Estimating the gradient and higher-order derivatives on quantum hardware

Abstract: For a large class of variational quantum circuits, we show how arbitrary-order derivatives can be analytically evaluated in terms of simple parameter-shift rules, i.e., by running the same circuit with different shifts of the parameters. As particular cases, we obtain parameter-shift rules for the Hessian of an expectation value and for the metric tensor of a variational state, both of which can be efficiently used to analytically implement second-order optimization algorithms on a quantum computer. We also consider the impact of statistical noise by studying the mean squared error of different derivative estimators. In the second part of this work, some of the theoretical techniques for evaluating quantum derivatives are applied to their typical use case: the implementation of quantum optimizers. We find that the performance of different estimators and optimizers is intertwined with the values of different hyperparameters, such as a step size or a number of shots. Our findings are supported by several numerical and hardware experiments, including an experimental estimation of the Hessian of a simple variational circuit and an implementation of the Newton optimizer.

• The paper extends the parameter-shift rule to compute higher-order derivatives, including the Hessian and Fubini-Study metric tensor, to support advanced quantum optimization.
• It introduces scaled estimators that mitigate statistical noise, providing unbiased gradient estimates with lower MSE compared to finite-difference methods.
• Experimental and numerical validations on IBM quantum hardware confirm the effectiveness of second-order methods like Newton’s and quantum natural gradient descent.

Estimating the Gradient and Higher-Order Derivatives on Quantum Hardware

The paper "Estimating the gradient and higher-order derivatives on quantum hardware" presents a study on the analytical evaluation of derivatives in variational quantum circuits using parameter-shift rules. This work is of particular importance for algorithms running on near-term quantum processors which frequently rely on the estimation of gradients and higher-order derivatives as part of optimization routines.

Key Contributions

1. Generalization of Parameter-Shift Rules: The authors extend the parameter-shift rule, commonly used for gradients, to arbitrary-order derivatives. This includes the Hessian and Fubini-Study metric tensor, providing a foundation for advanced optimization techniques in quantum algorithms.
2. Statistical Noise Consideration: In addition to theoretical development, the manuscript examines the impact of statistical noise on derivative estimators, which is crucial given the noisy nature of quantum hardware. The paper introduces a scaled parameter-shift estimator that offers reduced error in scenarios where noise dominates the gradient signal.
3. Comparison to Finite-Difference Methods: Even though parameter-shift rules are specific to circuits with involutory matrices, they often outperform the finite-difference method in scenarios where both can be applied. The analytic approach is unbiased and typically presents lower mean squared error (MSE) than finite-difference methods, especially for large numbers of shots.
4. Second-order Optimization: The work explores the use of these rules for implementing second-order optimization methods such as Newton's method and its diagonal approximation, and the quantum natural gradient descent algorithm. These methods offer potential advantages over first-order optimizers by better accounting for curvature in the cost function.
5. Experimental Validation: Through both numerical simulations and experiments on IBM quantum hardware, the paper demonstrates the efficacy of proposed methods, highlighting the real-world applicability of the advanced parameter-shift rules and second-order optimization procedures.

Implications and Future Work

The implications of this research are significant for the optimization of quantum variational algorithms, such as the Variational Quantum Eigensolver and Quantum Approximate Optimization Algorithm. By making higher-order derivatives accessible and practical, the pathway towards leveraging more complex and potentially more efficient optimizers is opened.

Moreover, the discussion on the statistical noise robustness provides a critical insight into practical quantum computing, where achieving reliable precision with limited shots is vital. It suggests that by judiciously scaling gradients, optimizers may remain functional even in the presence of notable noise—an essential trait when dealing with barren plateaus or parameter landscapes with small gradients.

Future work might focus on the extension and adaptation of these methods to circuits and gates beyond those involving only involutory matrices, potentially using stochastic techniques or other generalization approaches discussed within the study. Furthermore, a systematic comparison of second-order methods under various noise models and quantum architectures would be a valuable extension, potentially uncovering thresholds where these methods exhibit clear computational advantages.

Overall, this research contributes a comprehensive toolkit for derivative estimation and optimization on quantum hardware, marking a step forward in efficiently harnessing the computing power of near-term quantum devices.