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On Schrödingerization based quantum algorithms for linear dynamical systems with inhomogeneous terms

Shi Jin, Nana Liu, Chuwen Ma

TL;DR

The paper develops and analyzes Schrödingerization for general linear dynamical systems with inhomogeneous terms, transforming non-unitary dynamics into unitary dynamics in an extended space via a warped phase transform. It provides rigorous recovery criteria for extracting the original variables despite unstable modes, and derives error bounds for both discrete and continuous Fourier discretizations, including smoother initializations and a stretch transformation to mitigate positive eigenvalues. The authors extend the framework to non-autonomous systems and inhomogeneous sources, offering autonomous reformulations and detailed quantum-device implementation steps with cost and success-probability analyses. Numerical tests, including ill-posed backward-heat problems and Maxwell-like systems with sources, demonstrate stable recovery, convergence rates, and practical guidance on discretization choices. Overall, Schrödingerization offers a robust, generic approach to simulate and stabilize non-unitary, ill-posed linear dynamics on quantum platforms, with explicit recovery, error, and complexity guarantees.

Abstract

We analyze the Schrödingerization method for quantum simulation of a general class of non-unitary dynamics with inhomogeneous source terms. The Schrödingerization technique, introduced in [31], transforms any linear ordinary and partial differential equations with non-unitary dynamics into a system under unitary dynamics via a warped phase transition that maps the equations into a higher dimension, making them suitable for quantum simulation. This technique can also be applied to these equations with inhomogeneous terms modeling source or forcing terms, or boundary and interface conditions, and discrete dynamical systems such as iterative methods in numerical linear algebra, through extra equations in the system. Difficulty arises with the presence of inhomogeneous terms since they can change the stability of the original system. In this paper, we systematically study-both theoretically and numerically-the important issue of recovering the original variables from the Schrödingerized equations, even when the evolution operator contains unstable modes. We show that, even with unstable modes, one can still construct a stable scheme; however, to recover the original variable, one needs to use suitable data in the extended space. We analyze and compare both the discrete and continuous Fourier transforms used in the extended dimension and derive corresponding error estimates, which allow one to use the more appropriate transform for specific equations. We also provide a smoother initialization for the Schrödingerized system to gain higher-order accuracy in the extended space. We homogenize the inhomogeneous terms with a stretch transformation, making it easier to recover the original variable. Our recovery technique also provides a simple and generic framework to solve general ill-posed problems in a computationally stable way.

On Schrödingerization based quantum algorithms for linear dynamical systems with inhomogeneous terms

TL;DR

The paper develops and analyzes Schrödingerization for general linear dynamical systems with inhomogeneous terms, transforming non-unitary dynamics into unitary dynamics in an extended space via a warped phase transform. It provides rigorous recovery criteria for extracting the original variables despite unstable modes, and derives error bounds for both discrete and continuous Fourier discretizations, including smoother initializations and a stretch transformation to mitigate positive eigenvalues. The authors extend the framework to non-autonomous systems and inhomogeneous sources, offering autonomous reformulations and detailed quantum-device implementation steps with cost and success-probability analyses. Numerical tests, including ill-posed backward-heat problems and Maxwell-like systems with sources, demonstrate stable recovery, convergence rates, and practical guidance on discretization choices. Overall, Schrödingerization offers a robust, generic approach to simulate and stabilize non-unitary, ill-posed linear dynamics on quantum platforms, with explicit recovery, error, and complexity guarantees.

