Quantum algorithms for simulating linear differential equations have attracted growing interest, driven by applications ranging from Hamiltonian dynamics to general non-unitary dynamics. While time-independent cases are well studied, time-dependent non-unitary dynamics remains considerably less explored, and it is unclear how to systematically adapt existing solvers for time-independent systems to such problems. In this work, we address this gap by introducing an autonomization framework based on the clock-variable formulation, a technique originally developed for time-dependent Hamiltonian systems in~\cite{CJL23TimeSchr}. By lifting the original non-autonomous system to an autonomous transport-type equation on an extended space and applying the Fourier spectral discretization in the clock variable, we obtain an explicit time-independent linear system, together with a suitable initial state and a recovery map for the target solution. Crucially, this formulation decouples the treatment of time dependence from the choice of the quantum ODE solver, thereby enabling the direct application of existing solvers designed for time-independent systems to the resulting autonomous problem. We combine this framework with Schrödingerization and a Taylor-expansion-based quantum ODE solver. In the Schrödingerization-based combination, our complexity analysis shows that the precision dependence can scale as $\log^{5/4}(1/\varepsilon)$, improving upon the $\log^2(1/\varepsilon)$ scaling found in existing approaches. Numerical experiments validate the autonomization formulation and confirm the successful recovery of the target solution.
Quantum algorithms for simulating linear differential equations have attracted growing interest, driven by applications ranging from Hamiltonian dynamics to general non-unitary dynamics. While time-independent cases are well studied, time-dependent non-unitary dynamics remains considerably less explored, and it is unclear how to systematically adapt existing solvers for time-independent systems to such problems. In this work, we address this gap by introducing an autonomization framework based on the clock-variable formulation, a technique originally developed for time-dependent Hamiltonian systems in~\cite{CJL23TimeSchr}. By lifting the original non-autonomous system to an autonomous transport-type equation on an extended space and applying the Fourier spectral discretization in the clock variable, we obtain an explicit time-independent linear system, together with a suitable initial state and a recovery map for the target solution. Crucially, this formulation decouples the treatment of time dependence from the choice of the quantum ODE solver, thereby enabling the direct application of existing solvers designed for time-independent systems to the resulting autonomous problem. We combine this framework with Schrödingerization and a Taylor-expansion-based quantum ODE solver. In the Schrödingerization-based combination, our complexity analysis shows that the precision dependence can scale as log5/4(1/ε), improving upon the log2(1/ε) scaling found in existing approaches. Numerical experiments validate the autonomization formulation and confirm the successful recovery of the target solution.
We develop a Schrödingerization algorithm for quantum linear systems problems in two higher dimensions. The earlier LC-Schrödingerization approach represents the solution as a time integral of a homogeneous convection response and implements this integral by a linear combination of unitaries (LCU) over evolution times. We instead use Duhamel's principle to incorporate the integral into an inhomogeneous convection equation with zero initial data. Fourier projection in the convection variable and Schrödingerization in a second auxiliary variable then realize the solution without the separate LCU step. A joint choice of the kernel and recovery procedure gives an evolution time independent of the target accuracy, a uniformly bounded L2 kernel normalization, and accurate recovery on a fixed interval. We establish the periodization and discretization bounds and analyze the interval recovery probability. With block preconditioning, we retain a single logarithmic precision factor and obtain linear condition-number dependence without variable-time amplitude amplification. Under exact oracle access and given a valid constant-factor solution-norm estimate, the algorithm uses O(κAlog(1/ε)) queries to each original input oracle, with constant success probability and ℓ2 state error at most ε. The matrix-query bound matches the standard worst-case scaling when the supplied norm bounds are tight. UnitaryLab simulations on positive-definite and indefinite systems demonstrate the solution accuracy and probability gain from interval recovery.
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(κAlogε1) 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) with ∥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), 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)) and additional gate complexity O((αT+log(1/ε))2log(1/ε)), assuming coherent access to 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 give quantum algorithms for Hamiltonian simulation, linear differential equations, and linear systems with square-root dependence on sparsity. For an s-sparse Hermitian matrix with row and column Euclidean norms bounded by η, evolution for time t to error ε uses O(sτ+sτlog(1/ε)+log(1/ε)) queries, where τ=η∣t∣. The same bound holds for time-dependent Hamiltonians with coherent time-indexed sparse access and certified time discretisation. Our construction combines an exact entrywise representation with Cayley transducers. A joint lower bound nearly matches the time, sparsity, and precision dependence, ruling out a uniformly additive bound in sτ and log(1/ε). For dissipative linear differential equations we obtain O(gˉ[s(τ+τ)+1]log(Cgˉ/ε)) queries, where τ is the evolution time multiplied by a norm bound, gˉ bounds the ratio of combined input and source norms to the final-solution norm, and C is an absolute constant. We sharpen this bound by separating Hamiltonian and dissipative components, and treat general constant generators. Lower bounds capture joint parameter dependence for time-dependent equations and precision hardness for constant positive semidefinite dissipation. For linear systems, a Cayley walk and Chebyshev filter give O(κslog(1/ε)) queries when ∥A∥≤1 and ∥A−1∥≤κ, and we give an alternative proof of a matching joint lower bound. We also give gate-efficient implementations of time-independent Hamiltonian simulation and the dissipative ODE solver, including Lipschitz time dependence, retaining the query bounds with polylogarithmic overhead in inverse precision given efficient entry decoding, arithmetic, and required source preparations.
We translate the one-mode unitary-dilation framework for nonunitary linear dynamics into a gate-level hybrid oscillator--qubit architecture. An ancillary oscillator encodes the kernel through state preparation and postselection, while a qubit register simulates the discretized system operator. The construction applies to time-independent dynamics u˙=−(L+iH)u, including discretized partial differential equations, and removes the O(logMa) ancilla-qubit overhead of a discrete-variable (DV) Ma-term quadrature register. We bound the ideal state's squeezed-Fock coefficient-projection error, which decays superalgebraically in cutoff N for Schwartz-class kernels and at a stretched-exponential rate under stronger joint decay and smoothness assumptions. The finite squeezed-Fock state generically has stellar rank N−1, making N a measure of the oracle's non-Gaussian resource. For oscillator dimension NFock, a pth-order product formula reaches error εt in O(t1+1/pNFock(p+1)/(2p)εt−1/p) Trotter steps, up to generator-dependent commutator factors. A perturbation bound separates total scaled-map error from postselection probability. We benchmark Law--Eberly synthesis and assess variational SNAP+D state preparation on heat and advection--diffusion equations. At the prescribed DV sizing, the hybrid CV--DV route has smaller fixed-scale error in all ten instances, while the DV route accepts with fewer repetitions in every instance. These results quantify when a continuous qumode can replace a discretized ancilla register.