Schrödingerization in Quantum Simulation

• Schrödingerization is a mathematical and computational technique that maps non-unitary PDEs, ODEs, and linear systems into higher-dimensional unitary Hamiltonian systems using a warped phase transformation.
• It leverages Fourier analysis and block-encoding to enable quantum simulation with optimal resource scaling, providing exponential advantages over classical methods.
• The method finds applications in stabilizing ill-posed problems, enhancing quantum linear algebra, and solving time-fractional equations, which aids in regularization and error control.

Schrödingerization Technique

The Schrödingerization technique refers to a suite of mathematical and computational frameworks that systematically map general linear (and often non-unitary, dissipative, or even ill-posed) partial differential equations (PDEs), ordinary differential equations (ODEs), or linear algebraic systems into higher-dimensional unitary (Schrödinger-type) systems. This transformation enables the simulation of such problems using the full apparatus of quantum algorithms and Hamiltonian simulation, as well as providing new perspectives for classical scientific computing, stability analysis, and regularization. Core to this methodology is the warped phase transformation, an augmentation in which an auxiliary variable is introduced so that the original non-unitary dynamics become convection- or transport-type equations in extended space, which then translate—via Fourier analysis or block-encoding—into Hamiltonian systems with unitary evolution. The technique has achieved optimal resource scaling with respect to initialization and discretization in the auxiliary dimension, and delivers exponential advantages for quantum simulation of certain high-dimensional or fractional-order dynamical systems.

1. Mathematical Foundations and Warped Phase Transformation

At the heart of Schrödingerization lies the warped phase transformation, which analytically extends the dimension of the system by introducing an auxiliary variable $p$ and mapping the original state $u(t)$ to a new function $w(t,p)$ using a prefactor such as $e^{-p}$. Consider a linear evolution equation

$\frac{du}{dt} = A u + b,$

where $A$ is generally non-Hermitian. The Schrödingerization procedure rewrites $A=H_1 + i H_2$ with $H_1 = (A + A^\dagger)/2$ and $H_2 = (A - A^\dagger)/(2i)$, and defines $w(t,p)$ such that

$$\partial_t w(t,p) = -H_1 \partial_p w(t,p) + i H_2 w(t,p).$$

By taking the Fourier transform in $p$ (letting $P_\mu$ denote the momentum operator in $p$), one recasts the system as a Schrödinger equation: $$\frac{d}{dt} \tilde{w}(t) = -i \left(P_\mu \otimes H_1 + I \otimes H_2 \right) \tilde{w}(t).$$ This provides a direct embedding into a unitary framework suitable for quantum simulation (Jin et al., 2022, Jin et al., 22 Feb 2024, Jin et al., 28 Mar 2024, Jin et al., 21 Apr 2024, Jin et al., 1 May 2025, Yang et al., 19 Aug 2025, Jin et al., 22 Sep 2025).

Physical intuition for the transformation is that the auxiliary $p$-dimension absorbs the non-unitarity (e.g., dissipation or instability), resulting in a Hamiltonian or time-reversible evolution in $(t,p)$ which, under certain recovery maps (e.g., $u(t) = e^p w(t,p)$ for $p \gg 0$ or via integration over $p$), delivers the physical solution of the original system. Smooth initialization in $p$ is required for optimal convergence and resource scaling (Jin et al., 1 May 2025).

2. Quantum Simulation Protocols, Recovery Maps, and Error Analysis

The Schrödingerized system is discretized along the $p$-dimension (using uniform or spectral grids), and the equation is mapped either to a discrete or continuous Fourier basis. The maximal required Fourier mode $\mu_{\max}$ determines the cost of Hamiltonian simulation. Query complexity is fundamentally tied to the regularity of the initialization function $\psi(p)$: employing smooth extension techniques (e.g., mollifiers, high-order Hermite polynomial interpolation, or suitably decaying Fourier kernels) ensures that $\mu_{\max} = \mathcal{O}(\log(1/\varepsilon))$, yielding optimal scaling in matrix queries for reach accuracy $\varepsilon$ (Jin et al., 1 May 2025).

Resource scaling is dramatically improved over first-order or discontinuous initializations, which require mesh size polynomial in $1/\varepsilon$. In quantum simulation, the ancillary qubit requirement for the $p$-register is essentially $\log\log(1/\varepsilon)$ when spectral initialization is used (Jin et al., 28 Mar 2024).

Error analysis rigorously covers discretization in the $p$-direction, spectral truncation, and projection/recovery steps. For both discrete Fourier transform and continuous Fourier transform schemes, explicit bounds on $L^2$-norm errors are derived as functions of mesh size, time interval, operator norms, and maximal eigenvalues. Adaptivity to unstable systems is achieved by projecting the recovery to $p$-segments outside the influence of positive eigenmodes (Jin et al., 22 Feb 2024, Jin et al., 28 Mar 2024).

3. Applications to PDEs, ODEs, and Quantum Linear Algebra

Schrödingerization is broadly applicable across the following domains:

• Non-unitary Linear PDEs: Heat and diffusion equations, Fokker-Planck equations, convection-diffusion systems, and transport equations, even with boundary and interface conditions (Jin et al., 2022, Jin et al., 21 Apr 2024).
• Time-Fractional Equations: Through dimension extension (e.g., Caffarelli–Silvestre extension, AAA rational approximation), time-fractional (Caputo) problems are reduced to higher-dimensional local systems to which Schrödingerization is then applied (Jin et al., 22 Sep 2025).
• Ill-posed and Inverse Problems: The backward heat equation and convection with imaginary speed are stabilized by lifting, as the Hamiltonian system is time-reversible and stable in extended space, allowing stable computation (classically or quantumly) both forward and backward in time (Jin et al., 28 Mar 2024).
• Quantum Linear Systems and Preconditioning: Linear systems $A x = b$ are solved by mapping to evolution problems (ODEs) and extending via Schrödingerization. The solution is sought as the steady-state of the convolution system in the extended variable ($p$). LCHS (linear combination of Hamiltonian simulation) and BPX multilevel preconditioning are naturally compatible with this framework, enabling quantum algorithms that preserve or optimally reduce condition number (Yang et al., 19 Aug 2025).

