PaperlayerView exact revision on arXiv

Source-linked AI summary

A quantum-assisted framework for PDE-based Bayesian inverse problems

Dong An, Yinan Li, Pucheng Tang, Yunfeng Xiong

arXiv:2608.27028v1math.NA

TL;DR

Existing quantum PDE solvers often prepare solution states without efficiently recovering classical information, while Bayesian PDE inversion requires repeated data-misfit evaluations. The paper combines quantum PDE evolution and Hadamard-test overlap estimation with classical GPR-guided Bayesian optimization. Analysis and experiments support feasible parameter inversion under sampling noise, while the framework assumes efficient oracle access to observation states and normalized loss discards overall solution scale.

  • Problem

    Most quantum PDE solvers output quantum states, leaving efficient classical information recovery and quantum-compatible loss estimation insufficiently explored.

  • Method

    A quantum processor evaluates a normalized quantum-state loss through PDE evolution and Hadamard tests, while classical GPR selects Bayesian-optimization candidates from noisy evaluations.

  • Results

    The normalized quantum-state loss outperforms conventional L2-loss under quantum measurements, and experiments demonstrate feasible parameter inversion despite sampling noise.

  • Takeaways & Limitations

    The framework provides a quantum-assisted scheme for PDE-based inverse problems without reconstructing the full solution vector.

  • Takeaways & Limitations

    The framework assumes efficient oracle access to the observation state, and normalized loss discards overall norm information.

Abstract

from arXiv · show

Quantum computing offers potential advantages for solving partial differential equations (PDEs). However, most existing quantum PDE solvers primarily focus on preparing quantum states for solutions, while the efficient recovery of classical information from these states remains less explored. Motivated by the readout limitation, we propose a quantum-classical hybrid framework for Bayesian PDE inversion problems: The quantum processor evolves the PDE and evaluate the loss function with sampling noises, while the classical computer tunes the hyper-parameters in the Gaussian Process Regression to explore the next trial candidate. To match the quantum solvers for linear and semi-linear autonomous evolution PDEs, we suggest to use a normalized quantum-state loss as the data-misfit function and evaluate the new misfit by combining quantum PDE solvers with the Hadamard test, thereby allowing us to extract useful classical information using only a limited number of quantum state copies without reconstructing the full solution vector. The analysis of error propagation and overall complexity of loss evaluation under a prescribed accuracy shows that the new data-misfit function outperforms the conventional L2-loss under quantum measurements. Quantum circuit simulations of 1D and 2D linear convection diffusion equations under approximate and finite sampling loss evaluations, together with classical numerical experiments on a nonlinear forced viscous Burgers equation, demonstrate the feasibility of the proposed approach for parameter inversion even when the loss evaluations are affected by sampling noise. This framework may provide a viable quantum-assisted scheme for PDE-based inverse problems and elucidate the potential of quantum PDE algorithms in addressing a complete quantum-to-end optimization stack.

1. Introduction.

Quantum PDE solvers commonly produce solution quantum states, making classical readout costly. The paper proposes a noise-tolerant quantum-classical Bayesian inversion framework that estimates normalized misfits from quantum states while GPR guides parameter exploration.

  • Motivation: Existing quantum PDE frameworks primarily output quantum states rather than classical solution vectors, limiting direct use in applications.Quantum state tomography can incur prohibitive costs that may eliminate speedups gained during state preparation.
  • Problem setting: Bayesian PDE inversion infers unknown physical parameters through repeated evaluations of the forward PDE model against observed data.Its central computational task is evaluating a data-misfit function rather than reconstructing the full solution vector.
  • Framework: The proposed hybrid framework has the quantum processor evolve the PDE and evaluate noisy losses, while classical GPR tunes hyperparameters and selects trial candidates.GPR explicitly uses uncertainty in noisy results to support exploration.
  • Quantum-compatible loss: The method replaces conventional L2-loss with a normalized quantum-state loss and estimates it by combining quantum PDE solvers with the Hadamard test.The construction targets linear evolution PDEs and extends naturally to semi-linear PDEs through Carleman linearization.
  • Analysis and validation: Error and complexity analysis shows that the normalized quantum-state loss outperforms the physical loss when quantum measurement costs are included.Numerical experiments report feasible parameter inversion despite sampling noise in loss evaluations.

2. Classical Bayesian PDE inversion.

Classical Bayesian PDE inversion formulates parameter recovery from terminal observations as optimization of a PDE data-misfit. Bayesian optimization uses a GPR surrogate and acquisition function to select costly forward-model evaluations iteratively.

  • PDE inverse problem: The forward PDE maps model parameters m in a feasible set M to a terminal solution, with the operator F allowed to be linear or nonlinear.The initial condition is u0(x), and the evolution is specified by ∂t = F(u(x, t; m), t; m).
  • PDE inverse problem: Given discrete terminal observations uobs and known initial data and terminal time, inversion seeks the parameter m∗ that best fits the observations.The solution is represented at discrete spatial points as a vector u(t; m).
  • Objective function: The physical loss is the L2 misfit between the observed data and the parameter-dependent terminal solution.The objective is optimized over the feasible parameter set.
  • Bayesian optimization: Bayesian optimization constructs a probabilistic surrogate from evaluated samples and uses it to guide new trial points when objective evaluations are costly.Each evaluation may require a high-fidelity PDE simulation or experimental procedure.
  • Surrogate model by GPR: A Gaussian process models objective values with a mean function and covariance kernel, while GPR hyperparameters are fitted by minimizing negative log marginal likelihood.The covariance may combine an RBF signal kernel with a white-noise kernel.
  • Candidate selection: The acquisition function selects the next parameter by maximizing expected improvement over the current best observed objective value.After evaluation, the dataset and surrogate are updated until a stopping criterion such as iteration limit or acquisition convergence is reached.
  • Quantum-assisted extension: The proposed workflow retains classical GPR while replacing repeated high-fidelity objective evaluations with quantum-state operations that avoid reconstructing the complete solution.The overall workflow combines quantum objective evaluation with classical surrogate-based candidate selection.

3. Quantum-assisted Bayesian PDE inversion.

The framework combines quantum PDE solvers with classical Bayesian optimization to evaluate inverse-problem losses without reconstructing full classical solution vectors. It supports linear and semi-linear settings through normalized quantum-state losses and Hadamard-test measurements, with sampling complexity and error considerations guiding the design.

  • Hybrid inversion workflow: Quantum PDE solvers prepare terminal-solution states, while classical Gaussian Process Regression tunes hyper-parameters to select trial candidates for Bayesian inversion.The solver output may include auxiliary and garbage components, so successful-branch measurements are used in loss evaluation.
  • Quantum forward solvers: Linear autonomous, linear non-autonomous, and polynomial nonlinear PDEs are treated using matrix-exponential, evolution-operator, LCHS, Schrödingerization, or Carleman-linearization approaches.Carleman linearization lifts nonlinear dynamics into a truncated hierarchy whose first-order component approximates the physical solution.
  • Physical loss evaluation: The physical L2-loss preserves both terminal-solution magnitude and normalized profile but requires reconstructing quantum-measured norm and overlap quantities.The projected success probability supplies the approximate solution norm, while a Hadamard test estimates the observation overlap.
  • Quantum measurement: The Hadamard test estimates the real overlap between the observation state and solver output, using the probability of measuring its auxiliary qubit in |0⟩_H.At constant confidence, estimating the overlap to additive accuracy ϵ requires O(1/ϵ^2) repetitions.
  • Normalized loss evaluation: Normalized quantum-state losses avoid explicit recovery of the terminal solution's physical scaling by comparing normalized overlaps in the physical or lifted subspace.This reduces dependence on norm estimation, while the lifted loss compares the complete truncated Carleman hierarchy rather than only the original variable.
  • Scope of the loss: The normalized loss discards overall norm information, making it appropriate when inversion depends primarily on the normalized spatial profile rather than absolute solution scale.The lifted normalized loss may require fewer circuit repetitions than the projected normalized loss but incorporates higher-order tensor-product components.

4. Error and complexity analysis.

The analysis bounds approximation and sampling errors for normalized and physical losses, then characterizes oracle-query complexity for prescribed accuracy in Bayesian optimization evaluations.

  • Approximation error: The normalized-loss error is bounded in terms of the terminal-state approximation error and the observation norm under a closeness condition.The bound includes the factor 4ϵapp/((1−δ)∥uobs∥).
  • Approximation error: The physical-loss analysis likewise provides deterministic error bounds for the infinite-shot quantum approximation, including a bound under the same closeness condition.The corresponding bound is stated for the exact and approximate terminal solutions.
  • Sampling error: Finite-shot normalized-loss estimation uses concentration inequalities to control statistical error with prescribed failure probability and accuracy.The analysis combines Hoeffding and multiplicative Chernoff bounds with a union bound, requiring a lower bound on Hadamard-test samples.
  • Complexity analysis: Each projected-success measurement queries the state-preparation oracle once, whereas each Hadamard-test measurement queries both the preparation and observation oracles.The resulting oracle-query complexities for one BayesOpt loss evaluation are summarized in Table 2.
  • Complexity analysis: The same statistical-analysis strategy extends to the lifted normalized loss, while detailed gate complexity depends on the implementation of the quantum PDE solver.The solver-dependent scaling factor is denoted λ(m).

5. Numerical experiments.

The numerical experiments test quantum-assisted Bayesian inversion on linear convection-diffusion and nonlinear forced viscous Burgers equations. Quantum-circuit and classical simulations indicate feasible, stable parameter inversion under approximate, finite-shot, and prescribed solution-noise settings.

  • 5.1. 1-D convection-diffusion equation: The 1-D convection-diffusion experiment evaluates Taylor-truncated forward-solution accuracy and its effect on parameter inversion.The setup uses nTay = 5, nx = 16, and true parameter m∗= (0.3, 0.04).
  • 5.1. 1-D convection-diffusion equation: Finite-shot Hadamard-test simulations produce reasonable parameter estimates, demonstrating feasibility under measurement noise.One representative run with NH = 109 yields mopt = (0.2913, 0.0596), compared with the classical baseline mopt = (0.2993, 0.0343).
  • 5.2. 2-D convection-diffusion equation: The 2-D convection-diffusion experiment uses periodic boundaries, a 64×64 grid, true parameter m∗= (0.05, 1), and relative noise levels η ∈{0.1, 0.2, 0.3}.The noisy training data and subsequent BayesOpt evaluations use the prescribed relative noise levels.
  • 5.2. 2-D convection-diffusion equation: At η = 0.2, the reconstructed 2-D solution uses the mean GPR estimate mopt = (0.0530, 1.0000) over 10 independent runs and is compared pointwise with the exact solution.The figure includes the exact solution, reconstructed solution, and pointwise absolute error.
  • 5.3. Forced viscous Burgers equation: For the forced viscous Burgers equation, the nonlinear inverse problem treats Re as the unknown parameter and uses Carleman linearization with truncated Taylor approximation.The experiments examine Carleman truncation accuracy and then solve the inverse problem using the lifted formulation.
  • 5.3.2. PDE inversion under different loss functions: Both quantum-state loss functions yield reasonable inversion results, while the normalized loss has comparable performance to the physical loss and appears to produce larger surrogate uncertainty.The larger uncertainty may facilitate broader exploration of the parameter space.

6. Conclusion and discussion.

The framework is limited by observation-state preparation costs and retains classical surrogate modeling, while experiments report reasonable inversion results with both quantum-state losses.

  • Efficient oracle access to the observation state is assumed, but preparing arbitrary dense observational data may be costly.Sparse sensor observations and exploitable functional structure may enable specialized preparation techniques.
  • Both quantum-state loss functions yield reasonable inversion results within the Bayesian optimization procedure.Figure 6 compares the trained GPR surrogate and exact objective under different loss functions and settings.
  • The BayesOpt surrogate modeling remains classical in the present framework.Integrating quantum loss evaluation with quantum-enhanced surrogate modeling and optimization is identified as future work.

Appendix A. Quantum PDE solvers. We first consider an evolution PDE

The appendix formulates quantum PDE solvers from spatially discretized evolution PDEs, covering autonomous linear systems and representative linear non-autonomous and nonlinear examples.

  • After spatial discretization, the evolution PDE becomes a finite-dimensional dynamical system for the discrete solution vector u(t; m).The parameter m belongs to a feasible parameter set, while the differential operator may be linear or nonlinear and include source terms.
  • For linear autonomous systems, the terminal solution is uT(m) = eT A(m)u0, motivating block-encoding quantum algorithms for matrix-exponential evolution.A(m) is parameter-dependent, time-independent, and produced by spatial discretization.
  • The appendix uses LCHS for a linear non-autonomous equation and Carleman linearization with GQSP for a nonlinear equation.These examples illustrate how different quantum PDE solvers fit the proposed loss-function construction.

A.1. Linear non-autonomous evolution via LCHS.

LCHS represents nonunitary linear evolution through a continuous linear combination of Hamiltonian simulations, then prepares an approximate terminal state for normalized-loss evaluation.

  • LCHS expresses nonunitary evolution as a continuous linear combination of unitary Hamiltonian time evolutions.For each real k, H(τ; m) + kL(τ; m) is Hermitian, so the corresponding Uk(T, s; m) is unitary.
  • Finite implementations truncate and discretize the LCHS integral, encoding the approximate terminal solution in a successful ancillary branch.The coherent construction combines coefficient preparation, controlled simulations, and reversal of coefficient-state preparation.
  • The observation is amplitude-encoded and compared with the solver output through a Hadamard test to estimate their inner product.Only the successful solver branch contributes to the overlap with the observation state.
  • Taking the ratio of measured quantities removes the LCHS normalization factor and initial-state norm for normalized-loss evaluation.Physical-loss reconstruction instead retains the LCU normalization factor.
  • The loss-evaluation procedure is independent of LCHS's internal Hamiltonian-simulation structure once a coherent terminal-state output is prepared.The construction also extends to inhomogeneous systems by coherently combining initial-value and source contributions.

A.2. Carleman linearization via GQSP.

Carleman linearization lifts an autonomous quadratic nonlinear system into a truncated finite-dimensional linear system, which GQSP approximates through a normalized Taylor polynomial.

  • A.2. Carleman linearization via GQSP.: The nonlinear evolution has quadratic, linear, and time-independent forcing components: F2(m)u⊗2(t; m) + F1(m)u(t; m) + F0(m).The tensor-square state u⊗2 represents the quadratic contribution.
  • A.2. Carleman linearization via GQSP.: Carleman linearization introduces tensor powers of the original state and truncates the hierarchy at order N.The constant component incorporates affine forcing, while couplings to tensor powers above N are neglected.
  • A.2. Carleman linearization via GQSP.: The truncated nonlinear evolution becomes a finite-dimensional linear autonomous system with terminal lifted solution YN(T; m) = eT A(m)Y0.The first-order component of the lifted state approximates the original nonlinear state.
  • A.2. Carleman linearization via GQSP.: GQSP approximates the terminal evolution using a degree-n normalized Taylor polynomial compatible with block encoding of the generally non-Hermitian Carleman matrix.The scaling α(m) is chosen so that ∥A(m)/α(m)∥≤1.

A.2.1. Quantum algorithm for the forward solver.

The forward solver regularizes block-encoding and applies GQSP to implement polynomial approximations of the parameter-dependent evolution operator. Its coherent output can be used for normalized loss evaluation with an additional Hadamard test.

  • Regularized block-encoding: An n-regular block-encoding prevents unsuccessful branches from returning to the successful subspace during repeated polynomial applications.A counter register and incrementer implement this regularity.
  • GQSP polynomial transformation: GQSP uses single-qubit operations to implement bounded matrix polynomials for arbitrary square matrices encoded by the regularized construction.The polynomial must satisfy |P(z)| ≤1 on the relevant domain.
  • Circuit construction: The circuit for the parameter-dependent Carleman matrix uses a system register, counter register, block-encoding ancillas, and a GQSP processing qubit.The processing operations can be computed classically through numerical synthesis.
  • State preparation and output: The input state amplitude-encodes the normalized initial lifted vector, and the GQSP circuit produces a normalized approximation of the terminal lifted solution.The normalization factor is denoted λY(m).
  • Loss-evaluation interface: The coherent GQSP output can be used directly in normalized lifted-loss procedures with an additional Hadamard-test circuit.This avoids requiring full classical readout at the forward-solver stage.

A.2.2. Loss reconstruction from the GQSP output.

The GQSP output supports loss reconstruction through success-probability measurements and Hadamard-test overlaps. A flag register isolates the physical first-order Carleman component when losses are defined on the original variable.

  • Observation-state overlap: The normalized observation state is prepared with the flag enabled and ancillary registers in their successful zero states, so only the flagged physical component contributes to the Hadamard overlap.The projected success probability is obtained by jointly measuring the flag and GQSP ancillas.
  • Normalization cancellation: Combining the GQSP success probability with the overlap estimate eliminates the output normalization factor in the infinite-shot lifted normalized loss.This yields a loss directly from measurable circuit quantities.
  • Physical-component extraction: A flag register marks the physical block corresponding to the first-order Carleman component, allowing its overlap with an embedded observation state to be measured.The physical block occupies the system indices associated with the first-order component.
  • Measurement reconstruction: The Hadamard-test statistics provide the real part of the terminal-state and observation-state inner product used in normalized and physical loss formulas.Success probabilities supply the remaining quantities needed for reconstruction.
  • Loss-specific reconstruction: The lifted normalized loss is evaluated directly from the successful GQSP branch, while physical-space losses require identifying the first-order component.The latter selection uses a flag qubit and successful ancillary measurements.

Appendix B. Detailed results for the 1-D forced viscous Burgers equa-

The appendix evaluates BayesOpt for a 1-D forced viscous Burgers equation using three loss functions under an infinite-shot baseline. Across the cases, BayesOpt converges and normalized lifted-loss optimization remains comparable to physical-loss optimization.

  • Problem setting: The experiment considers three Reynolds-number and final-time cases for a discretization with nx = 16 and viscosity parameterized by the Reynolds number.The initial condition and boundary conditions are fixed for the experiment.
  • Compared losses: The benchmark compares physical, normalized lifted, and normalized losses while neglecting Hadamard-test sampling error.The objective function is defined from the selected loss function.
  • Reported results: Table 7 reports GPR-predicted parameters and corresponding relative L2 errors for the parameter-inversion experiments.Figures 8–10 provide baseline inversion results for the three losses.
  • Optimization behavior: BayesOpt converges in each case, with the GPR surrogate closely approximating the exact objective near the true parameter.The figure panels compare ground-truth and inversion solutions, optimization trajectories, and objective functions.
  • Comparison and implication: Although normalized lifted-loss objective values are more compressed, its optimization performance remains comparable to that of the physical loss.The results provide numerical support for nonlinear inversion through Carleman linearization and quantum-state-based normalized losses.
Loading 2608.27028v1…