Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.26268

Bosonic Encodings for Hermite-Galerkin Discretizations of High-Dimensional PDEs and Bayesian Inverse Problems

Alice Barthe

quant-phSubmitted Aug 26, 2026• Announced Aug 28, 2026• 0 citations
arXivHTMLPDF

Abstract

The Koopman-von Neumann framework has been proposed to design quantum algorithms for non-linear dynamics. It maps a non-linear ordinary differential equation to a linear partial differential equation (PDE) governing a probability amplitude. Previous works represents this amplitude in the Hermite-function basis, equivalently as a bosonic state, and truncates the total Hermite degree to obtain a representation over $Θ(m\log N)$ qubits, where $N$ is the number of variables and $m$ the truncation order. We extend this approach to a broader class of linear PDEs whose differential operators have a structured polynomial form. We prove convergence of the truncation for both time-dependent dynamics and gapped ground-state problems under explicit regularity and stability assumptions. We then introduce a qubit encoding that supports efficient block encodings of the truncated operators. Finally, we apply the framework to Bayesian inverse problems with Gaussian priors and observation noise, reducing posterior-state preparation to the preparation of a structured Hamiltonian's ground state.

Comments
33 pages, 0 figures

Related papers

Provably Efficient Quantum Algorithms for Solving Nonlinear Differential Equations Using Multiple Bosonic Modes Coupled with Qubits

Yu Gan, Hirad Alipanah, Jinglei Cheng +7 more

Quantum computers have long been expected to efficiently solve complex classical differential equations. Most digital, fault-tolerant approaches use Carleman linearization to map nonlinear systems to linear ones and then apply quantum linear-system solvers. However, provable speedups typically require digital truncation and full fault tolerance, rendering such linearization approaches challenging to implement on current hardware. Here we present an analog, continuous-variable algorithm based on coupled bosonic modes with qubit-based adaptive measurements that avoids Hilbert-space digitization. This method encodes classical fields as coherent states and, via Kraus-channel composition derived from the Koopman-von Neumann (KvN) formalism, maps nonlinear evolution to linear dynamics. Unlike many analog schemes, the algorithm is provably efficient: advancing a first-order, $L$-grid point, $d$-dimensional, order-$K$ spatial-derivative, degree-$r$ polynomial-nonlinearity, strongly dissipative partial differential equations (PDEs) for $T$ time steps costs $\mathcal{O}\left(T(\log L + d r \log K)\right)$. The capability of the scheme is demonstrated by using it to simulate the one-dimensional Burgers' equation and two-dimensional Fisher-KPP equation. The resilience of the method to photon loss is shown under strong-dissipation conditions and an analytic counterterm is derived that systematically cancels the dominant, experimentally calibrated noise. This work establishes a continuous-variable framework for simulating nonlinear systems and identifies a viable pathway toward practical quantum speedup on near-term analog hardware.

quant-pharXiv:2511.09939•Nov 14, 2025•0 citations

Continuous Variables Quantum Algorithm for Solving Ordinary Differential Equations

Alice Barthe, Michele Grossi, Jordi Tura +1 more

Solving Ordinary Differential Equations (ODE) is important for a broad range of domains, such as engineering, weather forecast, and finance. Most quantum algorithms proposed to solve them revolve around transforming ODEs into a system of linear equations, in order to benefit from the exponential advantage promised by the HHL algorithm. However, there are known limitations to this subroutine such as the linear scaling with condition number. In particular, for HHL-based ODE solvers, this dependency yields a complexity that generally grows exponentially with the integration time. In this paper, we present a scheme that does not rely on the HHL subroutine. We use the Koopman Von Neumann (KvN) formalism that maps arbitrary nonlinear dynamics to a Hilbert space where the dynamics are unitary. This effectively reduces the ODE problem to a Hamiltonian time evolution, which is a well-known problem in quantum computing. However, this comes at a cost as the KvN formalism is expressed in an infinite-dimensional Hilbert space, and involves nonphysical states with infinite energies. Previous works have tackled this by truncating the Hilbert space, and by approximating all the relevant states and operations on qubit-based systems. Instead of mapping to qubit-based computations, in this paper, we investigate the direct use of continuous-variable quantum computers for this problem. We provide an algorithm to compile a sequence of Gaussian and non-Gaussian continuous variables gates to solve an arbitrary one-dimensional polynomial ODE. We analyze the algorithm and propose that it is intrinsically better suited for solving so-called initial distribution problems, rather than initial condition problems. We propose the first steps towards a comprehensive complexity and specify which steps need to be developed for a complete analysis.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 17, 2023•5 citations

