Error and Resource Estimates of Variational Quantum Algorithms for Solving Differential Equations Based on Runge-Kutta Methods
arXiv:2412.12262 · doi:10.1063/5.0258074
Abstract
A focus of recent research in quantum computing has been on developing quantum algorithms for differential equations solving using variational methods on near-term quantum devices. A promising approach involves variational algorithms, which combine classical Runge-Kutta methods with quantum computations. However, a rigorous error analysis, essential for assessing real-world feasibility, has so far been lacking. In this paper, we provide an extensive analysis of error sources and determine the resource requirements needed to achieve specific target errors. In particular, we derive analytical error and resource estimates for scenarios with and without shot noise, examining shot noise in quantum measurements and truncation errors in Runge-Kutta methods. Our analysis does not take into account representation errors and hardware noise, as these are specific to the instance and the used device. We evaluate the implications of our results by applying them to two scenarios: classically solving a D ordinary differential equation and solving an option pricing linear partial differential equation with the variational algorithm, showing that the most resource-efficient methods are of order 4 and 2, respectively. This work provides a framework for optimizing quantum resources when applying Runge-Kutta methods, enhancing their efficiency and accuracy in both solving differential equations and simulating quantum systems.
57 pages, 9 figures
References in corpus (33)
- Supplementary information for "Quantum supremacy using a programmable superconducting processor"
- A variational eigenvalue solver on a quantum processor
- Quantum algorithm for solving linear systems of equations
- Surface codes: Towards practical large-scale quantum computation
- Quantum Circuit Learning
- The effect of data encoding on the expressive power of variational quantum machine learning models
- Efficient variational quantum simulator incorporating active error minimisation
- Parallel implementation of high-fidelity multi-qubit gates with neutral atoms
- Variational ansatz-based quantum simulation of imaginary time evolution
- Quantum computing with neutral atoms
- Theory of variational quantum simulation
- Variational quantum algorithms for nonlinear problems
- Fast quantum logic gates with trapped-ion qubits
- Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
- High-order quantum algorithm for solving linear differential equations
- Variational quantum simulation of general processes
- Variational Fast Forwarding for Quantum Simulation Beyond the Coherence Time
- Quantum algorithm for linear differential equations with exponentially improved dependence on precision
- Solving nonlinear differential equations with differentiable quantum circuits
- Hardware-efficient variational quantum algorithms for time evolution
- Adaptive Variational Quantum Dynamics Simulations
- An efficient quantum algorithm for the time evolution of parameterized circuits
- Subspace Variational Quantum Simulator
- Quantum algorithms for quantum dynamics: A performance study on the spin-boson model
- Quantum simulation of partial differential equations via Schrodingerisation
- Challenges of variational quantum optimization with measurement shot noise
- Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods
- A variational quantum algorithm for the Feynman-Kac formula
- Nonlinear dynamics as a ground-state solution on quantum computers
- Variational Quantum Time Evolution without the Quantum Geometric Tensor
- Error Bounds for Variational Quantum Time Evolution
- Quench dynamics of the Schwinger model via variational quantum algorithms
- Accelerating quantum imaginary-time evolution with random measurements