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… 

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References

SHOWING 1-10 OF 68 REFERENCES

Quantum Algorithm for Time-Dependent Hamiltonian Simulation by Permutation Expansion

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

  • Dong AnLin Lin
  • Computer Science
    ACM 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.

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

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

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

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

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

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

  • H. Krovi
  • Computer Science, Mathematics
    ArXiv
  • 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

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.
...