Quantum framework for parameterizing partial differential equations via diagonal block-encoding

Hiroshi Yano, Yuki Sato

We study a quantum-algorithmic framework for parameterizing partial differential equations (PDEs). For a broad class of problems in which the discretized parameter field admits a diagonal representation, block-encodings of diagonal matrices, or diagonal block-encodings, can be used to represent spatially varying coefficients with structured, potentially complicated profiles. This encoding enables efficient quantum simulation of forward PDEs and extends naturally to parameter-dependent settings. Such simulations are a key primitive for quantum algorithms for PDE-constrained optimization, where the goal is to identify optimal design parameters. We illustrate the framework numerically through forward simulation and parameter design for the two-dimensional wave equation with a Gaussian parameter profile.

quant-pharXiv:2603.01358•Mar 3, 2026•0 citations

A Quantum Path to Partial Differential Equations

Xiantao Li

Partial differential equations are a promising application area for fault-tolerant quantum algorithms, but the subject lies between two communities with different languages: numerical analysis and quantum computation. These lecture notes provide a numerically grounded introduction for readers entering from either field. Block encoding is the organizing principle: once a discretized differential operator is embedded in a unitary, primitives such as quantum singular value transformation, Hamiltonian simulation, linear combinations of unitaries, amplitude amplification, and measurement can be assembled into algorithms for elliptic, hyperbolic, and parabolic PDEs. Each chapter begins with a standard finite difference or finite element discretization and follows the full pipeline from the continuous PDE to quantum encoding, transformation, and extraction of a quantity of interest. Particular attention is paid to the factors governing end-to-end performance, including discretization error, state preparation, normalization, postselection, and measurement cost. A final chapter introduces nonlinear problems through Carleman and Koopman-von Neumann linearizations. The aim is not a comprehensive survey or a claim of universal quantum advantage, but a mathematically transparent entry point and a shared vocabulary for researchers in both communities.

quant-pharXiv:2607.09639•Jul 13, 2026•0 citations

How to Map Linear Differential Equations to Schrödinger Equations via Carleman and Koopman-von Neumann Embeddings for Quantum Algorithms

Yuki Ito, Yu Tanaka, Keisuke Fujii

Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed classical variables into a quantum state. While the Carleman and Koopman-von Neumann embeddings have been investigated so far, the class of problems that can be mapped to the Schrödinger equation is not well understood even for linear differential equations. In this work, we investigate the conditions for linear differential equations to be mapped to the Schrödinger equation and solved on a quantum computer. Interestingly, we find that these conditions are identical for both Carleman and Koopman-von Neumann embeddings. We also compute the computational complexity associated with estimating the expected values of an observable. This is done by assuming a state preparation oracle, block encoding of the mapped Hamiltonian via either Carleman or Koopman-von Neumann embedding, and block encoding of the observable using $O(\log M)$ qubits with $M$ is the mapped system size. Furthermore, we consider a general classical quadratic Hamiltonian dynamics and find a sufficient condition to map it into the Schrödinger equation. As a special case, this includes the coupled harmonic oscillator model [Babbush et al., \cite{babbush_exponential_2023}]. We also find a concrete example that cannot be described as the coupled harmonic oscillator but can be mapped to the Schrödinger equation in our framework. These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.

arXiv:2311.15628•Nov 27, 2023•0 citations

On the numerical limitations of dual Koopman von Neumann embeddings for solving conservative nonlinear ordinary differential equations on quantum computers

Thibault Fredon, Abhay K. Ram, Fabrice Debbasch +2 more

The simulation of nonlinear ordinary differential equations on quantum computers is inherently challenging, as quantum gates are linear operators on qubit states. In this paper, we put forth a Koopman-von Neumann (KvN) operator based algorithm for solving nonlinear ordinary differential equations on a quantum computer which overcomes the innate limitations of quantum operations. In this approach, a Liouville probability density is embedded into a wavefunction, the evolution of which is governed by an operator dual to the Koopman operator. The trajectories corresponding to the nonlinear differential equations are reconstructed from the average value of Koopman observables evaluated through quantum measurements. We specifically evaluate the computational limitations of solving nonlinear equations within this Liouville embedding framework. An Ehrenfest-type estimate is derived that relates the reconstruction error to the covariance of the transported density, highlighting the competition between linear stretching and Hessian-induced folding.This leads to a key stability criterion, which depends on the local Ehrenfest-Reynolds number. We discuss the effects of measurement-induced errors including those due to Hadamard-test sampling, amplitude estimation, and bias due to covariance in probabilistic Grover-type inference on the evaluation of trajectories. The limiting bounds on the accuracy of quantum computations are numerically verified for the Lotka-Volterra system and for the quartic oscillator. We observe a rapid growth in computational errors when the Ehrenfest-Reynolds number approaches the predicted threshold. The bounds resulting from our analysis provide key guidelines for selecting the width of the initial Gaussian probability distribution associated with a system of nonlinear ordinary differential equations.

