L2O-g†: 利用機器學習結合富比尼–施圖迪度量優化參數化量子電路

Abstract

量子電腦基於量子力學原理，如疊加和糾纏，提供了一種不同的計算範式。例如，Shor算法用於大數質因數分解和物理系統量子模擬等算法在時間和空間複雜度方面都表現出優於經典計算機的性能。在容錯量子計算機出現之前，變分量子算法（variational quantum algorithms, VQAs）作為噪聲中等規模量子（noisy intermediate-scale quantum, NISQ）機器中的重要算法。研究表明，VQAs能夠進行通用計算，包括量子機器學習、量子化學、解決大規模線性代數問題和組合優化等應用。

一個VQA由一個參數化的量子電路（PQC）組成，並使用古典計算機來優化基於量子計算機輸出的成本函數。優化VQA問題通常是一個NP難問題，需要精心選擇的初始化程序和優化器。傳統上，優化VQAs依賴於手工設計的算法，如一階方法梯度下降或二階方法BFGS算法。此外，手工設計的優化器需要仔細選擇初始參數並進行全局搜索以適應不同的優化問題。為了解決這一問題，Learn to Optimize (L2O) 使用一個小型神經網絡來替代手工設計的優化器，學習針對類似任務分布的最佳優化策略。這個方法利用神經網絡作為通用函數逼近器。以前的文獻顯示基於純古典學習優化器已經證明了它們在類似訓練集的有限案例中作為參數初始化器的性能。

在我們的工作中，我們提出了L2O-g†，一個quantum-aware學習優化器，在優化過程中動態調整參數空間優化和分布空間優化之間的平衡。 L2O-g† 利用富比尼-斯圖迪度量張量（Fubini-Study Metric Tensor, g†）和長短期記憶（LSTM），一種機器學習模型，在參數空間和分布空間優化之間動態調整平衡，自動適應問題的結構。我們從理論上推導出一個受到lookahead優化器啟發的更新方程，通過使用富比尼-斯圖迪度量張量表示的參數空間分布，在快速收斂和泛化之間取得平衡。具體來說，一個可學習的向量在參數空間和分布空間優化之間協調作用。

在實驗上，我們對一系列VQA問題進行了全面的實驗，包括變分量子特徵求解器（VQE）、量子近似優化算法（QAOA）和量子機器學習（QML）任務。我們的結果表明，L2O-g† 不僅在沒有任何超參數調優的情況下優於當前的最先進手工設計優化器（SOTA），而且在分布外泛化能力上也優於以前的L2O優化器，我們僅用一個通用PQC實例訓練L2O-g† 就實現了這一點。我們新提出的quantum-aware學習優化器L2O-g† 在應對VQAs的挑戰中展現了顯著的進步，使其成為NISQ時代的一個有價值的工具。實現代碼和數據可在以下網址獲取：https://github.com/Physics-Morris/L2O-g。

Quantum computers provide a different paradigm of computing based on quantum mechanical principles such as superposition and entanglement. Algorithms like Shor’s algorithm for factoring large numbers and quantum simulations of physical systems have demonstrated superior time and space complexity compared to classical computers. Before the advent of fault-tolerant quantum computers, variational quantum algorithms (VQAs) emerged as crucial algorithms in noisy intermediate-scale quantum (NISQ) machines. VQAs are capable of universal computing, including applications such as quantum machine learning, quantum chemistry, solving large-scale linear algebra problems, and combinatorial optimization. A VQA consists of a parameterized quantum circuit (PQC) and uses a classical computer to optimize the cost function based on the output of the quantum computer. Optimizing VQA problems is generally an NP-hard problem, requiring well-chosen initialization procedures and optimizers. Conventionally, optimizing VQAs relies on hand-designed algorithms such as first-order method gradient descent or second-order method BFGS algorithm. Additionally, a hand-designed optimizer requires a careful choice of initial parameters and a global search for hyperparameters for different optimization problems. To address this issue, learning to optimize (L2O) uses a small neural network to replace the handcrafted optimizer, learning the optimal optimization strategy tailored for a similar distribution of tasks. This approach leverages the intuition of a neural network as a universal function approximator. Several previous works using purely classical learned optimizers have demonstrated their ability to serve as parameter initializers on limited cases similar to their training sets.

In this work, we propose L2O-g†, a quantum-aware learned optimizer that dynamically adjusts the balance between parameter space optimization and distribution space optimization during the optimization process. L2O-g† leverages the Fubini-Study metric tensor (g†) and Long Short-Term Memory (LSTM), a machine learning model to dynamically adjust the balance between parameter space and distribution space optimization, adapting automatically to the problem's structure. The quantum-aware learned optimizer L2O-g† is designed specifically for optimizing PQCs to address the generalization problem for using purely classical learned optimizers. We theoretically derive the update equation inspired by the lookahead optimizer to incorporate quantum geometry of the optimization landscape in the learned optimizer by using the parameter space distribution represented by the Fubini-Study metric tensor to balance fast convergence and generalization. Specifically, a learnable vector balances the parameter space and distribution space optimization acting coordinate-wise on the parameters.

Empirically, we conducted comprehensive experiments across a range of VQA problems, including variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum machine learning (QML) tasks. Our results demonstrate that L2O-g† not only outperforms the current state-of-the-art (SOTA) hand-designed optimizer without any hyperparameter tuning but also shows strong out-of-distribution generalization compared to previous L2O optimizers. We achieve this by training L2O-g† on just a single generic PQC instance. Our novel quantum-aware learned optimizer, L2O-g†, presents an advancement in addressing the challenges of VQAs, making it a valuable tool in the NISQ era. The implementation and data are available at: https://github.com/Physics-Morris/L2O-g.