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

@article{Fang2022TimemarchingBQ, title={Time-marching based quantum solvers for time-dependent linear differential equations}, author={Di Fang and Lin Lin and Yu Tong}, journal={ArXiv}, year={2022}, volume={abs/2208.06941} }

The time-marching strategy, which propagates the solution from one time step to the next, is a natural strategy for solving time-dependent diﬀerential equations on classical computers, as well as for solving the Hamiltonian simulation problem on quantum computers. For more general linear diﬀerential equations dd t ) | ) i , | i = | ψ 0 i , a time-marching based quantum solver can suﬀer from exponentially vanishing success probability with respect to the number of time steps and is thus…

## 4 Citations

### Optimal Hamiltonian simulation for time-periodic systems

- Physics
- 2022

The implementation of time-evolution operators U ( t ), called Hamiltonian simulation, is one of the most promising usage of quantum computers. For time-independent Hamiltonians, qubitization has…

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

- Computer ScienceArXiv
- 2022

It is shown that ODEs in the absence of both types of “non-quantumness” are equivalent to quantum dynamics, and the conclusion is reached that quantum algorithms for quantum dynamics work best.

### Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

- MathematicsSSRN Electronic Journal
- 2022

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear…

### Infinite quantum signal processing

- Mathematics
- 2022

Quantum signal processing (QSP) represents a real scalar polynomial of degree d using a product of unitary matrices of size 2 × 2 , parameterized by ( d +1) real numbers called the phase factors.…

## References

SHOWING 1-10 OF 68 REFERENCES

### Quantum Algorithm for Time-Dependent Hamiltonian Simulation by Permutation Expansion

- PhysicsPRX Quantum
- 2021

We present a quantum algorithm for the dynamical simulation of time-dependent Hamiltonians. Our method involves expanding the interaction-picture Hamiltonian as a sum of generalized permutations,…

### Quantum Linear System Solver Based on Time-optimal Adiabatic Quantum Computing and Quantum Approximate Optimization Algorithm

- Computer ScienceACM Transactions on Quantum Computing
- 2022

We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with O(κ poly(log (κ ε))) runtime, where κ…

### Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Computing.

- Physics, Computer SciencePhysical review letters
- 2019

Two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state |x⟩ that is proportional to the solution of the system of linear equations Ax[over →]=b[ over →], yielding an exponential quantum speed-up under some assumptions.

### High-order quantum algorithm for solving linear differential equations

- Computer ScienceArXiv
- 2010

This work extends quantum simulation algorithms to general inhomogeneous sparse linear differential equations, which describe many classical physical systems, and examines the use of high-order methods to improve the efficiency.

### Quantum Spectral Methods for Differential Equations

- Computer ScienceCommunications in Mathematical Physics
- 2020

A quantum algorithm for linear ordinary differential equations based on so-called spectral methods is developed, an alternative to finite difference methods that approximates the solution globally.

### Variable time amplitude amplification and quantum algorithms for linear algebra problems

- Computer ScienceSTACS
- 2012

This paper generalizes quantum amplitude amplification to the case when parts of the algorithm that is being amplified stop at different times, and applies it to give two new quantum algorithms for linear algebra problems.

### Efficient phase-factor evaluation in quantum signal processing

- Computer Science
- 2020

This work presents an optimization based method that can accurately compute the phase factors using standard double precision arithmetic operations and demonstrates the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and the quantum linear system problems.

### Quantum Algorithm for Linear Differential Equations with Exponentially Improved Dependence on Precision

- Computer Science
- 2017

This work presents a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients that produces a quantum state that is proportional to the solution at a desired final time using a Taylor series.

### Improved quantum algorithms for linear and nonlinear differential equations

- Computer Science, MathematicsArXiv
- 2022

The present quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices including singular matrices and can handle any sparse matrix (that models dissipation) if it has a negative log-norm (including non- diagonalizableMatrices), whereas [2] and [3] additionally require normality.

### Exponential improvement in precision for simulating sparse Hamiltonians

- Computer ScienceForum of Mathematics, Sigma
- 2017

The algorithm is based on a significantly improved simulation of the continuous- and fractional- query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error.