quant-pharXiv:2609.28999•Sep 25, 2026•0 citations
The Koopman-von Neumann framework has been proposed to design quantum algorithms for non-linear dynamics. It maps a non-linear ordinary differential equation to a linear partial differential equation (PDE) governing a probability amplitude. Previous works represents this amplitude in the Hermite-function basis, equivalently as a bosonic state, and truncates the total Hermite degree to obtain a representation over Θ(mlog⁡N)Θ(m\log N)Θ(mlogN) qubits, where NNN is the number of variables and mmm the truncation order. We extend this approach to a broader class of linear PDEs whose differential operators have a structured polynomial form. We prove convergence of the truncation for both time-dependent dynamics and gapped ground-state problems under explicit regularity and stability assumptions. We then introduce a qubit encoding that supports efficient block encodings of the truncated operators. Finally, we apply the framework to Bayesian inverse problems with Gaussian priors and observation noise, reducing posterior-state preparation to the preparation of a structured Hamiltonian's ground state.
Quantum computers have long been expected to efficiently solve complex classical differential equations. Most digital, fault-tolerant approaches use Carleman linearization to map nonlinear systems to linear ones and then apply quantum linear-system solvers. However, provable speedups typically require digital truncation and full fault tolerance, rendering such linearization approaches challenging to implement on current hardware. Here we present an analog, continuous-variable algorithm based on coupled bosonic modes with qubit-based adaptive measurements that avoids Hilbert-space digitization. This method encodes classical fields as coherent states and, via Kraus-channel composition derived from the Koopman-von Neumann (KvN) formalism, maps nonlinear evolution to linear dynamics. Unlike many analog schemes, the algorithm is provably efficient: advancing a first-order, LLL-grid point, ddd-dimensional, order-KKK spatial-derivative, degree-rrr polynomial-nonlinearity, strongly dissipative partial differential equations (PDEs) for TTT time steps costs O(T(log⁡L+drlog⁡K))\mathcal{O}\left(T(\log L + d r \log K)\right)O(T(logL+drlogK)). The capability of the scheme is demonstrated by using it to simulate the one-dimensional Burgers' equation and two-dimensional Fisher-KPP equation. The resilience of the method to photon loss is shown under strong-dissipation conditions and an analytic counterterm is derived that systematically cancels the dominant, experimentally calibrated noise. This work establishes a continuous-variable framework for simulating nonlinear systems and identifies a viable pathway toward practical quantum speedup on near-term analog hardware.
Solving Ordinary Differential Equations (ODE) is important for a broad range of domains, such as engineering, weather forecast, and finance. Most quantum algorithms proposed to solve them revolve around transforming ODEs into a system of linear equations, in order to benefit from the exponential advantage promised by the HHL algorithm. However, there are known limitations to this subroutine such as the linear scaling with condition number. In particular, for HHL-based ODE solvers, this dependency yields a complexity that generally grows exponentially with the integration time. In this paper, we present a scheme that does not rely on the HHL subroutine. We use the Koopman Von Neumann (KvN) formalism that maps arbitrary nonlinear dynamics to a Hilbert space where the dynamics are unitary. This effectively reduces the ODE problem to a Hamiltonian time evolution, which is a well-known problem in quantum computing. However, this comes at a cost as the KvN formalism is expressed in an infinite-dimensional Hilbert space, and involves nonphysical states with infinite energies. Previous works have tackled this by truncating the Hilbert space, and by approximating all the relevant states and operations on qubit-based systems. Instead of mapping to qubit-based computations, in this paper, we investigate the direct use of continuous-variable quantum computers for this problem. We provide an algorithm to compile a sequence of Gaussian and non-Gaussian continuous variables gates to solve an arbitrary one-dimensional polynomial ODE. We analyze the algorithm and propose that it is intrinsically better suited for solving so-called initial distribution problems, rather than initial condition problems. We propose the first steps towards a comprehensive complexity and specify which steps need to be developed for a complete analysis.
We study a quantum-algorithmic framework for parameterizing partial differential equations (PDEs). For a broad class of problems in which the discretized parameter field admits a diagonal representation, block-encodings of diagonal matrices, or diagonal block-encodings, can be used to represent spatially varying coefficients with structured, potentially complicated profiles. This encoding enables efficient quantum simulation of forward PDEs and extends naturally to parameter-dependent settings. Such simulations are a key primitive for quantum algorithms for PDE-constrained optimization, where the goal is to identify optimal design parameters. We illustrate the framework numerically through forward simulation and parameter design for the two-dimensional wave equation with a Gaussian parameter profile.
Partial differential equations are a promising application area for fault-tolerant quantum algorithms, but the subject lies between two communities with different languages: numerical analysis and quantum computation. These lecture notes provide a numerically grounded introduction for readers entering from either field. Block encoding is the organizing principle: once a discretized differential operator is embedded in a unitary, primitives such as quantum singular value transformation, Hamiltonian simulation, linear combinations of unitaries, amplitude amplification, and measurement can be assembled into algorithms for elliptic, hyperbolic, and parabolic PDEs. Each chapter begins with a standard finite difference or finite element discretization and follows the full pipeline from the continuous PDE to quantum encoding, transformation, and extraction of a quantity of interest. Particular attention is paid to the factors governing end-to-end performance, including discretization error, state preparation, normalization, postselection, and measurement cost. A final chapter introduces nonlinear problems through Carleman and Koopman-von Neumann linearizations. The aim is not a comprehensive survey or a claim of universal quantum advantage, but a mathematically transparent entry point and a shared vocabulary for researchers in both communities.
Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed classical variables into a quantum state. While the Carleman and Koopman-von Neumann embeddings have been investigated so far, the class of problems that can be mapped to the Schrödinger equation is not well understood even for linear differential equations. In this work, we investigate the conditions for linear differential equations to be mapped to the Schrödinger equation and solved on a quantum computer. Interestingly, we find that these conditions are identical for both Carleman and Koopman-von Neumann embeddings. We also compute the computational complexity associated with estimating the expected values of an observable. This is done by assuming a state preparation oracle, block encoding of the mapped Hamiltonian via either Carleman or Koopman-von Neumann embedding, and block encoding of the observable using O(log⁡M)O(\log M)O(logM) qubits with MMM is the mapped system size. Furthermore, we consider a general classical quadratic Hamiltonian dynamics and find a sufficient condition to map it into the Schrödinger equation. As a special case, this includes the coupled harmonic oscillator model [Babbush et al., \cite{babbush_exponential_2023}]. We also find a concrete example that cannot be described as the coupled harmonic oscillator but can be mapped to the Schrödinger equation in our framework. These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.
The simulation of nonlinear ordinary differential equations on quantum computers is inherently challenging, as quantum gates are linear operators on qubit states. In this paper, we put forth a Koopman-von Neumann (KvN) operator based algorithm for solving nonlinear ordinary differential equations on a quantum computer which overcomes the innate limitations of quantum operations. In this approach, a Liouville probability density is embedded into a wavefunction, the evolution of which is governed by an operator dual to the Koopman operator. The trajectories corresponding to the nonlinear differential equations are reconstructed from the average value of Koopman observables evaluated through quantum measurements. We specifically evaluate the computational limitations of solving nonlinear equations within this Liouville embedding framework. An Ehrenfest-type estimate is derived that relates the reconstruction error to the covariance of the transported density, highlighting the competition between linear stretching and Hessian-induced folding.This leads to a key stability criterion, which depends on the local Ehrenfest-Reynolds number. We discuss the effects of measurement-induced errors including those due to Hadamard-test sampling, amplitude estimation, and bias due to covariance in probabilistic Grover-type inference on the evaluation of trajectories. The limiting bounds on the accuracy of quantum computations are numerically verified for the Lotka-Volterra system and for the quartic oscillator. We observe a rapid growth in computational errors when the Ehrenfest-Reynolds number approaches the predicted threshold. The bounds resulting from our analysis provide key guidelines for selecting the width of the initial Gaussian probability distribution associated with a system of nonlinear ordinary differential equations.