Single-component gradient rules for variational quantum algorithms

Abstract

Many near-term quantum computing algorithms are conceived as variational quantum algorithms, in which parameterized quantum circuits are optimized in a hybrid quantum-classical setup. Examples are variational quantum eigensolvers, quantum approximate optimization algorithms as well as various algorithms in the context of quantum-assisted machine learning. A common bottleneck of any such algorithm is constituted by the optimization of the variational parameters. A popular set of optimization methods work on the estimate of the gradient, obtained by means of circuit evaluations. We will refer to the way in which one can combine these circuit evaluations as gradient rules. This work provides a comprehensive picture of the family of gradient rules that vary parameters of quantum gates individually. The most prominent known members of this family are the parameter shift rule and the finite differences method. To unite this family, we propose a generalized parameter shift rule that expresses all members of the aforementioned family as special cases, and discuss how all of these can be seen as providing access to a linear combination of exact first- and second-order derivatives. We further prove that a parameter shift rule with one non-shifted evaluation and only one shifted circuit evaluation can not exist does not exist, and introduce a novel perspective for approaching new gradient rules.

arXiv comment9 pages, 0 figures

References in corpus (33)

• Quantum Computing in the NISQ era and beyond
• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• A variational eigenvalue solver on a quantum processor
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Supervised learning with quantum enhanced feature spaces
• Quantum computational advantage using photons
• The theory of variational hybrid quantum-classical algorithms
• A Quantum Approximate Optimization Algorithm
• Noisy intermediate-scale quantum (NISQ) algorithms
• Quantum Circuit Learning
• Parameterized quantum circuits as machine learning models
• Evaluating analytic gradients on quantum hardware
• Circuit-centric quantum classifiers
• The power of quantum neural networks
• Hartree-Fock on a superconducting qubit quantum computer
• The European Quantum Technologies Roadmap
• Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor
• Quantum Natural Gradient
• Quantum chemistry calculations on a trapped-ion quantum simulator
• qubit-ADAPT-VQE: An adaptive algorithm for constructing hardware-efficient ansatze on a quantum processor
• Differentiable Learning of Quantum Circuit Born Machine
• Hybrid Quantum-Classical Approach to Quantum Optimal Control
• Stochastic gradient descent for hybrid quantum-classical optimization
• Structure optimization for parameterized quantum circuits
• Solving nonlinear differential equations with differentiable quantum circuits
• Estimating the gradient and higher-order derivatives on quantum hardware
• Training Quantum Embedding Kernels on Near-Term Quantum Computers
• A variational toolbox for quantum multi-parameter estimation
• Variational-State Quantum Metrology
• Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule
• Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition
• Minimizing estimation runtime on noisy quantum computers
• Local, Expressive, Quantum-Number-Preserving VQE Ansatze for Fermionic Systems

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• Efficient Estimation and Sequential Optimization of Cost Functions in Variational Quantum Algorithms
• Quantum natural gradient with thermal-state initialization
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