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Quantum algorithm for time-dependent differential equations using Dyson series

Dominic W. Berry, Pedro C. S. Costa

TL;DR

The paper introduces a quantum algorithm to solve time-dependent linear differential equations by encoding the Dyson-series solution into a block-encoded linear system and solving it with an optimal quantum linear-system solver. The approach yields a complexity that scales nearly linearly with the total evolution time T and polylogarithmically with the desired precision ε, while achieving exponential speedups in dimension relative to classical methods, under amplitude-encoded outputs. A key feature is the use of a Dyson-series-based block encoding that avoids requiring smoothness or high-order derivative information, with conditioning controlled by the logarithmic norm of A(t). The method also provides a streamlined, simplified treatment of the time-independent case and improves log-factor dependencies over previous spectral approaches, making it a practical route for quantum-accelerated ODE solving in dissipative settings.

Abstract

Time-dependent linear differential equations are a common type of problem that needs to be solved in classical physics. Here we provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence of the complexity on the error and derivative. As usual, there is an exponential improvement over classical approaches in the scaling of the complexity with the dimension, with the caveat that the solution is encoded in the amplitudes of a quantum state. Our method is to encode the Dyson series in a system of linear equations, then solve via the optimal quantum linear equation solver. Our method also provides a simplified approach in the case of time-independent differential equations.

Quantum algorithm for time-dependent differential equations using Dyson series

TL;DR

The paper introduces a quantum algorithm to solve time-dependent linear differential equations by encoding the Dyson-series solution into a block-encoded linear system and solving it with an optimal quantum linear-system solver. The approach yields a complexity that scales nearly linearly with the total evolution time T and polylogarithmically with the desired precision ε, while achieving exponential speedups in dimension relative to classical methods, under amplitude-encoded outputs. A key feature is the use of a Dyson-series-based block encoding that avoids requiring smoothness or high-order derivative information, with conditioning controlled by the logarithmic norm of A(t). The method also provides a streamlined, simplified treatment of the time-independent case and improves log-factor dependencies over previous spectral approaches, making it a practical route for quantum-accelerated ODE solving in dissipative settings.

Abstract

Time-dependent linear differential equations are a common type of problem that needs to be solved in classical physics. Here we provide a quantum algorithm for solving time-dependent linear differential equations with logarithmic dependence of the complexity on the error and derivative. As usual, there is an exponential improvement over classical approaches in the scaling of the complexity with the dimension, with the caveat that the solution is encoded in the amplitudes of a quantum state. Our method is to encode the Dyson series in a system of linear equations, then solve via the optimal quantum linear equation solver. Our method also provides a simplified approach in the case of time-independent differential equations.
Paper Structure (17 sections, 2 theorems, 112 equations)

This paper contains 17 sections, 2 theorems, 112 equations.

Table of Contents

  1. Form of solution
  2. Solution to the homogeneous equation
  3. Particular solution
  4. The time-independent solution
  5. Encoding in linear equations
  6. Time-dependent equations
  7. Encoding
  8. Condition number
  9. Time-independent equations
  10. Block encoding
  11. Time-dependent block encoding
  12. Time-independent block encoding
  13. Complexity of solution
  14. Time-dependent complexity
  15. Time-independent complexity
  16. ...and 2 more sections

Key Result

Theorem 4.1

We are given an ordinary linear differential equation of the form where $\bm{b}(t) \in \mathbb{C}^{N}$ is a vector function of $t$, $A(t)\in \mathbb{C}^{N\times N}$ is a coefficient matrix with non-positive logarithmic norm, and $\bm{x}(t)\in \mathbb{C}^{N}$ is the solution vector as a function of $t$. The parameters of the differential equation are provided via u A quantum algorithm can provide

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 4.2