Solving nonlinear differential equations with differentiable quantum circuits

Abstract

We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations, and compute density, temperature and velocity profiles for the fluid flow in a convergent-divergent nozzle.

updated, close to the published version

References in corpus (127)

• Adam: A Method for Stochastic Optimization
• Quantum Computing in the NISQ era and beyond
• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Quantum Machine Learning
• A variational eigenvalue solver on a quantum processor
• Quantum algorithm for solving linear systems of equations
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Supervised learning with quantum enhanced feature spaces
• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• Barren plateaus in quantum neural network training landscapes
• DGM: A deep learning algorithm for solving partial differential equations
• The theory of variational hybrid quantum-classical algorithms
• Quantum support vector machine for big data classification
• Quantum machine learning in feature Hilbert spaces
• Quantum computational chemistry
• Quantum Circuit Learning
• Quantum Chemistry in the Age of Quantum Computing
• Quantum Convolutional Neural Networks
• Quantum principal component analysis
• Error mitigation for short-depth quantum circuits
• An introduction to quantum machine learning
• Parameterized quantum circuits as machine learning models
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Fourier Neural Operator for Parametric Partial Differential Equations
• Evaluating analytic gradients on quantum hardware
• Quantum algorithms: an overview
• An adaptive variational algorithm for exact molecular simulations on a quantum computer
• Circuit-centric quantum classifiers
• Quantum random access memory
• Scalable Quantum Simulation of Molecular Energies
• Simulating Hamiltonian dynamics with a truncated Taylor series
• The effect of data encoding on the expressive power of variational quantum machine learning models
• Hartree-Fock on a superconducting qubit quantum computer
• Efficient Learning for Deep Quantum Neural Networks
• Efficient variational quantum simulator incorporating active error minimisation
• Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
• Self-Verifying Variational Quantum Simulation of the Lattice Schwinger Model
• Quantum generative adversarial learning
• Variational ansatz-based quantum simulation of imaginary time evolution
• Practical Quantum Error Mitigation for Near-Future Applications
• Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
• Quantum Natural Gradient
• Quantum-enhanced machine learning
• Robust determination of molecular spectra on a quantum processor
• A generative modeling approach for benchmarking and training shallow quantum circuits
• Absence of Barren Plateaus in Quantum Convolutional Neural Networks
• Layerwise learning for quantum neural networks
• Variational quantum algorithms for nonlinear problems
• Towards Quantum Machine Learning with Tensor Networks
• Quantum Computation of Electronic Transitions using a Variational Quantum Eigensolver
• Subspace-search variational quantum eigensolver for excited states
• Active learning machine learns to create new quantum experiments
• Differentiable Learning of Quantum Circuit Born Machine
• Prediction by linear regression on a quantum computer
• Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers
• Efficient and Noise Resilient Measurements for Quantum Chemistry on Near-Term Quantum Computers
• Low-cost error mitigation by symmetry verification
• Error-mitigated digital quantum simulation
• High-order quantum algorithm for solving linear differential equations
• Variational quantum simulation of general processes
• Efficient Symmetry-Preserving State Preparation Circuits for the Variational Quantum Eigensolver Algorithm
• Variational Fast Forwarding for Quantum Simulation Beyond the Coherence Time
• Unsupervised Machine Learning on a Hybrid Quantum Computer
• A quantum linear system algorithm for dense matrices
• Training of Quantum Circuits on a Hybrid Quantum Computer
• Stochastic gradient descent for hybrid quantum-classical optimization
• Quantum algorithm for linear differential equations with exponentially improved dependence on precision
• Quantum algorithms and the finite element method
• Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing
• Variational algorithms for linear algebra
• High-precision quantum algorithms for partial differential equations
• Differentiable Quantum Architecture Search
• Quantum Algorithm for Simulating the Wave Equation
• Quantum noise protects quantum classifiers against adversaries
• The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine
• Estimating the gradient and higher-order derivatives on quantum hardware
• Variational Quantum Eigensolver with Fewer Qubits
• Quantum machine learning for quantum anomaly detection
• Towards simulating 2D effects in lattice gauge theories on a quantum computer
• Methodology for replacing indirect measurements with direct measurements
• Gate-efficient simulation of molecular eigenstates on a quantum computer
• Yao.jl: Extensible, Efficient Framework for Quantum Algorithm Design
• Quantum spectral methods for differential equations
• Quantum autoencoders to denoise quantum data
• An Adaptive Optimizer for Measurement-Frugal Variational Algorithms
• A variational toolbox for quantum multi-parameter estimation
• Machine learning for long-distance quantum communication
• Experimental realization of quantum algorithm for solving linear systems of equations
• Expressibility of the alternating layered ansatz for quantum computation
• Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems
• Calculating energy derivatives for quantum chemistry on a quantum computer
• Variational-State Quantum Metrology
• Learning and Inference on Generative Adversarial Quantum Circuits
• DiffEqFlux.jl - A Julia Library for Neural Differential Equations
• A Quantum Algorithm for Solving Linear Differential Equations: Theory and Experiment
• A quantum algorithm to train neural networks using low-depth circuits
• Adversarial quantum circuit learning for pure state approximation
• A state-averaged orbital-optimized hybrid quantum-classical algorithm for a democratic description of ground and excited states
• Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering cross section of a 2D target
• A Universal Training Algorithm for Quantum Deep Learning
• Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations
• Computational Investigations of the Lithium Superoxide Dimer Rearrangement on Noisy Quantum Devices
• A quantum algorithm to solve nonlinear differential equations
• Theory of analytical energy derivatives for the variational quantum eigensolver
• How will quantum computers provide an industrially relevant computational advantage in quantum chemistry?
• Measurement-based adaptation protocol with quantum reinforcement learning
• Near-term quantum algorithms for linear systems of equations
• Quantum Circuit Design for Solving Linear Systems of Equations
• Adaptive pruning-based optimization of parameterized quantum circuits
• Quantum inverse iteration algorithm for programmable quantum simulators
• Analysis and synthesis of feature map for kernel-based quantum classifier
• Simulating quantum chemistry in the seniority-zero space on qubit-based quantum computers
• Quantum Analytic Descent
• Quantum machine learning and quantum biomimetics: A perspective
• Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation
• Benchmarking adaptive variational quantum eigensolvers
• Advances in Quantum Deep Learning: An Overview
• MoG-VQE: Multiobjective genetic variational quantum eigensolver
• Toward Trainability of Quantum Neural Networks
• On the natural gradient for variational quantum eigensolver
• Non-linear quantum-classical scheme to simulate non-equilibrium strongly correlated fermionic many-body dynamics
• Setting up experimental Bell test with reinforcement learning
• Hybrid classical-quantum linear solver using Noisy Intermediate-Scale Quantum machines
• A diagrammatic approach to variational quantum ansatz construction
• Measurement-based adaptation protocol with quantum reinforcement learning in a Rigetti quantum computer
• Hybrid Quantum/Classical Derivative Theory: Analytical Gradients and Excited-State Dynamics for the Multistate Contracted Variational Quantum Eigensolver
• Quantum-Assisted Clustering Algorithms for NISQ-Era Devices

Cited by in corpus (82)

• Variational Quantum Algorithms
• Quantum computing for finance
• Higher Order Derivatives of Quantum Neural Networks with Barren Plateaus
• Variational Quantum algorithm for Poisson equation
• Variational Quantum Reinforcement Learning via Evolutionary Optimization
• Quantum Machine Learning: from physics to software engineering
• Near-Term Quantum Computing Techniques: Variational Quantum Algorithms, Error Mitigation, Circuit Compilation, Benchmarking and Classical Simulation
• Variational Quantum Algorithms for Computational Fluid Dynamics
• Neural Predictor based Quantum Architecture Search
• A measurement-based variational quantum eigensolver
• Barren plateaus in quantum tensor network optimization
• NISQ Computers: A Path to Quantum Supremacy
• Perceval: A Software Platform for Discrete Variable Photonic Quantum Computing
• Variational Quantum Solutions to the Advection-Diffusion Equation for Applications in Fluid Dynamics
• Quantum Computing: Towards Industry Reference Problems
• Generalized quantum circuit differentiation rules
• Quantum Kernel Methods for Solving Differential Equations
• Quantum Architecture Search via Deep Reinforcement Learning
• Quantum Variational Solving of Nonlinear and Multi-Dimensional Partial Differential Equations
• Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes
• Application of a variational hybrid quantum-classical algorithm to heat conduction equation
• Physics-Informed Neural Networks for Parametric Compressible Euler Equations
• Generalized Quantum Assisted Simulator
• Nonlinear dynamics as a ground-state solution on quantum computers
• FEqa: Finite Element Computations on Quantum Annealers
• Noisy intermediate-scale quantum simulation of the one-dimensional wave equation
• Quantum Gaussian Process Regression for Bayesian Optimization
• Quantum Algorithm for Solving the Advection Equation using Hamiltonian Simulation
• Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations
• Let Quantum Neural Networks Choose Their Own Frequencies
• Hybrid Quantum-Classical Graph Convolutional Network
• Reduction of finite sampling noise in quantum neural networks
• Reduced-order modeling of two-dimensional turbulent Rayleigh-Bénard flow by hybrid quantum-classical reservoir computing
• Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series
• Simulating fluid flows with quantum computing
• Efficient and fail-safe quantum algorithm for the transport equation
• Quantum algorithms for scientific computing
• Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems
• Quantum-Inspired Fluid Simulation of 2D Turbulence with GPU Acceleration
• What can we learn from quantum convolutional neural networks?
• Fully quantum algorithm for lattice Boltzmann methods with application to partial differential equations
• A quantum algorithm for the linear Vlasov equation with collisions
• Boundary Treatment for Variational Quantum Simulations of Partial Differential Equations on Quantum Computers
• Highly optimized quantum circuits synthesized via data-flow engines
• Single-component gradient rules for variational quantum algorithms
• Protocols for classically training quantum generative models on probability distributions
• Self-Adaptive Physics-Informed Quantum Machine Learning for Solving Differential Equations
• Quantum Algorithm for Solving a Quadratic Nonlinear System of Equations
• Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm
• Emergence of noise-induced barren plateaus in arbitrary layered noise models
• Protocols for Trainable and Differentiable Quantum Generative Modelling
• Qsun: an open-source platform towards practical quantum machine learning applications
• Variational Quantum Algorithms for Differential Equations on a Noisy Quantum Computer
• Physics-Informed Neural Networks for Transonic Flows around an Airfoil
• Solving Differential-Algebraic Equations in Power System Dynamic Analysis with Quantum Computing
• Solving Differential-Algebraic Equations in Power System Dynamic Analysis with Quantum Computing
• Towards Variational Quantum Algorithms for generalized linear and nonlinear transport phenomena
• Quantum homotopy analysis method with quantum-compatible linearization for nonlinear partial differential equations
• Iteration Complexity of Variational Quantum Algorithms
• Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems
• Partitioned Quantum Subspace Expansion
• Ground state-based quantum feature maps
• Quantum Extremal Learning
• Quantum circuits for partial differential equations in Fourier space
• Quantum Iterative Methods for Solving Differential Equations with Application to Computational Fluid Dynamics
• Correspondence between open bosonic systems and stochastic differential equations
• qlbm -- A Quantum Lattice Boltzmann Software Framework
• Multidimensional Quantum Generative Modeling by Quantum Hartley Transform
• Modeling Power Systems Dynamics with Symbolic Physics-Informed Neural Networks
• Quantum simulation and ground state preparation for the honeycomb Kitaev model
• Variational quantum algorithm based on Lagrange polynomial encoding to solve differential equations
• Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains
• A Hybrid Quantum-Classical Particle-in-Cell Method for Plasma Simulations
• Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations
• Efficient quantum algorithm for weighted partial sums and numerical integration
• Quantum Simulation-Based Optimization for Cooling System Design
• Quantum Model-Discovery
• Fully Quantum Lattice Gas Automata Building Blocks for Computational Basis State Encodings
• Vortex Detection from Quantum Data
• Quantum Chebyshev Probabilistic Models for Fragmentation Functions
• Quantum-Enhanced Neural Exchange-Correlation Functionals
• Error and Resource Estimates of Variational Quantum Algorithms for Solving Differential Equations Based on Runge-Kutta Methods