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arXiv:2608.07133

A unifying framework for quantum algorithms for time-dependent non-unitary dynamics

Xiaojing Dong, Chuwen Ma, Yizhe Peng, Yue Yu

quant-phSubmitted Aug 7, 2026• Announced Aug 10, 2026• 0 citations• Revised Sep 21, 2026 (v2)
v1 · Aug 7, 2026v2 · Sep 21, 2026
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Abstract

We present an improved autonomization method for quantum simulation of time-dependent homogeneous dissipative linear systems, combining a clock-variable reformulation with Schrödingerization to obtain a time-independent Hamiltonian system. To control both discretization error and recovery probability, we construct the clock profile from a compactly supported window function and a normalized Dirichlet kernel: the former imposes endpoint regularity, while the latter concentrates the sampled mass and keeps the discrete normalization bounded. For the subsequent Schrödingerization step, we use an exactly periodic version of an error-function initial profile, whose explicit Fourier coefficients allow direct error estimates and simplify the analysis of smooth initialization. By combining Fourier projection with weighted recovery, we retain a single logarithmic precision factor in the full algorithm. Under finite-order derivative bounds, sampled matrix access, and exact state-preparation access, we prepare the normalized solution at time $T$ to accuracy $\varepsilon$ with success probability $Θ(1)$ using $\mathcal O\!\bigl(gα_AT\log(gα_AT/\varepsilon)\bigr)$ matrix queries and $\mathcal O(g)$ queries to each state-preparation oracle, where $g=\|\boldsymbol{x}_0\|/\|\boldsymbol{x}(T)\|$ and $α_A$ is the matrix-oracle normalization. We illustrate the construction and its recovery probabilities through a numerical experiment on a time-dependent two-cavity system.

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We present an improved autonomization method for quantum simulation of time-dependent homogeneous dissipative linear systems, combining a clock-variable reformulation with Schrödingerization to obtain a time-independent Hamiltonian system. To control both discretization error and recovery probability, we construct the clock profile from a compactly supported window function and a normalized Dirichlet kernel: the former imposes endpoint regularity, while the latter concentrates the sampled mass and keeps the discrete normalization bounded. For the subsequent Schrödingerization step, we use an exactly periodic version of an error-function initial profile, whose explicit Fourier coefficients allow direct error estimates and simplify the analysis of smooth initialization. By combining Fourier projection with weighted recovery, we retain a single logarithmic precision factor in the full algorithm. Under finite-order derivative bounds, sampled matrix access, and exact state-preparation access, we prepare the normalized solution at time TTT to accuracy ε\varepsilonε with success probability Θ(1)Θ(1)Θ(1) using O ⁣(gαATlog⁡(gαAT/ε))\mathcal O\!\bigl(gα_AT\log(gα_AT/\varepsilon)\bigr)O(gαA​Tlog(gαA​T/ε)) matrix queries and O(g)\mathcal O(g)O(g) queries to each state-preparation oracle, where g=∥x0∥/∥x(T)∥g=\|\boldsymbol{x}_0\|/\|\boldsymbol{x}(T)\|g=∥x0​∥/∥x(T)∥ and αAα_AαA​ is the matrix-oracle normalization. We illustrate the construction and its recovery probabilities through a numerical experiment on a time-dependent two-cavity system.
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Quantum linear systems algorithms (QLSAs) aim to solve linear systems $A\bb{x}=\bb{b}$ exponentially faster than classical methods under certain conditions. In this work, we develop quantum algorithms for solving linear algebraic equations from an ODE-based perspective. Inspired by the linear combination of Hamiltonian simulation (LCHS) representation in the Fourier approach \cite{Childs2017QLSA}, we express the solution $\bb{x}$ as a linear combination of solutions to a system of linear convection equations, which become Schrödinger-type equations with unitary evolutions in the Fourier domain. We refer to this representation as LC-Schrödingerization. Based on this result, we construct an LCHS-based quantum algorithm with two LCHS instances: one for time-marching and one for numerical integration. The key construction uses the derivative of a Gaussian-smoothed hat function and recovers the solution over a fixed auxiliary interval. This permits a truncation time independent of the target accuracy and avoids the loss in success probability from selecting a single grid point. Periodization and explicit Fourier coefficients provide the corresponding projection error bounds. Under the stated oracle assumptions and given a constant-factor estimate of the solution norm, direct simulation of the select operators and block preconditioning achieve the optimal matrix-query complexity O(κAlog⁡1ε)\mathcal{O}(κ_A\log\frac1\varepsilon)O(κA​logε1​) without using variable-time amplitude amplification (VTAA). The same upper bound holds for queries to the right-hand-side preparation oracle.
We develop a quantum algorithm for slow analytic Hamiltonians H~(t)=H(t/T)\widetilde H(t)=H(t/T)H(t)=H(t/T) with ∥H(s)∥≤α\|H(s)\|\leqα∥H(s)∥≤α that achieves nearly additive query complexity and low gate overhead. Our main technical contribution is a periodic Gevrey extension of H(s)H(s)H(s), together with Fourier component decay and truncation bounds that enable an efficient finite-dimensional simulation. Combined with Floquet embedding and optimal time-independent Hamiltonian simulation technique, this gives query complexity O~ ⁣(αT+log⁡(1/ε))\widetilde{\mathcal O}\!\left(αT+\log(1/\varepsilon)\right)O(αT+log(1/ε)) and additional gate complexity O~ ⁣((αT+log⁡(1/ε))2log⁡(1/ε))\widetilde{\mathcal O}\!\left((αT+\log(1/\varepsilon))^2\log(1/\varepsilon)\right)O((αT+log(1/ε))2log(1/ε)), assuming coherent access to H′(s)H'(s)H′(s) and endpoint derivatives. For slow analytic control Hamiltonians, only block encodings of the time-independent control operators are required, with the same query complexity and lower gate overhead. Our method also extends to Gevrey Hamiltonians and improves the precision dependence for simulating slow analytic semi-dissipative linear differential equations.
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.
We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form A=L†LA=L^\dagger LA=L†L, a structure that arises naturally in standard discretizations of elliptic operators. Our main tool is the Kannai transform, which represents the diffusion semigroup e−ATe^{-AT}e−AT, where TTT is the final simulation time, as a Gaussian-weighted superposition of unitary wave propagators. For target accuracy ε\varepsilonε, this representation leads to a linear-combination-of-unitaries implementation with a Gaussian tail and yields query complexity O~(∥A∥ T log⁡(1/ε))\widetilde{O}(\sqrt{\|A\|\,T\,\log(1/\varepsilon)})O(∥A∥Tlog(1/ε)​), up to the standard dependence on state-preparation and output-norm factors, improving the scaling in ∥A∥\|A\|∥A∥, TTT, and ε\varepsilonε compared with generic Hamiltonian-simulation-based methods. We instantiate the method for the heat equation and biharmonic diffusion under non-periodic physical boundary conditions, and further use it as a subroutine for constant-coefficient linear parabolic surrogates arising in entropy-penalization schemes for the viscous Hamilton--Jacobi equations. In the long-time regime, under a spectral-gap assumption, the same framework gives a structured quantum linear solver by exploiting convergence to the steady state. For normalized positive definite systems σ(A)⊂[1,κ]σ(A)\subset[1,κ]σ(A)⊂[1,κ], the solver outputs an ε\varepsilonε-approximation to the state proportional to x=A−1b\mathbf{x}=A^{-1}\mathbf{b}x=A−1b with query complexity O~(∥b∥∥x∥κlog⁡2∥b∥ε∥x∥)\widetilde{O}\left(\frac{\|\mathbf{b}\|}{\|\mathbf{x}\|}\sqrtκ\log^2\frac{\|\mathbf{b}\|}{\varepsilon\|\mathbf{x}\|}\right)O(∥x∥∥b∥​κ​log2ε∥x∥∥b∥​).
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.