A summary of representative use cases is illustrated below:

Application Domain	Transformation/Algorithm	Reference
Non-unitary PDEs (heat, Fokker-Planck)	Warped phase transform, Fourier–spectral splitting	(Jin et al., 2022, Jin et al., 21 Apr 2024)
Time-fractional heat/gradient flows	Dimension lifting, rational approximation, Schrödingerization	(Jin et al., 22 Sep 2025)
Quantum linear systems & preconditioning	Auxiliary convection ODEs, spectral/LCU, BPX	(Yang et al., 19 Aug 2025, Jin et al., 1 May 2025)
Ill-posed inverse problems	Extended Hamiltonian system, stable recovery	(Jin et al., 28 Mar 2024)

4. Implementation Strategies and Circuit Complexity

Quantum algorithms implementing Schrödingerization proceed by:

1. Block-encoding of discretized spatial or auxiliary matrices for the main operator (e.g., Laplacian, drift, or coefficient matrices).
2. Register initialization with smooth/spectral extended data in the $p$-dimension using mollifiers or high-degree polynomials for near-exponential Fourier decay.
3. Hamiltonian simulation via standard techniques (truncated Dyson series, QSP, LCU) applied to the Hermitian system derived in the Fourier domain.
4. Recovery of the target state via projection or measurement in the $p$-register, possibly employing amplitude amplification to offset success-probability normalization factors introduced by the non-unitary extension (Jin et al., 21 Apr 2024).

The method allows efficient construction of time-splitting or operator-splitting circuits, with sub-evolutions in the extended Hamiltonian diagonalized efficiently either in the Fourier basis or, more efficiently for certain finite-difference matrices, using a Bell-basis decomposition with localized phase-shift gates on individual qubits (Jin et al., 21 Apr 2024). Spectral accuracy in the $p$-dimension reduces the grid size for the ancillary space to the minimal logarithmic requirement.

5. Stabilization, Regularization, and Eigenvalue Considerations

A salient feature of Schrödingerization is its ability to stabilize ill-posed and unstable systems by controlling the support in the extended $p$-dimension. For coefficient matrices or operators with positive eigenvalues (potentially leading to exponential blow-up), solution recovery is restricted to domains in $p$ beyond the reach of unstable modes, often formalized as $p \geq p^\diamond$ where $p^\diamond \geq \lambda_{\mathrm{max}}(H_1) T$. For dominant source terms, stretch transformations (rescaling $p$ according to the size of inhomogeneities) are employed to regularize the problem without degrading error bounds (Jin et al., 22 Feb 2024).

In the context of quantum linear systems and PDE problems, this technique allows for both the preservation of the underlying Hamiltonian structure and mitigation of condition-number blowup when combined with preconditioners (e.g., BPX), facilitating efficient use of quantum block-encoding and amplitude amplification (Yang et al., 19 Aug 2025).

6. Complexity Analysis and Quantum Advantage

Rigorous complexity analysis for Schrödingerization-based quantum algorithms demonstrates an exponential advantage in the inverse mesh size $h^{-1}$ in high dimensions $d$, as compared to classical approaches where the cost scales polynomially or exponentially with $d$ (classical work per step is at least $\mathcal{O}(N_t d h^{-(d+0.5)})$). The quantum approach, after discretization and block-encoding, achieves total query complexity (suppressing logarithmic factors) of $\tilde{\mathcal{O}}(T^2 d^4 h^{-8})$, with quantum complexity independent of $d$ in terms of $h^{-1}$ (Jin et al., 22 Sep 2025). This is primarily enabled by the efficient encoding of multidimensional data on quantum registers and the essentially optimal dimension scaling of the spectral $p$-register ensured by smooth initialization (Jin et al., 28 Mar 2024, Jin et al., 1 May 2025).

7. Implications, Generalizations, and Open Problems

The Schrödingerization framework unifies the treatment of non-unitary and unitary evolution, and is agnostic as to the source—be it dissipative, unstable, or nonlocal dynamics. It bridges quantum and classical computational paradigms by embedding non-Hamiltonian systems in higher-dimensional Hamiltonian form (with clear analogies across operator theory, functional analysis, and quantum information). This connection opens new approaches to regularization in inverse problems, control of decoherence and stabilization, and efficient Hamiltonian simulation for quantum PDE solvers.

Further generalizations include its applicability to non-Hermitian or time-dependent operators, integration of analog quantum simulation strategies, and compatibility with advanced quantum numerical linear algebra developments such as block-encoding and quantum singular value transformation. Open theoretical problems remain concerning the interpretive framework for minimal quantum resource cost (in $p$-extension), sharper error bounds under non-uniform mesh and for nontrivial boundary/interface conditions, and the exploration of multi-level and adaptive strategies for large-scale engineering problems.

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This comprehensive treatment delineates the mathematical, computational, and physical aspects of the Schrödingerization technique, as established and advanced in the cited literature (Jin et al., 2022, Jin et al., 22 Feb 2024, Jin et al., 28 Mar 2024, Jin et al., 21 Apr 2024, Jin et al., 1 May 2025, Yang et al., 19 Aug 2025, Jin et al., 22 Sep 2025).