Schrödingerization Technique

• Schrödingerization is a method that transforms non-unitary differential equations into unitary Schrödinger evolution by introducing an auxiliary phase variable and a warped phase transformation.
• It constructs a Hermitian Hamiltonian through matrix decomposition and Fourier discretization, ensuring precise error control and efficient simulation of complex systems.
• The technique underpins quantum algorithms that achieve significant speedup in simulating multiscale transport, elliptic problems, and weakly nonlinear dynamics.

Schrödingerization Technique

Schrödingerization refers to a systematic procedure that transforms linear (and, in extended variants, weakly nonlinear) ordinary or partial differential equations (ODEs/PDEs) with non-unitary dynamics into unitary Schrödinger-type evolution in an enlarged Hilbert space. The essential idea is to introduce an auxiliary continuous “phase” variable and carry out a warped phase transformation, thereby embedding the original dissipative or non-Hermitian evolution into a higher-dimensional Hermitian structure directly amenable to quantum Hamiltonian simulation. This allows direct application of quantum simulation primitives with rigorous control of error, complexity, and resource scaling, and underpins quantum algorithms capable of achieving polynomial or even exponential speedup over conventional classical solvers for a variety of scientific computing problems, including stiff multiscale transport systems, general PDEs with inhomogeneous terms, and quantum linear system solvers (He et al., 25 Jul 2025).

1. Mathematical Foundations: Warped Phase Transformation

Given a linear non-unitary system, such as $\frac{d}{dt}u(t) = A u(t) + f(t)$ with $A$ not necessarily skew-Hermitian, Schrödingerization proceeds as follows (Jin et al., 1 May 2025, Jin et al., 2024):

• Augmentation for inhomogeneity: Embed the original (possibly inhomogeneous) system into a larger homogeneous system by appending an auxiliary block: $$u_f(t) = \begin{pmatrix}u(t) \ r(t)\end{pmatrix}$$.
• Hermitian decomposition: Decompose the generator $A_f$ as $A_f = H_1 + i H_2$, with $H_1 = \tfrac12(A_f + A_f^\dagger)$ and $H_2 = \tfrac{1}{2i}(A_f - A_f^\dagger)$.
• Warped phase embedding: Introduce an auxiliary variable $p\in\mathbb R$ and define $w(t,p) = e^{-p}u_f(t)$. This yields the “warped-phase” PDE:

$$\frac{\partial}{\partial t} w(t,p) = -H_1\, \partial_p w(t,p) + i H_2\, w(t,p).$$

• Fourier spectral discretization: Restrict $p$ to a finite interval, discretize with $N_p$ Fourier modes or grid points, and transform via the discrete Fourier transform. The system becomes a time-independent Hermitian Hamiltonian ODE:

$\frac{d}{dt} w_h(t) = -i(H_{\text{eff}})\, w_h(t),$

with $H_{\text{eff}} = D_\mu \otimes H_1 - I \otimes H_2$ ($D_\mu$ the diagonalized momentum operator in $p$).

• Physical solution extraction: The original solution $u(t)$ is obtained by projecting the evolved wavefunction onto the $p=0$ (or $p \geq p^\diamond$) slice, possibly with amplitude amplification.

The core mechanism relies on lifting non-unitary dynamics to a strictly unitary evolution in an enlarged Hilbert space $\mathcal{H}_p \otimes \mathcal{H}_u$, which is then efficiently simulatable by standard quantum Hamiltonian simulation techniques (Jin et al., 2022, Jin et al., 11 May 2025).

2. Algorithmic Pipeline and Implementation

The quantum algorithmic workflow enabled by Schrödingerization consists of:

1. System Preprocessing and Discretization:
   • For PDEs: discretize spatial and (if necessary) velocity variables, yielding a high-dimensional but sparse ODE system.
   • For dynamic linear systems: reformulate iterative methods or time-evolution as discrete ODEs or affine maps.
   • For inhomogeneous problems: embed into homogeneous, higher-dimensional blocks as in section 1.
2. Warped Phase and Fourier Lifting:
   • Apply the warped phase embedding ($e^{-p}$-weight).
   • Fourier discretize auxiliary variables for spectrally accurate and periodic boundary treatment.
3. Hermitian Hamiltonian Construction:
   • Build the block-diagonal Hermitian Hamiltonian $H$ according to the splitting of $A$ into $H_1$ and $H_2$.
   • For multiscale or discretized iterative schemes, form the evolution generator (e.g., $C-I$ for iteration matrices).
4. Quantum Simulation:
   • Use oracles/block-encodings for the original matrix components.
   • Simulate the evolution $e^{-iH_{\text{eff}}t}$ via standard quantum Hamiltonian simulation algorithms—Berry–Childs product formulas, QSVT, quantum signal processing—exploiting optimal scaling with respect to simulation time and precision (Jin et al., 1 May 2025, Jin et al., 2022).
5. Solution Recovery:
   • Invert the Fourier transform on the $p$-register and project onto the appropriate $p$ domain.
   • Use amplitude amplification to boost success probability, especially in dissipative scenarios where the solution norm decays.

The pipeline is nearly black-box once the system matrix is available, and it is particularly advantageous for problems with multiscale, stiff, or ill-conditioned dynamics, as typical in kinetic, diffusion, and radiative transport equations (He et al., 25 Jul 2025).

3. Complexity, Query Scaling, and Error Control

The complexity of Schrödingerization-based quantum algorithms is fundamentally dictated by:

• Sparsity of Hamiltonian: Query cost scales linearly with the sparsity of the discretized system matrices (e.g., $O(N_v N_x^2 \log N_x)$ for multiscale transport with parabolic scaling) (He et al., 25 Jul 2025).
• Precision and Fourier modes: The number of Fourier modes $N_p$ in $p$ satisfies $N_p \sim O(\log(1/\varepsilon))$ for smooth solutions, and the norm of the Hamiltonian $\|H\|_{\max}$ scales as $O(N_p)$ or $O(\log N_x)$, securing polylogarithmic dependence on precision (Jin et al., 1 May 2025).
• Total runtime: For well-conditioned, preconditioned, or asymptotic preserving schemes, quantum query complexity can be made essentially polylogarithmic in the inverse error, outperforming classical methods that often scale polynomially or worse in system size or mesh refinement parameter (Jin et al., 11 May 2025, Yang et al., 19 Aug 2025).
• Error decomposition: Total $\ell_2$-error combines discretization, finite simulation time, and Fourier truncation, with theoretical guarantees available for each step through error estimates based on smoothness of initial data and spectral convergence (Jin et al., 1 May 2025, Jin et al., 2024).

A summary comparison of key complexity results for multiscale linear transport:

Algorithm	Query Complexity	$N_x$-dependence	$\varepsilon$-dependence
Quantum Schrödingerization	$\widetilde{O}(N_v N_x^2 \log N_x)$	Sublinear in $N_x^3$	Polylogarithmic ($O(\log^2(1/\varepsilon))$)
Classical	$O(N_v^2 N_x^3)$	Cubic ($N_x^3$)	Polynomial
Quantum HHL-type	$O(N_v^2 N_x^2 \log N_x)$	Quadratic ($N_x^2$)	Polylogarithmic

4. Treatment of Inhomogeneity, Boundaries, and Non-Autonomous Systems

Schrödingerization accommodates a range of physical and mathematical complications (Jin et al., 2024, Ma et al., 2024):

• Inhomogeneous terms: Handled by block augmentation, homogenization via stretch transformation when large sources threaten stability, and smooth windowing for error control in the auxiliary dimension.
• Boundary and source conditions: Incorporated into the block-matrix formalism and initial datum in the extended $p$-register. Smoother initialization (e.g., mollified cut-off, high-order interpolation) ensures higher-order accuracy in the Fourier spectral representation (Jin et al., 1 May 2025).
• Time-dependent (non-autonomous) systems: Employ "autonomization," adding an additional auxiliary register (e.g., $s$ for time-lifting), rendering the system time-independent in the enlarged space. This is pivotal for practical quantum circuit implementations of, e.g., Maxwell's equations with time-dependent sources (Ma et al., 2024).
• Nonlinear systems: By combining Carleman linearization with warped-phase Schrödingerization, general nonlinear (weakly) ODEs/PDEs are mapped—after finite tensor-power truncation and symmetrization—to high-dimensional linear Hermitian Hamiltonians, thereby supporting efficient simulation of nonlinear dynamics within quantum resource constraints (Muraleedharan et al., 22 May 2025, Sasaki et al., 3 Aug 2025).

