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SIAM Journal on Scientific Computing

Schrödingerization Based Computationally Stable Algorithms for Ill-Posed Problems in Partial Differential Equations

Shi Jin, Nana Liu, Chuwen Ma

Published Aug 25, 2025• 2 citations
Publisher

Abstract

Abstract. We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerization, introduced in [S. Jin, N. Liu, and Y. Yu, Phys. Rev. Lett., 133 (2024), 230602; S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerized equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the (constant and variable coefficient) backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods are applicable to both classical and quantum computers, and we also lay out quantum algorithms for these methods. Moreover, we introduce a smooth initialization for the Schrödingerized equation, which will lead to essentially spectral accuracy for the approximation in the extended space, if a spectral method is used. Consequently, the extra qubits needed due to the extra dimension, if a qubit-based quantum algorithm is used, for both well-posed and ill-posed problems, becomes almost [Formula: see text] where [Formula: see text] is the desired precision. This optimizes the complexity of the Schrödingerization based quantum algorithms for any nonunitary dynamical system introduced in [S. Jin, N. Liu, and Y. Yu, Phys. Rev. Lett., 133 (2024), 230602; S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603].

DOI
10.1137/24m1656116

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Abstract. We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerization, introduced in [S. Jin, N. Liu, and Y. Yu, Phys. Rev. Lett., 133 (2024), 230602; S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerized equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the (constant and variable coefficient) backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods are applicable to both classical and quantum computers, and we also lay out quantum algorithms for these methods. Moreover, we introduce a smooth initialization for the Schrödingerized equation, which will lead to essentially spectral accuracy for the approximation in the extended space, if a spectral method is used. Consequently, the extra qubits needed due to the extra dimension, if a qubit-based quantum algorithm is used, for both well-posed and ill-posed problems, becomes almost [Formula: see text] where [Formula: see text] is the desired precision. This optimizes the complexity of the Schrödingerization based quantum algorithms for any nonunitary dynamical system introduced in [S. Jin, N. Liu, and Y. Yu, Phys. Rev. Lett., 133 (2024), 230602; S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603].
We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerization, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerized equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods are applicable to both classical and quantum computers, and we also lay out quantum algorithms for these methods. Moreover, we introduce a smooth initialization for the Schrödingerized equation which will lead to essentially spectral accuracy for the approximation in the extended space, if a spectral method is used. Consequently, the extra qubits needed due to the extra dimension, if a qubit based quantum algorithm is used, for both well-posed and ill-posed problems, becomes almost log⁡log⁡1/ε\log\log {1/\varepsilon}loglog1/ε where ε\varepsilonε is the desired precision. This optimizes the complexity of the Schrödingerization based quantum algorithms for any non-unitary dynamical system introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603].
Abstract. We analyze the Schrödingerization method for quantum simulation of a general class of nonunitary dynamics with inhomogeneous source terms. The Schrödingerization technique, introduced in [S. Jin, N. Liu, and Y. Yu, Phys. Rev. Lett., 133 (2024), 230602], transforms any linear ordinary and partial differential equations with nonunitary 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.
Abstract. Quantum dynamics, typically expressed in the form of a time-dependent Schrödinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission of electrons, it is necessary to use artificial boundary conditions (ABCs) to confine the computation within a fixed domain. The introduction of ABCs alters the Hamiltonian structure of the dynamics, and existing quantum algorithms cannot be directly applied since the evolution is no longer unitary. The current paper utilizes a recently introduced Schrödingerization method that converts non-Hermitian dynamics into a Schrödinger form for the artificial boundary problems [S. Jin, N. Liu, and Y. Yu, Quantum Simulation of Partial Differential Equations via Schrödingerisation, preprint, arXiv:2212.13969 , 2022], [S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603]. We implement this method for three types of ABCs, including the complex absorbing potential technique, perfectly matched layer methods, and Dirichlet-to-Neumann approach. We analyze the query complexity of these algorithms and perform numerical experiments to demonstrate the validity of this approach. This helps to bridge the gap between available quantum algorithms and computational models for quantum dynamics in unbounded domains.
We study a recently introduced simple method [S. Jin, N. Liu, and Y. Yu, Quantum simulation of partial differential equations via Schr\"odingerisation, arXiv:2212.13969] for solving general linear partial differential equations with quantum simulation. This method converts linear partial differential equations into a Hamiltonian system, using a simple transformation called the warped phase transformation. Here we provide a more-in-depth technical discussion and expand on this approach in a more detailed and pedagogical way. We apply this to examples of partial differential equations, including heat, convection, Fokker-Planck, linear Boltzmann, and Black-Scholes equations. This approach can also be extended to general linear partial differential equations, including the Vlasov-Fokker-Planck equation and the Liouville representation equation for nonlinear ordinary differential equations. Extension to higher-order time derivatives is also possible.
We develop a quantum algorithm for linear algebraic equations $ A\bb{x} = \bb{b} $ from the perspective of Schrödingerization-form problems, which are characterized by a system of linear convection equations in one higher dimension. When $ A $ is positive definite, the solution $ \bb{x} $ can be interpreted as the steady-state solution to a system of linear ordinary differential equations (ODEs). This ODE system can be solved by using the linear combination of Hamiltonian simulation (LCHS) method in \cite{ACL2023LCH2}, which serves as the continuous implementation of the Fourier transform in the Schrödingerization method from \cite{JLY22SchrShort, JLY22SchrLong}. Schrödingerization transforms linear partial differential equations (PDEs) and ODEs with non-unitary dynamics into Schrödinger-type systems via the so-called warped phase transformation that maps the equation into one higher dimension. When $ A $ is a general Hermitian matrix, the inverse matrix can still be represented in the LCHS form in \cite{ACL2023LCH2}, but with a kernel function based on the Fourier approach in \cite{Childs2017QLSA}. Although this LCHS form provides the steady-state solution to a system of linear ODEs associated with the least-squares equation, applying Schrödingerization to this least-squares system is not appropriate, as it results in a much larger condition number. We demonstrate that in both cases, the solution $ \bb{x} $ can be expressed as the LCHS of Schrödingerization-form problems. We provide a detailed implementation and error analysis. Furthermore, we incorporate a block preconditioning technique to achieve nearly linear scaling in the condition number, thereby attaining near-optimal query complexity.
This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretisation of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method (Jin et al. 2022) -- it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems, we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions.