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Quantum simulation of partial differential equations via Schrodingerisation
Shi Jin, Nana Liu, Yue Yu
TL;DR
The paper addresses how to simulate general linear PDEs with quantum systems while avoiding time discretization and equation-specific transformations. It introduces Schrödingerisation, using a warped phase transformation to convert linear PDEs into real-time Schrödinger equations, and applies it to PDE solutions, ground and Gibbs states, and quantum states in random media. The method is presented as applicable to any linear PDE, while ground-state preparation carries an extra 1/ε factor compared with near-optimal schemes.
Problem
Quantum simulation of classical dynamics needs a generic way to represent linear PDE evolution as continuous-time Schrödinger evolution rather than relying on discretized-time primitives or heat-equation-specific imaginary time.
Method
Schrödingerisation uses the warped phase transformation to map any linear PDE to Schrödinger equations in real time and recover the original PDE solution as a quantum state.
Results
The approach is shown for general linear PDEs, heat-equation-based ground and Gibbs states, and quantum systems in random media in the semiclassical limit.
Takeaways & Limitations
Schrödingerisation provides a dynamical-equation-level connection between linear classical dynamics and quantum simulation, including applications to classical and quantum problems.
Takeaways & Limitations
Ground-state preparation has an extra factor 1/ε compared with near-optimal schemes.
Abstract
from arXiv · showhide
We present a simple new way - called Schrodingerisation - to simulate general linear partial differential equations via quantum simulation. Using a simple new transform, referred to as the warped phase transformation, any linear partial differential equation can be recast into a system of Schrodinger's equations - in real time - in a straightforward way. This can be seen directly on the level of the dynamical equations without more sophisticated methods. This approach is not only applicable to PDEs for classical problems but also those for quantum problems - like the preparation of quantum ground states, Gibbs states and the simulation of quantum states in random media in the semiclassical limit.
INTRODUCTION
The paper asks how quantum systems can simulate classical ODEs and PDEs continuously in time, then proposes Schrödingerisation as a generic real-time route from linear PDEs to Schrödinger equations. It motivates this route by limitations of discretized-time, imaginary-time, and more general unitary-dilation approaches.
- Quantum simulation of classical ODEs and PDEs is important for broader applications of quantum computing in science and engineering.
- Existing linear-PDE approaches may use quantum linear-system algorithms, but discretization in time prevents continuous preparation of solution states through Schrödinger evolution.
- Imaginary-time evolution for the heat equation produces different state evolutions, requires extra mapping resources, and does not extend beyond that equation.
- The paper asks whether a simpler generic method can obtain Schrödinger equations naturally from arbitrary linear dynamics.
- The warped phase transformation maps any linear PDE to Schrödinger equations in real time, allowing quantum simulation to prepare the original PDE solutions as quantum states.
- The approach is illustrated for the heat equation, ground-state and Gibbs-state preparation, linear radiative transport, and quantum systems in random media in the semiclassical limit.
SCHR¨ODINGERISATION OF THE HEAT EQUATION
For the heat equation, Schrödingerisation extends the solution into an auxiliary warped dimension and Fourier transforms it into uncoupled real-time Schrödinger equations. The resulting simulation can recover heat-equation states and support ground-state and Gibbs-state preparation, with an extra precision-dependent cost for ground states.
- Heat-equation transformation: The warped phase transformation extends the heat-equation solution into an extra dimension, where Fourier transformation yields uncoupled Schrödinger equations.
- Heat-equation transformation: The auxiliary Fourier mode η acts as a rescaling of time for the resulting Schrödinger equation while the rescaled time remains real-valued.
- Quantum simulation: The numerical procedure discretizes position and the auxiliary variables but not time, then evolves the transformed state with a Hermitian total Hamiltonian.
- Quantum simulation: The original heat-equation quantum state is recovered after inverse Fourier transformation by projecting onto positive auxiliary-momentum values or using amplitude amplification.
- Generalization: For Hermitian H, Schrödingerisation prepares states proportional to exp(−Ht) from an initial state, and the paper extends the method to non-Hermitian H.
- Ground and Gibbs states: Ground-state preparation converges exponentially with spectral gap Δ, but its complexity has an extra factor 1/ε compared with near-optimal schemes.
- Ground and Gibbs states: Gibbs states are prepared by applying the same approach to an extended Hamiltonian and tracing out one register.
SCHR¨ODINGERISATION OF LINEAR TRANSPORT EQUATION
Schrödingerisation applies the warped phase transformation to linear transport and general linear PDEs, yielding unitary Schrödinger dynamics that prepare encoded solution states. For transport equations, the approach also supports semiclassical quantum-state simulation without adding an extra O(1/ǫ) factor.
- Application: In the semiclassical limit, the Wigner transform converges to a particle probability density W governed by a linear transport equation for quantum systems in random media.The setting assumes a deterministic background potential, rapidly varying random fluctuations, and homogeneous isotropic scattering.
- Application: Schrödingerisation prepares a quantum state |W(t)⟩ whose amplitudes are proportional to the discretized transport density W(t).The discretization uses mesh points in position x and wavevector k.
- Complexity: For the transport equation, Schrödingerisation does not introduce an extra O(1/ǫ) factor because the transformed Hamiltonian retains the original max-norm order.The differential and scattering contributions remain of order 1/ǫ.
- Application: Moments of W recover semiclassical observables such as mass, momentum, and energy from fidelities with corresponding encoded states.The paper identifies g(x,k)=1, k, and k^2/2 with these observables and mentions swap tests for fidelity recovery.
- General formulation: The approach generalizes to any linear first-order-in-time PDE by decomposing its non-Hermitian system matrix into Hermitian components.Higher-order time derivatives can be converted to first-order systems by introducing additional variables.
- General formulation: The transformed dynamics are unitary because ηH + H̄ is Hermitian, with the resulting evolution generated by Htotal.A further warped phase transformation can cast the first-order operator explicitly into Schrödinger form.
- Complexity: The general formulation prepares |u(t)⟩ with query complexity Õ((∥u(0)∥/∥u(t)∥)st∥Htotal∥max) and additional two-qubit gates scaling as Õ(∥u(0)∥/∥u(t)∥dts∥Htotal∥max).These resources assume sparse access to the relevant matrices and an initial-state preparation unitary.
- Discussion: The paper presents Schrödingerisation as a dynamical-equation method for simulating solutions of any linear partial differential equation using quantum simulation.Further applications to other equations are deferred to a technical companion paper.
Appendix A: Warped phase transformation
The warped phase transformation embeds a PDE solution into an auxiliary p variable, extends the construction across p=0, and uses a Fourier transform to obtain independent real-time Schrödinger equations.
- Boundary treatment: No boundary condition is required at p=0 because Fourier modes in x show that the auxiliary solution convects from right to left.The domain may therefore be symmetrically extended to p<0 without changing the p>0 solution.
- Warped phase transformation: The auxiliary function is initialized with an exponentially weighted PDE state, w(0,x,p)=e^−|p|u0(x), on the extended p domain.The extension to p<0 preserves the solution in the p>0 region.
- Numerical implementation: A finite computational p interval can impose zero endpoint values when L is sufficiently large because of exponential decay in |p|.The paper points to a technical companion paper for numerical justification.
- Fourier representation: Fourier transforming w with respect to p produces ˜w(t,x,η), with one Schrödinger equation for every η.The transformed equation is identified as i∂t ˜w=−η∇²x ˜w.
Appendix B: Proof of Theorem 3
The proof constructs the encoded auxiliary state, simulates its Hamiltonian evolution, and projects onto positive p to recover the PDE solution with stated complexity and success probability.
- Hamiltonian simulation: The auxiliary state |˜w(t)⟩ is prepared by simulating Htotal to time t, assuming the original matrix H is Hermitian.The construction uses logarithmically many qubits in the discretized dimension and p register.
- Complexity: The Hamiltonian-simulation cost is Õ(st∥Htotal∥max), with ∥Htotal∥max=∥H∥max∥D∥max and ∥D∥max=O(N).Here N=O(1/ǫ), while logarithmic factors are suppressed.
- State recovery: Projecting onto positive-p states retrieves |u(t)⟩ with probability approximately N(∥u(t)∥/∥w(t)∥)^2.The normalization contribution satisfies ∥exp(−p)∥²=O(N).
- State recovery: Amplitude amplification raises the recovery probability using the reflection oracle Q, at an additional query cost proportional to Õ(∥w(t)∥/(√N∥u(t)∥)).The oracle combines reflections about the auxiliary state and the positive-p projector.
- Final complexity: Given |u(0)⟩, the total query and gate complexity is Õ((∥u(0)∥/∥u(t)∥)st∥H∥max/ǫ).This result combines preparation, Hamiltonian simulation, and recovery of the target state.
Appendix C: Preparation of quantum ground state via Schr¨odingerisation
Schrödingerisation prepares ground states by realizing non-unitary evolution through a directly simulated Schrödinger system. For a gapped Hamiltonian, the ground-state component emerges exponentially, with complexity determined by the spectral gap, initial overlap, and target accuracy.
- Motivation: Imaginary-time evolution requires indirect state mapping because the unitary and non-unitary solutions agree only in the stationary limit.The indirect procedure uses tomographic measurements at every small time step.
- Ground-state emergence: Non-unitary evolution suppresses higher-energy components, causing the ground state to emerge as t approaches infinity for gapped Hamiltonians.The convergence rate is exponential when E1 > E0.
- Schrödingerisation: Schrödingerisation directly produces a state proportional to exp(−Ht)|u(0)⟩ and defines the required evolution time through fidelity at least 1 − ǫ.The evolution time is the minimum time needed to reach the target fidelity with the true ground state.
- Complexity: The total query and gate complexity for ground-state preparation is ˜O(s∥H∥max/(|α0|∆ǫ)).This bound is stated for preparing the ground state of H.
- Gibbs states: Gibbs-state preparation follows the same proof pattern, with the evolution time proportional to 1/T.Here T denotes the Gibbs-state temperature.
Appendix D: Transport equation and quantum systems in random media
The transport-equation construction maps semiclassical quantum dynamics in random media into a unitarily simulable system. After warped-phase and Fourier transformations, discretization yields a Hermitian Hamiltonian whose simulation prepares the target Wigner-function state with stated resource costs.
- Transport equation: The transport equation models scalar energy density f(t, x, k) under periodic boundary conditions.The wave frequency is ω = ω(x, k), with k on the unit sphere.
- Physical setting: The semiclassical Wigner function of a quantum system in a random medium solves a transport equation.The random potential contains a slowly varying background and a rapidly varying fluctuation.
- Transformation: The warped phase transformation V = exp(−p)W converts the transport equation into an equation involving derivatives with respect to p.The transformed equation retains the scattering and total-cross-section terms.
- Fourier transformation: Fourier transformation in x and p introduces conjugate variables ξ and η, after which the transformed function obeys a system suitable for Schrödingerisation.The resulting equations are Schrödinger equations for each η because ηH + H̄ is Hermitian.
- Discretization: Discretization of ξ, k, and η forms a vector representation of the transformed function, with grid sizes controlled by ∆ξ = 2/J, ∆k = 2/K, and ∆η = 2/N.The vector is assembled from the corresponding discretized grid components.
- Hamiltonian construction: The discretized dynamics are governed by a Hermitian Hamiltonian Htotal = L[ξ,k] ⊗1 −1 ⊗Σ ⊗D + 1 ⊗σ ⊗D.The matrix L[ξ,k] encodes the ξ·k transport factor, while D and Σ encode transformed momentum and total scattering terms.
- Quantum-state preparation: The transformed vector evolves unitarily, enabling preparation of a quantum state whose amplitudes are proportional to the Wigner function in the random medium.Fourier transforms preserve norms, supporting the relation between the transformed state and the target Wigner-function state.
- Resource costs: The query and gate complexities for preparing |W(t)⟩ are respectively ˜O((∥W(0)∥/∥W(t)∥)st∥Htotal∥max) and ˜O((∥W(0)∥/∥W(t)∥)dst∥Htotal∥max).The sparsity s is dominated by scattering sparsity, so more off-diagonal scattering terms increase preparation cost.