Stochastic gradient descent for hybrid quantum-classical optimization

Abstract: Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with $k$ measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of $k$. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases $k$-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain "doubly stochastic" gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.

• The paper formalizes stochastic gradient descent in hybrid quantum models, showing that single measurement outcomes can ensure verified convergence.
• It introduces doubly stochastic gradient descent that samples both measurement outcomes and Hamiltonian terms to reduce computational overhead.
• Empirical results on VQE, QAOA, and quantum classifiers demonstrate that adaptive learning and hyper-parameter tuning yield efficient optimization on NISQ devices.

Stochastic Gradient Descent for Hybrid Quantum-Classical Optimization

In the field of hybrid quantum-classical optimization, the parameterized quantum circuits have become instrumental in harnessing the potential of noisy intermediate-scale quantum (NISQ) devices. This paper, authored by Sweke et al., explores the application of stochastic gradient descent (SGD) within this context, particularly focusing on the challenges and methodologies related to evaluating expectation values derived from quantum circuits.

Key Contributions

The paper explores several critical contributions to the field of quantum computing and optimization:

1. Formalization of Stochastic Gradient Descent: It establishes a formal link between gradient descent optimizers and stochastic gradient descent in the context of variational quantum eigensolvers (VQE), quantum approximate optimization algorithms (QAOA), and quantum classifiers. The authors show that even singular measurement outcomes can suffice for constructing SGD optimizers with verifiable convergence, offering a significant reduction in computational overhead.
2. Doubly Stochastic Gradient Descent: A novel approach introduced in this work is the "doubly stochastic" gradient descent. This involves sampling not only the measurement outcomes but also the terms of a Hamiltonian or data-set instances, which further economizes on computational resources while maintaining the robustness and efficiency of the optimization process.
3. Convergence Guarantees: The authors provide rigorous convergence guarantees for these SGD frameworks, encompassing both doubly stochastic and single-shot settings. This theoretical rigor offers confidence in the practical application of these methodologies in near-term quantum devices.
4. Empirical Evaluations: Through numerical experiments on benchmark tasks in VQE, QAOA, and quantum machine learning, the paper demonstrates that treating stochastic parameters as hyper-parameters can yield state-of-the-art results with comparatively fewer quantum circuit evaluations and measurements.

Numerical Results and Observations

The empirical analyses reveal several insights:

• Efficiency: The numerical results underline the potential for significant efficiency gains, particularly in scenarios such as quantum classifiers and VQE settings, where sampling strategies can dramatically reduce the number of quantum operations necessary.
• Trade-offs: The paper acknowledges the trade-off between the number of measurements (shots) used and the variance of the estimator, which affects convergence speed and solution quality. It highlights that adaptive strategies in learning rates can mitigate some of these trade-offs, enhancing final solution quality even with smaller shot numbers.
• Practical Implementations: The study proposes practical strategies for leveraging the devised stochastic frameworks, including adaptive learning strategies and the potential for hyper-parameter tuning, to optimize performance on near-term quantum devices.

Implications and Future Directions

The implications of this research are significant for both theoretical and practical developments in quantum computing:

• Reduction in Quantum Resources: The ability to achieve optimization with fewer quantum operations opens avenues for more complex and large-scale quantum applications, potentially speeding up the development of quantum algorithms applicable to real-world problems.
• Generalizability: The principles outlined in this paper could serve as a groundwork for further extensions and adaptations in other quantum-classical hybrid models, expanding applications across different quantum computing paradigms.
• Future Research: Potential future research may focus on exploring different classes of loss functions, incorporating noise models into these stochastic frameworks, and challenging the applicability of these strategies under various quantum circuit architectures.

In conclusion, this work contributes valuable insights into the practical application and theoretical underpinnings of stochastic gradient descent within hybrid quantum-classical systems, laying a foundation for further exploration and innovation in leveraging NISQ devices for optimization tasks.