Abstract

We analyze the Schrödingerization method for quantum simulation of a general class of non-unitary dynamics with inhomogeneous source terms. The Schrödingerization technique, introduced in [31], transforms any linear ordinary and partial differential equations with non-unitary dynamics into a system under unitary dynamics via a warped phase transition that maps the equations into a higher dimension, making them suitable for quantum simulation. This technique can also be applied to these equations with inhomogeneous terms modeling source or forcing terms, or boundary and interface conditions, and discrete dynamical systems such as iterative methods in numerical linear algebra, through extra equations in the system. Difficulty arises with the presence of inhomogeneous terms since they can change the stability of the original system. In this paper, we systematically study-both theoretically and numerically-the important issue of recovering the original variables from the Schrödingerized equations, even when the evolution operator contains unstable modes. We show that, even with unstable modes, one can still construct a stable scheme; however, to recover the original variable, one needs to use suitable data in the extended space. We analyze and compare both the discrete and continuous Fourier transforms used in the extended dimension and derive corresponding error estimates, which allow one to use the more appropriate transform for specific equations. We also provide a smoother initialization for the Schrödingerized system to gain higher-order accuracy in the extended space. We homogenize the inhomogeneous terms with a stretch transformation, making it easier to recover the original variable. Our recovery technique also provides a simple and generic framework to solve general ill-posed problems in a computationally stable way.
Paper Structure (26 sections, 14 theorems, 129 equations, 4 figures, 2 tables)

This paper contains 26 sections, 14 theorems, 129 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A review of Schrödingerization for general linear dynamical systems
  3. The discrete Fourier transform for Schrödingerization
  4. The continuous Fourier transform for Schrödingerization
  5. Recovery of the original variables and implementations on quantum devices
  6. Recovery of the original variable
  7. Implementation on a quantum device
  8. Computation cost for the measurements
  9. Turning a non-autonomous system into an autonomous one
  10. The discrete Fourier transform
  11. The continuous Fourier transform
  12. Error estimates for the discretizations of the Schrödingerized system
  13. Error estimates for \ref{['eq:schro_dis']}
  14. Error estimates for \ref{['eq:schro_conti_computer']}
  15. Complexity analysis
  16. ...and 11 more sections

Key Result

Theorem 3.1

\newlabelthm:recovery u0 Assume the eigenvalues of $H_1$ satisfy eq:eigenvalues H1, the solution of eq:ODE can be recovered by where $p^{\Diamond}\geq \lambda_{\max}^+(H_1) T$, or recovered by using the integration,

Figures (4)

  • Figure 1: Left: error of $\|{\boldsymbol{u}} - {\boldsymbol{w}}_h^c(p)e^{p}\|/\|{\boldsymbol{u}}\|$ with respect to $p$ with ${\boldsymbol{w}}_h^c$ computed by \ref{['eq:whc']}. Right: the recovery from Schrödingerization by choosing $p>p^{\diamond} = \lambda_n(H_1) T\approx 6$.
  • Figure 2: Left: error of $\|{\boldsymbol{u}} - {\boldsymbol{w}}_h^d(p)e^{p}\|$ with respect to $p$ with ${\boldsymbol{w}}_h^d$ defined in \ref{['eq:whd(t,x,p)']}. Right: the recovery from Schrödingerization by choosing $p>p^{\diamond} = \frac{\pi^2}{4}T\approx 247$.
  • Figure 3: The first row: the error of discrete Fourier transform defined by $\|{\boldsymbol{w}}_h^d e^{p}-{\boldsymbol{u}}\|/\|{\boldsymbol{u}}\|$ for Schrödingerization, with $\triangle p = \frac{\pi}{2^5}$ and $\triangle t = \frac{1}{2^5}$. The second row: the error of continuous Fourier transform defined by $\|{\boldsymbol{w}}_h^c e^p-{\boldsymbol{u}}\|/\|{\boldsymbol{u}}\|$ for Schrödingerization, with $X = 160$ and $\triangle t = \frac{1}{2^5}$.
  • Figure 4: The first row: the error of discrete Fourier transform defined by $\|{\boldsymbol{w}}_h^d e^{p}-{\boldsymbol{u}}\|$ for Schrödingerization, with $\triangle p = \frac{\pi}{2^5}$ and $\triangle t = \frac{1}{2^5}$. The second row: the error of continuous Fourier transform defined by $\|{\boldsymbol{w}}_h^c e^p-{\boldsymbol{u}}\|$ for Schrödingerization, with $X = 160$ and $\triangle t = \frac{1}{2^5}$.

Theorems & Definitions (29)

  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • Proof 2
  • Lemma 4.2
  • Proof 3
  • Lemma 4.3
  • Proof 4
  • ...and 19 more