This paper investigates quantum simulation algorithms for the Liouville equation in geometrical optics with partial transmission and reflection at sharp interfaces, based on the Schrödingerization method. By means of a warped phase transformation in one higher dimension, the Schrödingerization method converts linear partial differential equations into a system of Schrödinger-type equations with unitary evolution, thereby rendering them suitable for quantum simulation. In this work, the Schrödingerization method is combined with a Hamiltonian-preserving scheme that incorporates partial transmission and reflection into the numerical flux. A main difficulty is that the interface treatment in the classical scheme relies on threshold-dependent "if/else" procedures, making it highly nontrivial to reformulate the method in a matrix form suitable for quantum simulation. To overcome this difficulty, we encode the interface conditions into a partial transmission and reflection matrix prepared a priori, rather than during the time evolution. We present detailed constructions of the resulting quantum algorithms and show through complexity analysis that the proposed methods achieve polynomial quantum advantage in the precision parameter $ε$ over their classical counterparts.
This paper investigates quantum simulation algorithms for the Liouville equation in geometrical optics with partial transmission and reflection at sharp interfaces, based on the Schrödingerization method. By means of a warped phase transformation in one higher dimension, the Schrödingerization method converts linear partial differential equations into a system of Schrödinger-type equations with unitary evolution, thereby rendering them suitable for quantum simulation. In this work, the Schrödingerization method is combined with a Hamiltonian-preserving scheme that incorporates partial transmission and reflection into the numerical flux. A main difficulty is that the interface treatment in the classical scheme relies on threshold-dependent "if/else" procedures, making it highly nontrivial to reformulate the method in a matrix form suitable for quantum simulation. To overcome this difficulty, we encode the interface conditions into a partial transmission and reflection matrix prepared a priori, rather than during the time evolution. We present detailed constructions of the resulting quantum algorithms and show through complexity analysis that the proposed methods achieve polynomial quantum advantage in the precision parameter ε over their classical counterparts.
We develop quantum simulation algorithms for the Liouville equation of classical mechanics with discontinuous potential. Such discontinuities represent potential barriers at which classical particles undergo energy preserving transmission or reflection, and the resulting interface conditions must be incorporated into the numerical flux. We combine Hamiltonian-preserving schemes by Jin and Wen in Commun. Math. Sci. 3(3), 285-315 (2005) with the Schrödingerization method, which embeds the resulting nonunitary semi-discrete dynamics into a unitary Schrödinger type system in one additional auxiliary variable [arXiv:2212.14703, arXiv:2212.13969]. For one-, two-, and n-dimensional problems with grid aligned interfaces, we construct sparse matrix representations of the transmission and reflection fluxes using step and hat functions, derive the corresponding Hamiltonians of the Schrödingerized systems, and analyze their sparse-access query complexity. In the sparse-access oracle model, the resulting algorithms have a polynomial dependence on the inverse accuracy and avoid the exponential dependence on the phase-space dimension suffered by classical grid based Hamiltonian-preserving schemes, up to the cost of implementing the oracles and the postselection overhead. We also describe the postselected recovery of the physical solution state and the quantum readout of macroscopic observables such as density and averaged velocity through overlap estimation. Numerical experiments based on classical simulation of the Schrödingerized dynamics validate the proposed formulation and illustrate the correct transmission/reflection behavior at potential barriers.
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.
We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the d-dimensional fractional equation is reformulated as a local partial differential equation in d+1 dimensions. We propose a quantum algorithm for the finite element discretization of this local problem, by capturing the steady-state of the corresponding differential equations using the Schrödingerization approach from \cite{JLY22SchrShort, JLY22SchrLong, analogPDE}. The Schrödingerization technique transforms general linear partial and ordinary differential equations into Schrödinger-type systems, making them suitable for quantum simulation. This is achieved through the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of the method and conduct a comprehensive complexity analysis, which can show up to exponential advantage -- with respect to the inverse of the mesh size in high dimensions -- compared to its classical counterpart. Specifically, while the classical method requires O(d1/233d/2h−d−2) operations, the quantum counterpart requires O(d33d/2h−2.5) queries to the block-encoding input models, with the quantum complexity being independent of the dimension d in terms of the inverse mesh size h−1. Numerical experiments are conducted to verify the validity of our formulation.
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.
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.
In this paper, we present quantum algorithms for a class of highly-oscillatory transport equations, which arise in semi-classical computation of surface hopping problems and other related non-adiabatic quantum dynamics, based on the Born-Oppenheimer approximation. Our method relies on the classical nonlinear geometric optics method, and the recently developed Schrödingerisation approach for quantum simulation of partial differential equations. The Schrödingerisation technique can transform any linear ordinary and partial differential equations into Hamiltonian systems evolving under unitary dynamics, via a warped phase transformation that maps these equations to one higher dimension. We study possible paths for better recoveries of the solution to the original problem by shifting the bad eigenvalues in the Schrödingerized system. Our method ensures the uniform error estimates independent of the wave length, thus allowing numerical accuracy, in maximum norm, even without numerically resolving the physical oscillations. Various numerical experiments are performed to demonstrate the validity of this approach.