Quantum algorithms for computing observables of nonlinear partial differential equations

Abstract

We construct quantum algorithms to compute physical observables of nonlinear PDEs with M initial data. Based on an exact mapping between nonlinear and linear PDEs using the level set method, these new quantum algorithms for nonlinear Hamilton-Jacobi and scalar hyperbolic PDEs can be performed with a computational cost that is independent of M, for arbitrary nonlinearity. Depending on the details of the initial data, it can also display up to exponential advantage in both the dimension of the PDE and the error in computing its observables. For general nonlinear PDEs, quantum advantage with respect to M is possible in the large M limit.

References in corpus (13)

• Quantum algorithm for solving linear systems of equations
• Efficient quantum algorithm for dissipative nonlinear differential equations
• Optimal Quantum Measurements of Expectation Values of Observables
• Quantum algorithm and circuit design solving the Poisson equation
• Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications
• High-precision quantum algorithms for partial differential equations
• Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics
• Improved quantum algorithms for linear and nonlinear differential equations
• On applications of quantum computing to plasma simulations
• Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations
• Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations
• Tight Bound for Estimating Expectation Values from a System of Linear Equations
• Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Cited by in corpus (9)

• Quantum Circuits for partial differential equations via Schrödingerisation
• Efficient explicit gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations
• Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling
• Quantum Dynamics Simulation of the Advection-Diffusion Equation
• Quantum homotopy analysis method with quantum-compatible linearization for nonlinear partial differential equations
• Quantum circuits for partial differential equations in Fourier space
• An Efficient Decomposition of the Carleman Linearized Burgers' Equation
• Quantum Simulation-Based Optimization for Cooling System Design
• Randomized adiabatic quantum linear solver algorithm with optimal complexity scaling and detailed running costs