DeepAI

# A theory of quantum differential equation solvers: limitations and fast-forwarding

We study the limitations and fast-forwarding of quantum algorithms for solving linear ordinary differential equation (ODE) systems with particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the ODE is inhomogeneous. On the one hand, for generic homogeneous linear ODEs, by proving worst-case lower bounds, we show that quantum algorithms suffer from computational overheads due to two types of “non-quantumness”: real part gap and non-normality of the coefficient matrix. We then show that ODEs in the absence of both types of “non-quantumness” are equivalent to quantum dynamics, and reach the conclusion that quantum algorithms for quantum dynamics work best. We generalize our results to the inhomogeneous case and find that existing generic quantum ODE solvers cannot be substantially improved. To obtain these lower bounds, we propose a general framework for proving lower bounds on quantum algorithms that are amplifiers, meaning that they amplify the difference between a pair of input quantum states. On the other hand, we show how to fast-forward quantum algorithms for solving special classes of ODEs which leads to improved efficiency. More specifically, we obtain quadratic to exponential improvements in terms of the evolution time T and the spectral norm of the coefficient matrix for the following classes of ODEs: inhomogeneous ODEs with a negative definite coefficient matrix, inhomogeneous ODEs with a coefficient matrix having an eigenbasis that can be efficiently prepared on a quantum computer and eigenvalues that can be efficiently computed classically, and the spatially discretized inhomogeneous heat equation and advection-diffusion equation. We give fast-forwarding algorithms that are conceptually different from existing ones in the sense that they neither require time discretization nor solving high-dimensional linear systems.

• 13 publications
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08/14/2022

### Time-marching based quantum solvers for time-dependent linear differential equations

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### Quantum advantage for differential equation analysis

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### Local lower bounds on characteristics of quantum and classical systems

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### Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix

Learning about physical systems from quantum-enhanced experiments, relyi...
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### Analysis of GMRES for Low-Rank and Small-Norm Perturbations of the Identity Matrix

In many applications, linear systems arise where the coefficient matrix ...
08/30/2020

### Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions

Preconditioning is the most widely used and effective way for treating i...
05/21/2020

### Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics

Fox and Lu introduced a Langevin framework for discrete-time stochastic ...