QuACK: Accelerating Gradient-Based Quantum Optimization with Koopman Operator Learning

TL;DR

This paper tackles the gradient-cost bottleneck in gradient-based variational quantum algorithms by introducing QuACK, which leverages Koopman operator learning to linearize and predict gradient dynamics. The method alternates between actual gradient steps on quantum hardware and Koopman-based predictions (DMD and neural DMD) to forecast future updates, reducing the number of costly gradient evaluations. The authors establish theoretical connections to quantum natural gradient and overparameterization theory, analyze stability, and derive speedup bounds. Empirical results across quantum Ising models, LiH chemistry, and quantum machine learning demonstrate dramatic accelerations, including over 200x in overparameterized regimes, over 10x in smooth regimes, and over 3x in non-smooth regimes, with robustness to measurement and hardware noise. These findings highlight the practical potential of Koopman-based acceleration for quantum optimization and pave the way for integrating more advanced neural-DMD architectures.

Abstract

Quantum optimization, a key application of quantum computing, has traditionally been stymied by the linearly increasing complexity of gradient calculations with an increasing number of parameters. This work bridges the gap between Koopman operator theory, which has found utility in applications because it allows for a linear representation of nonlinear dynamical systems, and natural gradient methods in quantum optimization, leading to a significant acceleration of gradient-based quantum optimization. We present Quantum-circuit Alternating Controlled Koopman learning (QuACK), a novel framework that leverages an alternating algorithm for efficient prediction of gradient dynamics on quantum computers. We demonstrate QuACK's remarkable ability to accelerate gradient-based optimization across a range of applications in quantum optimization and machine learning. In fact, our empirical studies, spanning quantum chemistry, quantum condensed matter, quantum machine learning, and noisy environments, have shown accelerations of more than 200x speedup in the overparameterized regime, 10x speedup in the smooth regime, and 3x speedup in the non-smooth regime. With QuACK, we offer a robust advancement that harnesses the advantage of gradient-based quantum optimization for practical benefits.

Figures (16)

• Figure 1: QuACK: Quantum-circuit Alternating Controlled Koopman Operator Learning. (a) Parameterized quantum circuits process information; loss function is evaluated via quantum measurements. Parameter updates for the quantum circuit are computed by a classical optimizer. (b) Optimization history forms a time series, the computational cost of which is proportional to the number of parameters. (c) Koopman operator learning finds an embedding of data with approximately linear dynamics from time series in (b). (d) Koopman operator predicts parameter updates with computational cost independent of the number of parameters. Loss from predicted parameters is evaluated, and optimal parameters are used as starting point for the next iteration.
• Figure 2: Performance of our QuACK with the standard DMD in the following cases. (\ref{['subfig:qng']}) For quantum natural gradient, with short training (4 steps per piece) and long prediction (40 steps per piece), DMD accurately predicts the intrinsic dynamics of quantum optimization, and QuACK has 20.18x speedup. (\ref{['subfig:speedup_op']}) In the overparameterization regime, QuACK has >200x speedup with 2-5 qubits. (\ref{['subfig:speedup_smooth']}) In smooth optimization regimes, QuACK has >10x speedup with 2-12 qubits.
• Figure 3: Experimental results for (\ref{['subfig:lih']}) LiH molecule with 10 qubits using Adam (\ref{['subfig:adam']}) Quantum Ising model with 12 qubits using Adam (\ref{['fig:qml']}) test accuracy of binary classification in QML. The solid piecewise curves are true gradient steps, and the dashed lines connecting them indicate when the DMD prediction is applied to find $\boldsymbol{\theta} (t_{\mathrm{opt}})$ in our controlled scheme. Our QuACK with all the DMD methods bring acceleration, with maximum speedups (\ref{['subfig:lih']}) 4.63x (\ref{['subfig:adam']}) 3.24x (\ref{['fig:qml']}) 4.38x.
• Figure 4: Noisy quantum optimization with $n_{\mathrm{shots}} = 100$. (a) 10-qubit shot noise system (b) 5-qubit FakeManila.
• Figure 5: Neural network architectures for our neural DMD approaches. The factor $(d+1)$ in the dimension $(d+1) n_{\mathrm{params}}$ is due to the sliding window embedding $\mathbf{\Phi}$. (a) MLP bottleneck architecture with MSE loss for training. (b) CNN bottleneck architecture that operates on simulations as temporal and parameters as channel dimensions.

Theorems & Definitions (10)

• Theorem 5.1
• proof
• Theorem 5.2
• proof
• Corollary 5.3
• Theorem 5.4
• proof
• Corollary D.1
• proof
• proof