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Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and limit sets

P. Singkanipa, D. A. Lidar

TL;DR

The paper addresses trainability challenges in variational quantum algorithms by analyzing how noise affects gradients and convergence. It extends the noise analysis from unital maps to general CPTP maps, introducing HS-contractive non-unital noise and the concepts of noise-induced limit sets (NILS). By generalizing the parameter shift rule to noisy settings and providing rigorous gradient-bounding bounds, it shows that NIBPs are unavoidable for unital noise but may be avoided in HS-contractive non-unital scenarios, albeit with a persistent NILS. Numerical simulations corroborate the theory, highlighting distinct behaviors for depolarizing versus amplitude-damping noise and emphasizing the need for error mitigation strategies in practical VQA implementations.

Abstract

Variational quantum algorithms (VQAs) hold much promise but face the challenge of exponentially small gradients. Unmitigated, this barren plateau (BP) phenomenon leads to an exponential training overhead for VQAs. Perhaps the most pernicious are noise-induced barren plateaus (NIBPs), a type of unavoidable BP arising from open system effects, which have so far been shown to exist for unital noise maps. Here, we generalize the study of NIBPs to more general completely positive, trace-preserving maps, investigating the existence of NIBPs in the unital case and a class of non-unital maps we call Hilbert-Schmidt (HS)-contractive. The latter includes amplitude damping. We identify the associated phenomenon of noise-induced limit sets (NILS) of the VQA cost function and prove its existence for both unital and HS-contractive non-unital noise maps. Along the way, we extend the parameter shift rule of VQAs to the noisy setting. We provide rigorous bounds in terms of the relevant variables that give rise to NIBPs and NILSs, along with numerical simulations of the depolarizing and amplitude-damping maps that illustrate our analytical results.

Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and limit sets

TL;DR

The paper addresses trainability challenges in variational quantum algorithms by analyzing how noise affects gradients and convergence. It extends the noise analysis from unital maps to general CPTP maps, introducing HS-contractive non-unital noise and the concepts of noise-induced limit sets (NILS). By generalizing the parameter shift rule to noisy settings and providing rigorous gradient-bounding bounds, it shows that NIBPs are unavoidable for unital noise but may be avoided in HS-contractive non-unital scenarios, albeit with a persistent NILS. Numerical simulations corroborate the theory, highlighting distinct behaviors for depolarizing versus amplitude-damping noise and emphasizing the need for error mitigation strategies in practical VQA implementations.

Abstract

Variational quantum algorithms (VQAs) hold much promise but face the challenge of exponentially small gradients. Unmitigated, this barren plateau (BP) phenomenon leads to an exponential training overhead for VQAs. Perhaps the most pernicious are noise-induced barren plateaus (NIBPs), a type of unavoidable BP arising from open system effects, which have so far been shown to exist for unital noise maps. Here, we generalize the study of NIBPs to more general completely positive, trace-preserving maps, investigating the existence of NIBPs in the unital case and a class of non-unital maps we call Hilbert-Schmidt (HS)-contractive. The latter includes amplitude damping. We identify the associated phenomenon of noise-induced limit sets (NILS) of the VQA cost function and prove its existence for both unital and HS-contractive non-unital noise maps. Along the way, we extend the parameter shift rule of VQAs to the noisy setting. We provide rigorous bounds in terms of the relevant variables that give rise to NIBPs and NILSs, along with numerical simulations of the depolarizing and amplitude-damping maps that illustrate our analytical results.
Paper Structure (38 sections, 13 theorems, 119 equations, 4 figures, 1 algorithm)

This paper contains 38 sections, 13 theorems, 119 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. VQA and PSR
  4. Nice operator basis and Schatten $p$-norms
  5. The coherence vector
  6. Unital maps
  7. Non-unital maps
  8. Parameter shift rule in the presence of noise
  9. The noiseless PSR case
  10. Control noise
  11. Random unitary noise
  12. Bounds on $\|\boldsymbol{h}\|$
  13. Cost Function Concentration and Noise-Induced Limit Sets
  14. Cost function concentration under non-unital noise
  15. Noise-induced limit set
  16. ...and 23 more sections

Key Result

Proposition 1

The CPTP map $\rho'=\mathcal{N}(\rho)$ is equivalent to the affine coherence vector transformation where $M\in \mathcal{M}(d^2-1, \mathbb{R})$ and $\boldsymbol{c}\in \mathbb{R}^{d^2-1}$ have elements given by

Figures (4)

  • Figure 1: Mean and variance ($\log_{10}$ scale) of the magnitude of the cost function gradient for depolarizing (left) and amplitude-damping (right) maps with noise probability $p=0.3$ as a function of layers. Error bars in the upper plots represent the range between the maximum and minimum values.
  • Figure 2: Mean and variance ($\log_{10}$ scale) of the magnitude of the cost function gradient for depolarizing (left) and amplitude-damping (right) maps as a function of noise probability $p$ in a VQA with 20 layers. Error bars in the upper plots represent the range between the maximum and minimum values.
  • Figure 3: Final cost function averaged over $50$ random $n$-qubit Hamiltonians with zero ground state energy under depolarizing (red up-triangles) and amplitude-damping (yellow down-triangles) maps as a function of noise probability $p$ using VQA with $L=5$ layers. The solid black line at zero denotes the true minimum of the Hamiltonians used in this simulation. The error bars are the standard deviation of the final cost. The dashed black line is the predicted NILS value in the large circuit depth limit in the unital case, \ref{['eq:C_NILS-unital']}.
  • Figure 4: Magnitude and variance of the gradient of the cost functions for depolarizing (red) and amplitude-damping (orange) maps as a function of the number of qubits for noise probabilities $p=0.1$, $0.2$, and $0.3$ in the left, middle, and right plots, respectively. Error bars in the top row represent the range of the values.

Theorems & Definitions (25)

  • Proposition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Definition 1
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • Corollary 1
  • ...and 15 more