5. Applications and Extensions

Schrödingerization has been demonstrated in a range of settings:

• Multiscale linear transport equations: First-principles quantum algorithms for diffusive/kinetic equations, achieving state-of-the-art $N_x$-scaling and absolute stability under asymptotic-preserving discretizations (He et al., 25 Jul 2025).
• Quantum linear systems solvers: Provides a momentum-accelerated framework with linear-in-$\kappa$ query complexity, avoiding the overhead of VTAA or qubitization-based approaches (Hu et al., 20 Sep 2025, Yang et al., 19 Aug 2025).
• Fractional Poisson and elliptic problems: Dimension-lifting (e.g., Caffarelli–Silvestre extension) combined with Schrödingerization achieves exponential mesh-parameter speedup in high dimensions (Jin et al., 2 May 2025).
• Quantum preconditioning and FEM: Leverages multilevel preconditioners (e.g., BPX) to obtain polylogarithmic scaling in target error, independent of mesh size (Jin et al., 11 May 2025).
• Maxwell's equations and electromagnetic simulation: Construction of explicit quantum circuits for time-dependent Maxwell systems with boundary conditions, powered by warped-phase plus autonomization, and achieving polynomial acceleration over FDTD (Ma et al., 2024).
• Nonlinear dynamical systems: With Carleman linearization, extends the framework to time-dependent and weakly nonlinear ODEs/PDEs, allowing many nonlinear classical systems to be simulated efficiently on quantum devices (Muraleedharan et al., 22 May 2025, Sasaki et al., 3 Aug 2025).

The approach is nearly black-box once the discrete generator is known and admits flexibility with regard to discretization choices (AP, micro-macro, spectral, finite-element, etc.) (He et al., 25 Jul 2025, Yang et al., 19 Aug 2025).

6. Limitations, Optimality, and Scope for Generalization

Several limitations and caveats accompany Schrödingerization:

• Auxiliary space overhead: Nonsmooth or highly oscillatory initial data in the auxiliary $p$ direction inflates the required number of Fourier modes $N_p$; large source terms can lead to large $p^\diamond$ for stable recovery, mitigated by the stretch transformation (Jin et al., 2024).
• Recovery success probability: For dissipative problems, the solution norm may decay, resulting in probabilistic overhead for extraction that may require amplitude amplification techniques (Jin et al., 2022).
• Sparsity preservation: The quantum complexity is proportional to system sparsity; systems with dense or full-rank scattering kernels may lose the sparseness advantage (He et al., 25 Jul 2025).
• Nonlinear extension: For strong nonlinearities, Carleman truncation order $k$ can become large, resulting in exponential dimension growth; quantum advantage holds when nonlinearity and simulation time remain within controlled bounds (Muraleedharan et al., 22 May 2025, Sasaki et al., 3 Aug 2025).
• Error optimality: Fully optimal dependence on precision is attainable with mollified cut-off or highly smooth initialization; polynomial or Fourier-kernel schemes yield near-optimal scaling (Jin et al., 1 May 2025).

The core methodology is extensible to time-dependent systems via further dimensional lifting, generalized discretizations, and a wide class of analytic nonlinearities. It provides a unified analytical and computational framework for quantum scientific computation across transport, elliptic, parabolic, and kinetic regimes (He et al., 25 Jul 2025, Jin et al., 2 May 2025, Yang et al., 19 Aug 2025).

7. Theoretical Guarantees and Empirical Validation

The Schrödingerization framework is supported by detailed theoretical guarantees and validated by numerical experiments:

• Stability: Unconditional stability holds under CFL conditions for AP schemes, with error and cost independent of stiffness or mean free path parameter $\varepsilon$ (He et al., 25 Jul 2025).
• Error analysis: Decomposition of error into time-stepping, spatial discretization, and quantum simulation, with explicit scaling strategies (e.g., set $\tau = O(h^2)$, $h=O(1/N_x)$, and $N_p=O(\log(1/\delta))$ for target error $\delta$).
• Optimal scaling: Proven attainment of the information-theoretic lower bound in matrix queries, i.e., $\widetilde{O}(\alpha_H T \log(1/\varepsilon))$ for non-unitary linear dynamics, when smooth cutoff initializations are employed (Jin et al., 1 May 2025).
• Empirical results: Extensive benchmarking for elliptic, kinetic, and radiative transfer equations, confirming accuracy, scaling, and speedup over classical and prior quantum methods (Jin et al., 2 May 2025, Yang et al., 19 Aug 2025).

Collectively, these results establish Schrödingerization as a general, efficient, and theoretically sound technique for quantum simulation of linear and weakly nonlinear differential systems across a broad spectrum of scientific computing tasks.