arxiv: 2605.00794 · v1 · submitted 2026-05-01 · 🪐 quant-ph

Recognition: unknown

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

Hsuan-Cheng Wu , Xiantao Li

Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:20 UTC · model grok-4.3

classification 🪐 quant-ph

keywords differential-algebraic equationsquantum simulationStokes equationsprojected dynamicsquantum Zeno effectblock encodingHamiltonian simulationconstrained PDEs

0 comments

The pith

A dilation framework embeds non-Hermitian DAE dynamics into projected Schrödinger evolution for quantum simulation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to handle differential-algebraic equations on quantum computers by enlarging the Hilbert space and using a projector to enforce constraints. This turns the evolution into the form i d/dt Ψ = P H P Ψ that can be simulated using existing quantum techniques like block encodings and quantum singular value transformation. Applied to the unsteady Stokes equations, it preserves the structure of the incompressibility constraint enforced by pressure. Sympathetic readers care because this provides a route for quantum algorithms to address constrained physical systems that appear in fluid dynamics and other areas, potentially offering efficiency gains over classical methods for large problems.

Core claim

The authors show that constrained linear DAEs can be lifted to a projected Schrödinger-type dynamics i d/dt Ψ(t) = P tH P Ψ(t) on an enlarged space, where this identifies the DAE with quantum Zeno dynamics and permits quantum simulation via Hamiltonian methods, demonstrated on discretizations of the unsteady Stokes flow.

What carries the argument

The dilation framework that embeds the generally non-Hermitian constrained evolution into a projected Schrödinger-type dynamics on an enlarged Hilbert space.

Load-bearing premise

The load-bearing premise is that the orthogonal projector onto the lifted constraint subspace can be efficiently constructed via QSVT and that low-energy spectral cutoffs remain valid without introducing prohibitive approximation errors.

What would settle it

Simulating a small instance of the discretized unsteady Stokes equations on a quantum device or classical emulator and observing whether the solution accuracy matches the predicted bounds from the spectral cutoffs and projector approximation would test the claim.

Figures

Figures reproduced from arXiv: 2605.00794 by Hsuan-Cheng Wu, Xiantao Li.

**Figure 1.** Figure 1: Equivalence between the dilation of the Schur complement system and the full DAE. view at source ↗

**Figure 2.** Figure 2: Error from the dilation technique described in view at source ↗

**Figure 3.** Figure 3: Velocity and pressure from the original discretized Stokes equation. Here we make the view at source ↗

**Figure 4.** Figure 4: Recovered velocity and pressure from the dilated DAE. view at source ↗

read the original 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 Schr\"odinger-type dynamics on an enlarged Hilbert space, \[ i\frac{d}{dt}\Psi(t)=P\tH P\Psi(t), \] 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper maps linear DAEs to projected Hamiltonian dynamics via a dilation that links to Zeno evolution, then applies it to discrete unsteady Stokes, but the key projector construction for the incompressibility constraint lacks an explicit efficient block encoding.

read the letter

The main point is that Wu and Li give a dilation that embeds a linear DAE into an enlarged space where the evolution becomes i d/dt Ψ = P H P Ψ with H Hermitian and P the orthogonal projector onto the constraint subspace. This turns the problem into something that looks like Hamiltonian simulation plus a projector, which they tie to quantum Zeno dynamics and standard tools like block encodings and QSVT. They then work out the version for structure-preserving discretizations of the unsteady Stokes equations, where pressure enforces the discrete divergence-free condition, and they discuss low-energy spectral cutoffs plus rough cost comparisons to classical projection methods. That framework is new; prior quantum ODE simulation papers do not directly handle the algebraic constraints and compatibility conditions that define DAEs. The Stokes application shows a concrete setting where the idea might matter for fluid problems. The construction itself is laid out clearly enough in the abstract and early sections to see the logic. The soft spot is exactly where the stress-test note flags it. For the quantum advantage to be real, P has to admit a block encoding whose cost is polylog in the spatial degrees of freedom. In the Stokes case P is the L2 projection onto the kernel of the discrete divergence operator, which classically comes from solving a Poisson problem. The paper does not supply an explicit block-encoding scheme or query-complexity bound for this P, nor a full error analysis that relates the low-energy cutoff to solution smoothness and the DAE index. Without those, the claimed simulation cost remains unverified and could be dominated by the projector step. This paper is for people already working on quantum algorithms for PDEs and constrained systems. A reader who wants to extend Hamiltonian simulation past pure ODEs will find a usable starting point. It deserves a serious referee because the core dilation is worth checking in detail, even though the Stokes implementation needs more work to be convincing.

Referee Report

2 major / 1 minor

Summary. This paper proposes a dilation framework for quantum simulation of linear differential-algebraic equations (DAEs) by embedding the generally non-Hermitian constrained evolution into projected Schrödinger-type dynamics i d/dt Ψ(t) = P H P Ψ(t) on an enlarged Hilbert space, where H is Hermitian and P is the orthogonal projector onto the lifted constraint subspace. This identifies the DAE evolution with quantum Zeno dynamics and enables block encodings, QSVT-based projector construction, and Hamiltonian simulation. The framework is applied to structure-preserving discretizations of the unsteady Stokes equations (with pressure enforcing the discrete incompressibility constraint), deriving the corresponding projected Hamiltonian formulation, identifying low-energy spectral cutoffs motivated by solution smoothness, and discussing the resulting quantum simulation costs in comparison with classical projection-type methods.

Significance. If the dilation accurately reproduces the original DAE dynamics and the projector P admits an efficient block encoding, this work could provide a conceptually novel route to quantum simulation of constrained PDEs such as unsteady Stokes flow, leveraging existing quantum primitives for Hamiltonian simulation. The Zeno-dynamics identification is a useful link that may generalize to other constrained systems. However, the practical significance is currently limited by the lack of explicit constructions and error bounds for the Stokes application.

major comments (2)

[Abstract and Stokes application section] Abstract and Stokes application section: the claim that the framework enables efficient use of block encodings and QSVT-based projector construction for the discrete incompressibility constraint lacks an explicit block-encoding scheme or query complexity bound. The projector P onto ker(div_h) is the L2-orthogonal projection onto the discrete divergence-free subspace; standard classical realizations solve a discrete Poisson problem, and no derivation is supplied showing a sparse or efficiently block-encodable representation whose cost does not dominate the Hamiltonian simulation.
[Discussion of low-energy spectral cutoffs] Discussion of low-energy spectral cutoffs: low-energy spectral cutoffs motivated by solution smoothness are identified, but no rigorous error analysis is provided relating the cutoff to the smoothness of the Stokes solution and the index of the DAE. This leaves the approximation error in the projected dynamics unquantified and weakens the cost comparison with classical methods.

minor comments (1)

[Abstract] The notation for the Hermitian operator (written as H or tH in the abstract) should be made consistent and defined clearly on first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below, providing clarifications and indicating revisions made to strengthen the presentation without overstating the results.

read point-by-point responses

Referee: [Abstract and Stokes application section] Abstract and Stokes application section: the claim that the framework enables efficient use of block encodings and QSVT-based projector construction for the discrete incompressibility constraint lacks an explicit block-encoding scheme or query complexity bound. The projector P onto ker(div_h) is the L2-orthogonal projection onto the discrete divergence-free subspace; standard classical realizations solve a discrete Poisson problem, and no derivation is supplied showing a sparse or efficiently block-encodable representation whose cost does not dominate the Hamiltonian simulation.

Authors: We agree that the manuscript does not supply an explicit block-encoding construction or query-complexity bound for the projector P onto the discrete divergence-free subspace in the Stokes discretization. The core contribution is the dilation that recasts the DAE as projected Hermitian dynamics, thereby making the problem amenable to existing quantum primitives (block encodings of the Hamiltonian together with QSVT-based projectors) whenever such an encoding for P can be obtained. We have revised the abstract and the Stokes-application section to replace the phrasing “enables efficient use” with “makes accessible the use of,” and we have added an explicit remark that constructing a sparse or efficiently block-encodable representation of P (whose cost must be compared with the Hamiltonian simulation) remains an open question whose resolution is outside the scope of the present work. This revision preserves the conceptual link while accurately reflecting the current state of the analysis. revision: partial
Referee: [Discussion of low-energy spectral cutoffs] Discussion of low-energy spectral cutoffs: low-energy spectral cutoffs motivated by solution smoothness are identified, but no rigorous error analysis is provided relating the cutoff to the smoothness of the Stokes solution and the index of the DAE. This leaves the approximation error in the projected dynamics unquantified and weakens the cost comparison with classical methods.

Authors: We acknowledge that the identification of low-energy spectral cutoffs is motivated by heuristic considerations of solution smoothness rather than a rigorous error bound that relates the cutoff threshold to the regularity of the Stokes solution and the index of the underlying DAE. In the revised manuscript we have inserted a short paragraph in the discussion section that states this limitation explicitly and notes that a quantitative error analysis would require approximation-theoretic estimates for the discrete Stokes operator; such an analysis is left for future investigation. The cost comparison with classical projection methods is accordingly presented as qualitative, highlighting only the structural advantage of the projected-Hamiltonian formulation. revision: partial

Circularity Check

0 steps flagged

No circularity; dilation framework and Stokes application are self-contained

full rationale

The paper defines a new dilation that maps the DAE to the projected dynamics i d/dt Ψ = P H P Ψ and then applies standard block-encoding/QSVT primitives to it. No equation reduces to a prior fitted quantity, no parameter is renamed as a prediction, and no load-bearing step relies on a self-citation chain whose content is unverified. The low-energy cutoff discussion is motivated by solution smoothness rather than being forced by the construction itself. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the embeddability of the DAE into projected Hermitian dynamics and the efficient quantum implementation of the projector; no free parameters or new physical entities are introduced.

axioms (1)

domain assumption The constrained DAE evolution can be identified with quantum Zeno-type dynamics on the dilated space.
Invoked to justify use of block encodings and QSVT.

pith-pipeline@v0.9.0 · 5508 in / 1212 out tokens · 29486 ms · 2026-05-09T19:20:05.224676+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

66 extracted references · 6 canonical work pages

[1]

J. W. Kamman and R. L. Huston. Dynamics of constrained multibody systems.Journal of Applied Mechanics, 51(4):899–903, 12 1984

1984
[2]

An augmented lagrangian formulation for the equations of motion of multibody systems subject to equality constraints

Elias Paraskevopoulos, Nikolaos Potosakis, and Sotirios Natsiavas. An augmented lagrangian formulation for the equations of motion of multibody systems subject to equality constraints. Procedia Engineering, 199:747–752, 2017

2017
[3]

Unsteady viscous flows and stokes’s first problem.Inter- national Journal of Thermal Sciences, 49(5):820–828, 2010

YS Muzychka and MM Yovanovich. Unsteady viscous flows and stokes’s first problem.Inter- national Journal of Thermal Sciences, 49(5):820–828, 2010

2010
[4]

John Wiley & Sons, 2009

Paul G Huray.Maxwell’s equations. John Wiley & Sons, 2009

2009
[5]

Springer Berlin Heidelberg New York, 1996

Gerhard Wanner and Ernst Hairer.Solving ordinary differential equations II, volume 375. Springer Berlin Heidelberg New York, 1996

1996
[6]

Projected implicit runge–kutta methods for differential- algebraic equations.SIAM Journal on Numerical Analysis, 28(4):1097–1120, 1991

Uri M Ascher and Linda R Petzold. Projected implicit runge–kutta methods for differential- algebraic equations.SIAM Journal on Numerical Analysis, 28(4):1097–1120, 1991

1991
[7]

Differential/algebraic equations are not ode’s.SIAM Journal on Scientific and Statistical Computing, 3(3):367–384, 1982

Linda Petzold. Differential/algebraic equations are not ode’s.SIAM Journal on Scientific and Statistical Computing, 3(3):367–384, 1982

1982
[8]

Rabier and Werner C

Patrick J. Rabier and Werner C. Rheinboldt. Theoretical and numerical analysis of differential- algebraic equations. InSolution of Equations inR n (Part 4), Techniques of Scientific Com- puting (Part4), Numerical Methods for Fluids (Part 2), volume 8 ofHandbook of Numerical Analysis, pages 183–540. Elsevier, 2002. 27

2002
[9]

Simultaneous numerical solution of differential-algebraic equations.IEEE trans- actions on circuit theory, 18(1):89–95, 1971

Charles Gear. Simultaneous numerical solution of differential-algebraic equations.IEEE trans- actions on circuit theory, 18(1):89–95, 1971

1971
[10]

Differential algebraic equations, indices, and integral algebraic equa- tions.SIAM Journal on Numerical Analysis, 27(6):1527–1534, 1990

Charles William Gear. Differential algebraic equations, indices, and integral algebraic equa- tions.SIAM Journal on Numerical Analysis, 27(6):1527–1534, 1990

1990
[11]

Hamiltonian Simulation Using Linear Combinations of Unitary Operations

Andrew M Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of unitary operations.arXiv preprint arXiv:1202.5822, 2012

work page Pith review arXiv 2012
[12]

Exponential improvement in precision for simulating sparse Hamiltonians

Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Exponential improvement in precision for simulating sparse Hamiltonians. InProceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 283–292, 2014

2014
[13]

Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

Andr´ as Gily´ en, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st annual ACM SIGACT symposium on theory of computing, pages 193– 204, 2019

2019
[14]

Theory of trotter error with commutator scaling.Physical Review X, 11(1):011020, 2021

Andrew M Childs, Yuan Su, Minh C Tran, Nathan Wiebe, and Shuchen Zhu. Theory of trotter error with commutator scaling.Physical Review X, 11(1):011020, 2021

2021
[15]

Time-dependent unbounded hamiltonian simulation with vector norm scaling.Quantum, 5:459, 2021

Dong An, Di Fang, and Lin Lin. Time-dependent unbounded hamiltonian simulation with vector norm scaling.Quantum, 5:459, 2021

2021
[16]

Linear combination of Hamiltonian simulation for nonuni- tary dynamics with optimal state preparation cost.Physical Review Letters, 131(15):150603, 2023

Dong An, Jin-Peng Liu, and Lin Lin. Linear combination of Hamiltonian simulation for nonuni- tary dynamics with optimal state preparation cost.Physical Review Letters, 131(15):150603, 2023

2023
[17]

Hamiltonian simulation in zeno subspaces.arXiv preprint arXiv:2405.13589, 2024

Kasra Rajabzadeh Dizaji, Ariq Haqq, Alicia B Magann, and Christian Arenz. Hamiltonian simulation in zeno subspaces.arXiv preprint arXiv:2405.13589, 2024

work page arXiv 2024
[18]

Quantum signal processing.IEEE Signal Processing Magazine, 19(6):12–32, 2002

Yonina C Eldar and Alan V Oppenheim. Quantum signal processing.IEEE Signal Processing Magazine, 19(6):12–32, 2002

2002
[19]

Quantum algorithm for linear systems of equations.Physical review letters, 103(15):150502, 2009

Aram W Harrow, Avinatan Hassidim, and Seth Lloyd. Quantum algorithm for linear systems of equations.Physical review letters, 103(15):150502, 2009

2009
[20]

Near-term quantum algorithms for linear systems of equations.arXiv preprint arXiv:1909.07344, 2019

Hsin-Yuan Huang, Kishor Bharti, and Patrick Rebentrost. Near-term quantum algorithms for linear systems of equations.arXiv preprint arXiv:1909.07344, 2019

work page arXiv 1909
[21]

Experimental realization of quantum algorithm for solving linear systems of equations.Physical Review A, 89(2):022313, 2014

Jian Pan, Yudong Cao, Xiwei Yao, Zhaokai Li, Chenyong Ju, Hongwei Chen, Xinhua Peng, Sabre Kais, and Jiangfeng Du. Experimental realization of quantum algorithm for solving linear systems of equations.Physical Review A, 89(2):022313, 2014

2014
[22]

Dervovic, M

Danial Dervovic, Mark Herbster, Peter Mountney, Simone Severini, Na¨Iri Usher, and Leonard Wossnig. Quantum linear systems algorithms: a primer.arXiv preprint arXiv:1802.08227, 2018

work page arXiv 2018
[23]

Preconditioned quantum linear system algorithm.Physical review letters, 110(25):250504, 2013

B David Clader, Bryan C Jacobs, and Chad R Sprouse. Preconditioned quantum linear system algorithm.Physical review letters, 110(25):250504, 2013. 28

2013
[24]

Quantum spectral methods for differential equations

Andrew M Childs and Jin-Peng Liu. Quantum spectral methods for differential equations. Communications in Mathematical Physics, 375(2):1427–1457, 2020

2020
[25]

Efficient quantum algorithm for dissipative nonlinear differential equations

Jin-Peng Liu, Herman Øie Kolden, Hari K Krovi, Nuno F Loureiro, Konstantina Trivisa, and Andrew M Childs. Efficient quantum algorithm for dissipative nonlinear differential equations. Proceedings of the National Academy of Sciences, 118(35):e2026805118, 2021

2021
[26]

Koopman–von neumann approach to quantum simulation of nonlinear classical dynamics.Physical Review Research, 2(4):043102, 2020

Ilon Joseph. Koopman–von neumann approach to quantum simulation of nonlinear classical dynamics.Physical Review Research, 2(4):043102, 2020

2020
[27]

Dominic W. Berry. High-order quantum algorithm for solving linear differential equations. Journal of Physics A: Mathematical and Theoretical, 47(10):105301, 2014

2014
[28]

Quantum machine learning.Nature, 549(7671):195–202, 2017

Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learning.Nature, 549(7671):195–202, 2017

2017
[29]

An introduction to quantum machine learning.Contemporary Physics, 56(2):172–185, 2015

Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione. An introduction to quantum machine learning.Contemporary Physics, 56(2):172–185, 2015

2015
[30]

Challenges and opportunities in quantum machine learning.Nature computational science, 2(9):567–576, 2022

Marco Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J Coles. Challenges and opportunities in quantum machine learning.Nature computational science, 2(9):567–576, 2022

2022
[31]

Power of data in quantum machine learning.Nature communications, 12(1):2631, 2021

Hsin-Yuan Huang, Michael Broughton, Masoud Mohseni, Ryan Babbush, Sergio Boixo, Hart- mut Neven, and Jarrod R McClean. Power of data in quantum machine learning.Nature communications, 12(1):2631, 2021

2021
[32]

Qkan: quantum kolmogorov-arnold networks with applications in machine learning and multivariate state preparation.npj Quantum Information, 2026

Petr Ivashkov, Po-Wei Huang, Kelvin Koor, Lirand¨ e Pira, and Patrick Rebentrost. Qkan: quantum kolmogorov-arnold networks with applications in machine learning and multivariate state preparation.npj Quantum Information, 2026

2026
[33]

From linear differential equations to unitaries: A moment-matching dilation frame- work with near-optimal quantum algorithms.arXiv preprint arXiv:2507.10285, 2025

Xiantao Li. From linear differential equations to unitaries: A moment-matching dilation frame- work with near-optimal quantum algorithms.arXiv preprint arXiv:2507.10285, 2025

work page arXiv 2025
[34]

Schr¨ odingerisation for quantum simulation of classical dynamics.Physical Review A, 108:032603, 2023

Shi Jin, Nai-Hui Liu, and Hongyu Yu. Schr¨ odingerisation for quantum simulation of classical dynamics.Physical Review A, 108:032603, 2023

2023
[35]

Quantum simulation of partial differential equations via Schr¨ odingerization.Phys

Shi Jin, Nana Liu, and Yue Yu. Quantum simulation of partial differential equations via Schr¨ odingerization.Phys. Rev. Lett., 133:230602, Dec 2024

2024
[36]

Quantum zeno dynamics: mathematical and physical aspects.Journal of Physics A: Mathematical and Theoretical, 41(49):493001, 2008

Paolo Facchi and Saverino Pascazio. Quantum zeno dynamics: mathematical and physical aspects.Journal of Physics A: Mathematical and Theoretical, 41(49):493001, 2008

2008
[37]

Unification of random dynamical decoupling and the quantum zeno effect.New Journal of Physics, 24(6):063027, 2022

Alexander Hahn, Daniel Burgarth, and Kazuya Yuasa. Unification of random dynamical decoupling and the quantum zeno effect.New Journal of Physics, 24(6):063027, 2022

2022
[38]

Quantum zeno effect.Physical Review A, 41(5):2295, 1990

Wayne M Itano, Daniel J Heinzen, John J Bollinger, and David J Wineland. Quantum zeno effect.Physical Review A, 41(5):2295, 1990

1990
[39]

Dynamical quantum zeno effect.Physical Review A, 50(6):4582, 1994

Saverio Pascazio and Mikio Namiki. Dynamical quantum zeno effect.Physical Review A, 50(6):4582, 1994. 29

1994
[40]

Quantum zeno effect by general measurements.Physics reports, 412(4):191–275, 2005

Kazuki Koshino and Akira Shimizu. Quantum zeno effect by general measurements.Physics reports, 412(4):191–275, 2005

2005
[41]

Simulating dynamics of rlc circuits with a quantum differential-algebraic equations solver, 2026

Arkopal Dutt, Anirban Chowdhury, Kristan Temme, and Hari Krovi. Simulating dynamics of rlc circuits with a quantum differential-algebraic equations solver, 2026

2026
[42]

Hamiltonian simulation by qubitization.Quantum, 3:163, 2019

Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by qubitization.Quantum, 3:163, 2019

2019
[43]

Experimental realization of quantum zeno dynam- ics.Nature Communications, 5:3194, 2014

Florian Sch¨ afer, Ivette Herrera, Sujit Cherukattil, Chiara Lovecchio, Francesco Saverio Catal- iotti, Filippo Caruso, and Augusto Smerzi. Experimental realization of quantum zeno dynam- ics.Nature Communications, 5:3194, 2014

2014
[44]

Reiserer, Peter C

Norbert Kalb, Andreas A. Reiserer, Peter C. Humphreys, Jacob J. W. Bakermans, Sten J. Kamerling, Naomi H. Nickerson, Simon C. Benjamin, Daniel J. Twitchen, Matthew Markham, and Ronald Hanson. Experimental creation of quantum zeno subspaces by repeated multi-spin projections in diamond.Nature Communications, 7:13111, 2016

2016
[45]

P. M. Harrington, J. T. Monroe, and K. W. Murch. Quantum zeno effects from measurement controlled qubit-bath interactions.Physical Review Letters, 118:240401, 2017

2017
[46]

J. J. W. H. Sørensen and Klaus Mølmer. Quantum control with measurements and quantum zeno dynamics.Physical Review A, 98:062317, 2018

2018
[47]

A quantum spectral frame- work for solving pdes, 2026

Chih-Kang Huang, Giacomo Antonioli, and Fr´ ed´ eric Barbaresco. A quantum spectral frame- work for solving pdes, 2026

2026
[48]

Ruehli, and Pierce A

Chung-Wen Ho, Albert E. Ruehli, and Pierce A. Brennan. The modified nodal approach to network analysis.IEEE Transactions on Circuits and Systems, 22(6):504–509, 1975

1975
[49]

World Scientific, Singapore, 2008

Ricardo Riaza.Differential-Algebraic Systems: Analytical Aspects and Circuit Applications. World Scientific, Singapore, 2008

2008
[50]

Harlow and J

Francis H. Harlow and J. Eddie Welch. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface.The Physics of Fluids, 8(12):2182–2189, 1965

1965
[51]

Alexandre J. Chorin. Numerical solution of the navier–stokes equations.Mathematics of Computation, 22(104):745–762, 1968

1968
[52]

Finite element approximation of the nonstationary navier–stokes problem

John G Heywood and Rolf Rannacher. Finite element approximation of the nonstationary navier–stokes problem. i. regularity of solutions and second-order error estimates for spatial discretization.SIAM Journal on Numerical Analysis, 19(2):275–311, 1982

1982
[53]

Finite-element approximation of the nonstationary navier–stokes problem

John G Heywood and Rolf Rannacher. Finite-element approximation of the nonstationary navier–stokes problem. part iv: error analysis for second-order time discretization.SIAM Journal on Numerical Analysis, 27(2):353–384, 1990

1990
[54]

American Mathematical Society, 2024

Roger Temam.Navier–Stokes equations: theory and numerical analysis, volume 343. American Mathematical Society, 2024. 30

2024
[55]

Springer, Berlin, 1986

Vivette Girault and Pierre-Arnaud Raviart.Finite Element Methods for Navier–Stokes Equa- tions: Theory and Algorithms. Springer, Berlin, 1986

1986
[56]

An overview of projection methods for incom- pressible flows.Computer Methods in Applied Mechanics and Engineering, 195(44–47):6011– 6045, 2006

Jean-Luc Guermond, Peter Minev, and Jie Shen. An overview of projection methods for incom- pressible flows.Computer Methods in Applied Mechanics and Engineering, 195(44–47):6011– 6045, 2006

2006
[57]

Error estimates for semidiscrete finite element approximations of the stokes equations under minimal regularity assumptions.Journal of scientific computing, 16(3):287–317, 2001

LS Hou. Error estimates for semidiscrete finite element approximations of the stokes equations under minimal regularity assumptions.Journal of scientific computing, 16(3):287–317, 2001

2001
[58]

Finite difference method for stokes equations: Mac scheme.University of California, Irvine https://www

LONG Chen. Finite difference method for stokes equations: Mac scheme.University of California, Irvine https://www. math. uci. edu/chenlong/226/MACStokes. pdf, 2016

2016
[59]

An overview of projection methods for in- compressible flows.Computer methods in applied mechanics and engineering, 195(44-47):6011– 6045, 2006

Jean-Luc Guermond, Peter Minev, and Jie Shen. An overview of projection methods for in- compressible flows.Computer methods in applied mechanics and engineering, 195(44-47):6011– 6045, 2006

2006
[60]

A new mixed finite element formulation and the mac method for the stokes equations.SIAM Journal on Numerical Analysis, 35(2):560–571, 1998

Houde Han and Xiaonan Wu. A new mixed finite element formulation and the mac method for the stokes equations.SIAM Journal on Numerical Analysis, 35(2):560–571, 1998

1998
[61]

Hamiltonian simulation by uniform spectral amplification

Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by uniform spectral amplifica- tion.arXiv preprint arXiv:1707.05391, 2017

work page arXiv 2017
[62]

Hamiltonian simulation for low-energy states with optimal time dependence.Quantum, 8:1449, 2024

Alexander Zlokapa and Rolando D Somma. Hamiltonian simulation for low-energy states with optimal time dependence.Quantum, 8:1449, 2024

2024
[63]

Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface.Physics of fluids, 8(12):2182, 1965

Francis H Harlow, J Eddie Welch, et al. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface.Physics of fluids, 8(12):2182, 1965

1965
[64]

The mac method.Computers & Fluids, 37(8):907–930, 2008

Sean McKee, Murilo F Tom´ e, Valdemir G Ferreira, Jos´ e A Cuminato, Antonio Castelo, FS Sousa, and Norberto Mangiavacchi. The mac method.Computers & Fluids, 37(8):907–930, 2008

2008
[65]

Stability and superconvergence of mac scheme for stokes equations on nonuniform grids.SIAM Journal on Numerical Analysis, 55(3):1135–1158, 2017

Hongxing Rui and Xiaoli Li. Stability and superconvergence of mac scheme for stokes equations on nonuniform grids.SIAM Journal on Numerical Analysis, 55(3):1135–1158, 2017

2017
[66]

Error analysis of a fully discrete consistent splitting mac scheme for time dependent stokes equations.Journal of Computational and Applied Mathematics, 421:114892, 2023

Xiaoli Li and Jie Shen. Error analysis of a fully discrete consistent splitting mac scheme for time dependent stokes equations.Journal of Computational and Applied Mathematics, 421:114892, 2023. 31

2023