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arXiv:2605.00794

Quantum Simulation of Differential-Algebraic Equations with Applications to Unsteady Stokes Flow

Hsuan-Cheng Wu, Xiantao Li

quant-phSubmitted May 1, 2026• Announced May 4, 2026• 0 citations
v1 · May 1, 2026v2 · May 18, 2026
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Abstract

Differential-algebraic equations (DAEs) arise naturally in constrained dynamical systems, but their algebraic constraints and hidden compatibility conditions make them more subtle than standard ordinary differential equations. This paper initiates a quantum-algorithmic study of constrained linear DAEs. We introduce a dilation framework that embeds the generally non-Hermitian constrained evolution into a projected Schrödinger-type dynamics on an enlarged Hilbert space, \[ i\frac{d}{dt}Ψ(t)=P\tH PΨ(t), \] where $\tH$ is Hermitian and $P$ is the orthogonal projector onto the lifted constraint subspace. This identifies the DAE evolution with a quantum Zeno-type dynamics and enables the use of block encodings, QSVT-based projector construction, and Hamiltonian simulation. We apply the framework to structure-preserving discretizations of the unsteady Stokes equations, where the pressure enforces the discrete incompressibility constraint. We derive the corresponding projected Hamiltonian formulation, identify low-energy spectral cutoffs motivated by solution smoothness, and discuss the resulting quantum simulation cost in comparison with classical projection-type methods. The results provide a first step toward understanding the potential intersection of quantum algorithms, DAEs, and constrained PDE dynamics.

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Differential-algebraic equations (DAEs) arise naturally in constrained dynamical systems, but their algebraic constraints and hidden compatibility conditions make them more subtle than standard ordinary differential equations. This paper initiates a quantum-algorithmic study of constrained linear DAEs. We introduce a dilation framework that embeds the generally non-Hermitian constrained evolution into a projected Schrödinger-type dynamics on an enlarged Hilbert space, [ i\frac{d}{dt}Ψ(t)=P\tH PΨ(t), ] where \tH\tH\tH is Hermitian and PPP is the orthogonal projector onto the lifted constraint subspace. This identifies the DAE evolution with a quantum Zeno-type dynamics and enables the use of block encodings, QSVT-based projector construction, and Hamiltonian simulation.
We apply the framework to structure-preserving discretizations of the unsteady Stokes equations, where the pressure enforces the discrete incompressibility constraint. We derive the corresponding projected Hamiltonian formulation, identify low-energy spectral cutoffs motivated by solution smoothness, and discuss the resulting quantum simulation cost in comparison with classical projection-type methods. The results provide a first step toward understanding the potential intersection of quantum algorithms, DAEs, and constrained PDE dynamics.
This paper presents a rigorous derivation pipeline for the quantum simulation of sparse polynomial nonlinear ordinary differential equations. The framework proceeds in three stages: (i) mapping the nonlinear dynamics to an exact infinite-dimensional linear system via Carleman–Koopman lifting; (ii) applying a graph-theoretic truncation and compression scheme that exploits the sparsity and locality of the dependency graph; and (iii) implementing Schr\"odingerisation for the resulting class of dissipative, Hermitian positive definite generators to yield a unitary-overlap formula compatible with standard quantum primitives. The study provides a comprehensive mathematical construction of these steps, establishing the algebraic and analytical foundations required for structure-preserving quantum simulation.
Quantum simulation has primarily focused on unitary dynamics, while many physical and engineering systems can be modeled by linear ordinary differential equations whose generators include non-Hermitian terms. Recent studies have shown that such equations, which give rise to nonunitary dynamics, can be embedded into a larger unitary framework via dilation techniques. However, their concrete realization on quantum circuits remains underexplored. In this paper we present a concrete pipeline that connects the dilation formalism with explicit quantum circuit constructions. On the analytical side, building on the recent dilation framework, we introduce a discretization of the continuous dilation operator that is tailored for quantum implementation. This construction ensures an exactly skew-Hermitian ancillary generator, which allows the moment conditions to be satisfied without imposing artificial constraints. We prove that the resulting scheme achieves a global error bound of order O(M−3/2)O(M^{-3/2})O(M−3/2), up to exponentially small boundary effects. This error can be suppressed by refining the discretization, where MMM denotes the discretization parameter. On the algorithmic side, we demonstrate that the dilation triple (Fh,∣rh⟩,⟨lh∣)(F_h, |r_h\rangle, \langle l_h|)(Fh​,∣rh​⟩,⟨lh​∣) can be efficiently implemented on quantum circuits. Using linear combinations of unitaries, QFT-adder operators, and quantum singular value transformation, the framework requires resources ranging from O(log⁡M)O(\log M)O(logM) to O((log⁡M)2)O((\log M)^2)O((logM)2), depending on the stage of the pipeline.
We develop a quantum algorithm for linear algebraic equations $ A\bb{x} = \bb{b} $ from the perspective of Schrödingerization-form problems, which are characterized by a system of linear convection equations in one higher dimension. When $ A $ is positive definite, the solution $ \bb{x} $ can be interpreted as the steady-state solution to a system of linear ordinary differential equations (ODEs). This ODE system can be solved by using the linear combination of Hamiltonian simulation (LCHS) method in \cite{ACL2023LCH2}, which serves as the continuous implementation of the Fourier transform in the Schrödingerization method from \cite{JLY22SchrShort, JLY22SchrLong}. Schrödingerization transforms linear partial differential equations (PDEs) and ODEs with non-unitary dynamics into Schrödinger-type systems via the so-called warped phase transformation that maps the equation into one higher dimension. When $ A $ is a general Hermitian matrix, the inverse matrix can still be represented in the LCHS form in \cite{ACL2023LCH2}, but with a kernel function based on the Fourier approach in \cite{Childs2017QLSA}. Although this LCHS form provides the steady-state solution to a system of linear ODEs associated with the least-squares equation, applying Schrödingerization to this least-squares system is not appropriate, as it results in a much larger condition number. We demonstrate that in both cases, the solution $ \bb{x} $ can be expressed as the LCHS of Schrödingerization-form problems. We provide a detailed implementation and error analysis. Furthermore, we incorporate a block preconditioning technique to achieve nearly linear scaling in the condition number, thereby attaining near-optimal query complexity.
We introduce a quantum algorithm for simulating the dynamics of electrical circuits consisting of resistors, inductors and capacitors (aka RLC circuits) along with power sources. Given oracle access to the connectivity of the circuit and values of the electrical elements, our algorithm prepares a quantum state that encodes voltages and current values either at a specified time or the history of their evolution over a time-interval. For an RLC circuit with NNN components, our algorithm runs in time polylog(N)\textsf{polylog}(N)polylog(N) under mild assumptions on the connectivity of the circuit and values of its components. This provides an exponential speed-up over classical algorithms that take poly(N)\textsf{poly}(N)poly(N) time in the worst-case. Our algorithm can be used to estimate energy across a set of components or dissipated power in polylog(N)\textsf{polylog}(N)polylog(N) time, a problem that we prove is BQP-hard and therefore unlikely to be efficiently solved by classical algorithms. The main challenge in simulating the dynamics of RLC circuits is that they are governed by differential-algebraic equations (DAEs), a coupled system of differential equations with hidden algebraic constraints. Consequentially, existing quantum algorithms for ordinary differential equations cannot be directly utilized. We therefore develop a quantum DAE solver for simulating the time-evolution of linear DAEs. For RLC circuits, we employ modified nodal analysis to create a system of DAEs compatible with our quantum algorithm. We establish BQP-hardness by demonstrating that any network of classical harmonic oscillators, for which an energy-estimation problem is known to be BQP-hard, is a special case of an LC circuit. Our work gives theoretical evidence of quantum advantage in simulating RLC circuits and we expect that our quantum DAE solver will find broader use in the simulation of dynamical systems.
We introduce a universal dilation framework to simulate arbitrary non-Hermitian dynamics x ˙ = L ( t ) x on unitary quantum processors. By imposing algebraic moment-matching conditions on an ancilla, our framework generates diverse families of dilation schemes that unify existing methods and broaden the design space for hardware-aware implementations. A tight-binding dilation that maps the ancilla to a 1D lattice with nearest-neighbor hopping is derived, overcoming the connectivity bottlenecks of dilation schemes. This construction handles general dynamics, including gain, without rescaling and achieves near-optimal complexity, as demonstrated by benchmarks on dissipative wave propagations.
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.