General parameter-shift rules for quantum gradients

Abstract

Variational quantum algorithms are ubiquitous in applications of noisy intermediate-scale quantum computers. Due to the structure of conventional parametrized quantum gates, the evaluated functions typically are finite Fourier series of the input parameters. In this work, we use this fact to derive new, general parameter-shift rules for single-parameter gates, and provide closed-form expressions to apply them. These rules are then extended to multi-parameter quantum gates by combining them with the stochastic parameter-shift rule. We perform a systematic analysis of quantum resource requirements for each rule, and show that a reduction in resources is possible for higher-order derivatives. Using the example of the quantum approximate optimization algorithm, we show that the generalized parameter-shift rule can reduce the number of circuit evaluations significantly when computing derivatives with respect to parameters that feed into many gates. Our approach additionally reproduces reconstructions of the evaluated function up to a chosen order, leading to known generalizations of the Rotosolve optimizer and new extensions of the quantum analytic descent optimization algorithm.

arXiv comment26 pages, 5 figures; Publication version in Quantum

References in corpus (57)

• A variational eigenvalue solver on a quantum processor
• Variational Quantum Algorithms
• Supervised learning with quantum enhanced feature spaces
• A Quantum Approximate Optimization Algorithm
• Quantum machine learning in feature Hilbert spaces
• Quantum Circuit Learning
• Parameterized quantum circuits as machine learning models
• Evaluating analytic gradients on quantum hardware
• An adaptive variational algorithm for exact molecular simulations on a quantum computer
• Circuit-centric quantum classifiers
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices
• Efficient variational quantum simulator incorporating active error minimisation
• Variational ansatz-based quantum simulation of imaginary time evolution
• Quantum circuits with many photons on a programmable nanophotonic chip
• Quantum autoencoders for efficient compression of quantum data
• Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor
• Quantum Natural Gradient
• Transfer learning in hybrid classical-quantum neural networks
• Continuous-variable quantum neural networks
• Quantum-assisted quantum compiling
• Quantum Approximate Optimization Algorithm for MaxCut: A Fermionic View
• Layerwise learning for quantum neural networks
• Hierarchical quantum classifiers
• Subspace-search variational quantum eigensolver for excited states
• Differentiable Learning of Quantum Circuit Born Machine
• Hybrid Quantum-Classical Approach to Quantum Optimal Control
• Variational quantum algorithms for discovering Hamiltonian spectra
• PennyLane: Automatic differentiation of hybrid quantum-classical computations
• Stochastic gradient descent for hybrid quantum-classical optimization
• Structure optimization for parameterized quantum circuits
• Variational Quantum State Diagonalization
• Sequential minimal optimization for quantum-classical hybrid algorithms
• Hardware-efficient variational quantum algorithms for time evolution
• Estimating the gradient and higher-order derivatives on quantum hardware
• Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer
• Efficient variational simulation of non-trivial quantum states
• Methodology for replacing indirect measurements with direct measurements
• Fisher Information in Noisy Intermediate-Scale Quantum Applications
• 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
• Universal discriminative quantum neural networks
• On the Universality of the Quantum Approximate Optimization Algorithm
• A quantum algorithm to train neural networks using low-depth circuits
• Quantum approximate optimization is computationally universal
• A Feasible Approach for Automatically Differentiable Unitary Coupled-Cluster on Quantum Computers
• Variational Hamiltonian Diagonalization for Dynamical Quantum Simulation
• Analytic gradients in variational quantum algorithms: Algebraic extensions of the parameter-shift rule to general unitary transformations
• Generalized quantum circuit differentiation rules
• Quantum Analytic Descent
• Abrupt Transitions in Variational Quantum Circuit Training
• Classical and Quantum Bounded Depth Approximation Algorithms
• Variational quantum algorithm for molecular geometry optimization
• A Jacobi Diagonalization and Anderson Acceleration Algorithm For Variational Quantum Algorithm Parameter Optimization
• Single-component gradient rules for variational quantum algorithms
• Calculus on parameterized quantum circuits
• Local, Expressive, Quantum-Number-Preserving VQE Ansatze for Fermionic Systems

Cited by in corpus (28)

• The Variational Quantum Eigensolver: a review of methods and best practices
• Quantum Machine Learning: from physics to software engineering
• Near-Term Quantum Computing Techniques: Variational Quantum Algorithms, Error Mitigation, Circuit Compilation, Benchmarking and Classical Simulation
• Recent advances for quantum classifiers
• Generalization despite overfitting in quantum machine learning models
• Quantum Kernel Methods for Solving Differential Equations
• Analytical nonadiabatic couplings and gradients within the state-averaged orbital-optimized variational quantum eigensolver
• Constrained Optimization via Quantum Zeno Dynamics
• Optimizing quantum circuits with Riemannian gradient flow
• Fock State-enhanced Expressivity of Quantum Machine Learning Models
• Training variational quantum circuits with CoVaR: covariance root finding with classical shadows
• Inference-Based Quantum Sensing
• Optimizing the information extracted by a single qubit measurement
• Quantum Advantage Seeker with Kernels (QuASK): a software framework to speed up the research in quantum machine learning
• Guaranteed efficient energy estimation of quantum many-body Hamiltonians using ShadowGrouping
• Randomized adaptive quantum state preparation
• Measurement-induced entanglement phase transitions in variational quantum circuits
• Faster variational quantum algorithms with quantum kernel-based surrogate models
• Sequential optimal selection of a single-qubit gate and its relation to barren plateau in parameterized quantum circuits
• Hybrid Quantum-Classical Autoencoders for End-to-End Radio Communication
• Reduce&chop: Shallow circuits for deeper problems
• Information flow in parameterized quantum circuits
• Differentiable quantum computational chemistry with PennyLane
• Diagrammatic Analysis for Parameterized Quantum Circuits
• Iteration Complexity of Variational Quantum Algorithms
• Calculation of the moscovium ground-state energy by quantum algorithms
• Addition and Differentiation of ZX-diagrams
• "Proper" Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions