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Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits

Kaining Zhang, Liu Liu, Min-Hsiu Hsieh, Dacheng Tao

TL;DR

The paper addresses barren plateaus in deep variational quantum algorithms by introducing Gaussian initializations with variance $\mathcal{O}(\frac{1}{L})$, which provably bound the gradient norm from below by a polynomial inverse in $N$ and $L$. It extends the theory from local to global observables and to correlated gate structures, via Theorems 4.1 and 4.2, and analyzes the impact of measurement noise on gradient estimation. Empirical results on the Heisenberg model and LiH molecule validate the theoretical claims, showing faster convergence and robustness to noise compared to uniform or zero initializations. The work also characterizes the gradient-estimation cost, showing $\mathcal{O}(\frac{L}{\epsilon})$ measurements suffice for reliable gradients, enabling practical training of deeper quantum circuits.

Abstract

Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for very shallow circuits. Experimental results verify our theoretical findings in the quantum simulation and quantum chemistry.

Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits

TL;DR

The paper addresses barren plateaus in deep variational quantum algorithms by introducing Gaussian initializations with variance , which provably bound the gradient norm from below by a polynomial inverse in and . It extends the theory from local to global observables and to correlated gate structures, via Theorems 4.1 and 4.2, and analyzes the impact of measurement noise on gradient estimation. Empirical results on the Heisenberg model and LiH molecule validate the theoretical claims, showing faster convergence and robustness to noise compared to uniform or zero initializations. The work also characterizes the gradient-estimation cost, showing measurements suffice for reliable gradients, enabling practical training of deeper quantum circuits.

Abstract

Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for very shallow circuits. Experimental results verify our theoretical findings in the quantum simulation and quantum chemistry.
Paper Structure (17 sections, 7 theorems, 89 equations, 8 figures)

This paper contains 17 sections, 7 theorems, 89 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Notations and quantum computing basics
  4. Framework of general VQAs
  5. Theoretical results about Gaussian initialized VQAs
  6. Independent parameters with local observables
  7. Correlated parameters with global observables
  8. Experiments
  9. Heisenberg model
  10. Quantum chemistry
  11. Conclusions
  12. Additional Experiments
  13. Heisenberg model
  14. Quantum chemistry
  15. Technical Lemmas
  16. ...and 2 more sections

Key Result

Theorem 4.1

Consider the $L$-layer $N$-qubit variational quantum circuit $V(\bm{\theta})$ defined in Figure gidqnn_local_circuit and the cost function $f(\bm{\theta}) = {\rm Tr} \left[ \sigma_{\bm{i}} V(\bm{\theta}) \rho_{{\rm in}} V(\bm{\theta})^{\dag}\right]$, where the observable $\sigma_{\bm{i}}$ follows th where $S$ is the number of non-zero elements in $\bm{i}$, and the index $\bm{j}=(j_1,j_2,\cdots,j_N

Figures (8)

  • Figure 1: The quantum circuit framework for the local observable case. The circuit performs $L$ layers of single qubit rotations and CZ layers on the input state $\rho_{\text{in}}$, followed by a $R_X$ layer and a $R_Y$ layer. In the $\ell$-th single qubit layer, we employ the gate $e^{-i\theta_{\ell,n} G_{\ell,n}}$ for all qubits $n \in [N]$, where $G_{\ell,n}$ is a Hermitian unitary, which anti-commutes with $\sigma_3$ for $\ell \in [L]$. In each $\text{CZ}_\ell$ layer, CZ gates are employed between arbitrary qubit pairs. The measurement is performed on $S$ qubits where the observable acts nontrivially on these qubits.
  • Figure 2: Numerical results of finding the ground energy of the Heisenberg model. The first row shows training results with the gradient descent optimizer, where Figures \ref{['gidqnn_exp_heisenberg_gd_loss_1']} and \ref{['gidqnn_exp_heisenberg_gd_loss_2']} illustrate the loss function corresponding to Eq.(\ref{['gidqnn_heisenberg_ham_eq']}) during the optimization with accurate and noisy gradients, respectively. Figures \ref{['gidqnn_exp_heisenberg_gd_grad_1']} and \ref{['gidqnn_exp_heisenberg_gd_grad_2']} show the $\ell_2$ norm of corresponding gradients.The second row shows training results with the Adam optimizer, where Figures \ref{['gidqnn_exp_heisenberg_adam_loss_1']} and \ref{['gidqnn_exp_heisenberg_adam_loss_2']} illustrate the loss function with accurate and noisy gradients, respectively. Figures \ref{['gidqnn_exp_heisenberg_adam_grad_1']} and \ref{['gidqnn_exp_heisenberg_adam_grad_2']} show the $\ell_2$ norm of corresponding gradients.Each line denotes the average of $5$ rounds of optimizations.
  • Figure 3: Numerical results of finding the ground energy of the molecule LiH. The first and second rows show training results with the gradient descent and the Adam optimizer, respectively. The left, the middle, and the right columns show results using accurate gradients, noisy gradients with adaptive-distributed noises, and noisy gradients with constant-distributed noises. The variance of noises in the middle line (Figures \ref{['gidqnn_exp_chem_gd_loss_2']} and \ref{['gidqnn_exp_chem_adam_loss_2']}) follows Eq. (\ref{['gidqnn_chem_noise_gamma']}), while the variance of noises in the right line (Figures \ref{['gidqnn_exp_chem_gd_loss_3']} and \ref{['gidqnn_exp_chem_adam_loss_3']}) is $0.001$. Each line denotes the average of $5$ rounds of optimizations.
  • Figure 4: Numerical results of finding the ground state energy of the Heisenberg model using the noiseless gradient descent. Figures \ref{['gidqnn_app_exp_heisenberg_multi_gd_loss_240']}-\ref{['gidqnn_app_exp_heisenberg_multi_gd_loss_360']} show the loss during optimizations for different $L \in \{14,18,22\}$ using the circuit 1 in the main text. For each $L$, we adopt Gaussian initializaions with different variances $0.01\gamma,0.1\gamma,\gamma,10\gamma,100\gamma$, where the value $\gamma$ follows the formulation in Theorem 4.1. Figures \ref{['gidqnn_app_exp_heisenberg_multi_gd_gradnorm_240']}-\ref{['gidqnn_app_exp_heisenberg_multi_gd_gradnorm_360']} show the $\ell_2$ norm of corresponding gradients during the optimization. Each line illustrates the average of $5$ rounds of independent experiments.
  • Figure 5: Numerical results of finding the ground state energy of the Heisenberg model using the noisy gradient descent. Figures \ref{['gidqnn_app_exp_heisenberg_multi_gd_loss_240_noisy']}-\ref{['gidqnn_app_exp_heisenberg_multi_gd_loss_360_noisy']} show the loss during optimizations for different $L \in \{14,18,22\}$ using the circuit 1 in the main text. For each $L$, we adopt Gaussian initializaions with different variances $0.01\gamma,0.1\gamma,\gamma,10\gamma,100\gamma$, where the value $\gamma$ follows the formulation in Theorem 4.1. Figures \ref{['gidqnn_app_exp_heisenberg_multi_gd_gradnorm_240_noisy']}-\ref{['gidqnn_app_exp_heisenberg_multi_gd_gradnorm_360_noisy']} show the $\ell_2$ norm of corresponding gradients during the optimization. Each line illustrates the average of $5$ rounds of independent experiments.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 4.1
  • Theorem 4.2
  • Corollary 4.3
  • Lemma B.1
  • proof
  • Lemma B.2
  • proof
  • Lemma B.3
  • proof
  • Lemma B.4
  • ...and 3 more