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, $i\frac{d}{dt}Ψ(t)=P\widehat H PΨ(t),$ where $\widehat H$ 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. For Stokes, the Zeno-reduced generator has the projected square factorization $S_h=-Π_hΔ_hΠ_h=(G_hΠ_h)^\dagger(G_hΠ_h),$ which can be represented through a Gaussian moment dilation and implemented as a Gaussian superposition of unitary Zeno evolutions generated by a first-order square-root Hamiltonian. In the generic sparse-access model, this gives a simulation-stage cost $\widetilde O(h^{-2}\sqrt t)$ , up to the usual postselection factor for preparing the normalized dissipative state. The results provide a first step toward understanding the intersection of quantum algorithms, DAEs, constrained PDE dynamics, and square-root Gaussian dilations.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, \[ i\frac{d}{dt}Ψ(t)=P\widehat H PΨ(t), \] where $\widehat H$ 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. For Stokes, the Zeno-reduced generator has the projected square factorization \[ S_h=-Π_hΔ_hΠ_h=(G_hΠ_h)^\dagger(G_hΠ_h), \] which can be represented through a Gaussian moment dilation and implemented as a Gaussian superposition of unitary Zeno evolutions generated by a first-order square-root Hamiltonian. In the generic sparse-access model, this gives a simulation-stage cost $\widetilde O(h^{-2}\sqrt t)$, up to the usual postselection factor for preparing the normalized dissipative state. The results provide a first step toward understanding the intersection of quantum algorithms, DAEs, constrained PDE dynamics, and square-root Gaussian dilations.

quant-pharXiv:2605.00794May 4, 20260 citations

Einstein and quantum mechanics

Evan Wilson

International Journal of Quantum ChemistryJun 18, 200919 citations

A Constraint between Noncommutative Parameters of Quantum Theories in Noncommutative Space

Jian-zu Zhang A B

In two-dimensional noncommutive space for the case of both position - position and momentum - momentum noncommuting, a constraint between noncommutative parameters is investigated. The related topic of guaranteeing Bose - Einstein statistics in noncommutive space in the general case are elucidated: Bose - Einstein statistics is guaranteed by the deformed Heisenberg - Weyl algebra itself, independent of dynamics. A special character of a dynamical system is represented by a constraint between noncommutative parameters. The general feature of the constraint for any system is a direct proportionality between noncommutative parameters with a proportional coefficient depending on characteristic parameters of the system under study. The constraint for a harmonic oscillator is illustrated.In two-dimensional noncommutive space for the case of both position - position and momentum - momentum noncommuting, a constraint between noncommutative parameters is investigated. The related topic of guaranteeing Bose - Einstein statistics in noncommutive space in the general case are elucidated: Bose - Einstein statistics is guaranteed by the deformed Heisenberg - Weyl algebra itself, independent of dynamics. A special character of a dynamical system is represented by a constraint between noncommutative parameters. The general feature of the constraint for any system is a direct proportionality between noncommutative parameters with a proportional coefficient depending on characteristic parameters of the system under study. The constraint for a harmonic oscillator is illustrated.

arXiv:quant-ph/0701166Jan 23, 20070 citations

Quantum Mechanics with Constraints

H. Jensen, H. Koppe

Elsevier eBooksJan 1, 197212 citations

Emergence of Matter and Quantum Dynamics from the Universal Constraint

Diana Haskins

This note derives both matter and quantum dynamics from a single universal constraint. Admissibility fixes the minimal excitation to a two-component structure and enforces antisymmetry, leading to spinorial degrees of freedom. The same constraint, through its representation and semiclassical expansion, yields the Wheeler–DeWitt equation and the Schrödinger equation as a necessary consistency condition. In this framework, matter and quantum mechanics are not fundamental inputs, but emergent features of the admissible state space and its constraint algebra.This note derives both matter and quantum dynamics from a single universal constraint. Admissibility fixes the minimal excitation to a two-component structure and enforces antisymmetry, leading to spinorial degrees of freedom. The same constraint, through its representation and semiclassical expansion, yields the Wheeler–DeWitt equation and the Schrödinger equation as a necessary consistency condition. In this framework, matter and quantum mechanics are not fundamental inputs, but emergent features of the admissible state space and its constraint algebra.

Apr 7, 20260 citations

Quantum mechanics with constraints

H Jensen, Heinz Koppe

Annals of PhysicsApr 1, 1971335 citations

i\frac{d}{dt}Ψ(t)=P\widehat H PΨ(t),

where

\widehat H

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. For Stokes, the Zeno-reduced generator has the projected square factorization

S_h=-Π_hΔ_hΠ_h=(G_hΠ_h)^\dagger(G_hΠ_h),

which can be represented through a Gaussian moment dilation and implemented as a Gaussian superposition of unitary Zeno evolutions generated by a first-order square-root Hamiltonian. In the generic sparse-access model, this gives a simulation-stage cost

\widetilde O(h^{-2}\sqrt t)

, up to the usual postselection factor for preparing the normalized dissipative state. The results provide a first step toward understanding the intersection of quantum algorithms, DAEs, constrained PDE dynamics, and square-root Gaussian dilations.