This paper contains 13 sections, 8 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variational Quantum Algorithms
  4. Quantum Noise
  5. Theoretical Analysis
  6. Perturbed observable
  7. Coherent noise
  8. Incoherent noise
  9. Special cases
  10. Numerics
  11. Conclusions and future works
  12. Conclusions
  13. Future Works

Key Result

Theorem 1

Let nominalproblem have a compact parameter space $\Theta$ and suppose there exists $M_1, M_2 > 0$ such that $||\tilde{O}(\theta)||<M_1$ and $||\nabla\tilde{O}(\theta)||<M_2$ for each $\theta \in \Theta$. Then there exist constants $C, \bar{\epsilon}>0$ and $\alpha \in (0,1)$ such that Furthermore, if $U(\theta)$ is locally surjective,

Figures (3)

  • Figure 1: Visual representation of the set of unperturbed optimal parameters $\Theta^*$ and the set $\tilde{\Theta}^*_\epsilon$.
  • Figure 2: Distance of the perturbed optimal parameter from the nominal one over the perturbation level for coherent errors
  • Figure 3: Distance of the perturbed optimal parameter from the nominal one over the perturbation level for incoherent errors

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 7 more