ABSTRACT Nagy's unitary dilation theorem in the operator theory asserts the possibility of dilating a contraction into a unitary operator. When used in quantum computing, its practical implementation primarily relies on block‐encoding techniques, based on finite‐dimensional scenarios. In this study, we delve into the recently devised Schrödingerization approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of , which arises in wide‐ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerization approach is adaptable to both finite‐ and infinite‐dimensional cases, in both countable and uncountable domains. For quantum systems lying in infinite‐dimensional Hilbert space, the dilation involves adding a single infinite dimensional mode, and this is the continuous‐variable version of the Schrödingerization procedure which makes it suitable for analog quantum computing. Furthermore, by discretizing continuous variables, the Schrödingerization method can also be effectively employed in finite‐dimensional scenarios suitable for qubit‐based quantum computing.
ABSTRACT Nagy's unitary dilation theorem in the operator theory asserts the possibility of dilating a contraction into a unitary operator. When used in quantum computing, its practical implementation primarily relies on block‐encoding techniques, based on finite‐dimensional scenarios. In this study, we delve into the recently devised Schrödingerization approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of , which arises in wide‐ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerization approach is adaptable to both finite‐ and infinite‐dimensional cases, in both countable and uncountable domains. For quantum systems lying in infinite‐dimensional Hilbert space, the dilation involves adding a single infinite dimensional mode, and this is the continuous‐variable version of the Schrödingerization procedure which makes it suitable for analog quantum computing. Furthermore, by discretizing continuous variables, the Schrödingerization method can also be effectively employed in finite‐dimensional scenarios suitable for qubit‐based quantum computing.
Nagy's unitary dilation theorem in operator theory asserts the possibility of dilating a contraction into a unitary operator. When used in quantum computing, its practical implementation primarily relies on block-encoding techniques, based on finite-dimensional scenarios. In this study, we delve into the recently devised Schrödingerisation approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of V(t)=exp(−At), which arises in wide-ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerisation approach is adaptable to both finite and infinite-dimensional cases, in both countable and uncountable domains. For quantum systems lying in infinite dimensional Hilbert space, the dilation involves adding a single infinite dimensional mode, and this is the continuous-variable version of the Schrödingerisation procedure which makes it suitable for analog quantum computing. Furthermore, by discretising continuous variables, the Schrödingerisation method can also be effectively employed in finite-dimensional scenarios suitable for qubit-based quantum computing.
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 present a novel new way-called Schrödingerization-to simulate general (quantum and nonquantum) systems of linear ordinary and partial differential equations (PDEs) via quantum simulation. We introduce a new transform, referred to as the warped phase transformation, where any linear-including nonautonamous-system of ordinary or partial differential equation can be recast into a system of Schrödinger's equations, in real time, in a straightforward way. This approach is not only applicable to PDEs for classical problems but is also useful for quantum problems, including the preparation of quantum ground states and Gibbs thermal states, the simulation of quantum states in random media in the semiclassical limit, simulation of Schrödinger's equation in a bounded domain with artificial boundary conditions, and other non-Hermitian physics. This formulation is versatile enough to be applicable in a simple way to both digital quantum simulation as well as to analog quantum simulation, and using either qubits or continuous-variable quantum systems (qumodes).
This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrodingerization method-it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrodinger-type equations. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrodingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrodingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrodingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique. We discuss the relationship between operator splitting in the Schrodingerization method and its application directly to the original problem, illustrating how the Schrodingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.
This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrödingerization method – it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrödinger-type equations. It does so via the so-called warped phase transformation that maps the equation into one higher dimension. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrödingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrödingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrödingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique, and give explicitly its corresponding quantum circuit. We discuss the relationship between operator splitting in the Schrödingerization method and its application directly to the original problem, illustrating how the Schrödingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.
Abstract Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case, it corresponds to having a time-dependent Hamiltonian. Designing quantum algorithms for time-dependent Hamiltonian is generally much more complicated than time-independent Hamiltonian problems, due to the challenge in handling the time-ordering; moreover, almost all existing such algorithms are developed for qubit-based quantum devices. Here we propose an alternative formalism based on Sambe-Howland’s clock that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e. quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrödingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients.