We present a Hilbert space framework in which observed reality is defined as the image of a noncommutative operator cascade acting on an admissible configuration space. The cascade consists of three physically motivated filters: formation, interaction, and persistence, with formation decomposed into a hard projector (structural admissibility) and a soft weighting operator (thermodynamic and dynamical accessibility). An ordering commutator is introduced to quantify the noncommutativity of the cascade, along with a scalar diagnostic that measures the degree of structural competition between filters. Using a constrained three-qubit model, we show that the constraint operator determines whether the system exhibits nontrivial filter tension or collapses into an effectively commuting structure. In one sector, the filters act on misaligned subspaces and produce a weighted superposition; in another, they align and reduce observed reality to a single configuration. This provides a minimal, computable model of constraint-dependent structural selection in Hilbert space and establishes noncommutativity as a structural indicator of complexity. This work is part of the Carroll Wells Gradient (CWG) framework, which explores how persistence, constraint, and information shape structure across physical systems.We present a Hilbert space framework in which observed reality is defined as the image of a noncommutative operator cascade acting on an admissible configuration space. The cascade consists of three physically motivated filters: formation, interaction, and persistence, with formation decomposed into a hard projector (structural admissibility) and a soft weighting operator (thermodynamic and dynamical accessibility). An ordering commutator is introduced to quantify the noncommutativity of the cascade, along with a scalar diagnostic that measures the degree of structural competition between filters. Using a constrained three-qubit model, we show that the constraint operator determines whether the system exhibits nontrivial filter tension or collapses into an effectively commuting structure. In one sector, the filters act on misaligned subspaces and produce a weighted superposition; in another, they align and reduce observed reality to a single configuration. This provides a minimal, computable model of constraint-dependent structural selection in Hilbert space and establishes noncommutativity as a structural indicator of complexity. This work is part of the Carroll Wells Gradient (CWG) framework, which explores how persistence, constraint, and information shape structure across physical systems.

Apr 7, 20260 citations

Generalization of the Berezin Formula to the Noncommutative Case

A. G. Tokarev

Mathematical NotesJan 1, 20010 citations

Two examples in noncommutative probability

Dror Bar-Natan

Foundations of PhysicsJan 1, 19890 citations

Expanding the reach of quantum optimization with fermionic embeddings

Andrew Zhao, Nicholas C. Rubin

Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.

arXiv:2301.01778QuantumAug 28, 20240 citations

A note on the stabilizer formalism via noncommutative graphs

Roy Araiza, Jihong Cai, Yushan Chen +5 more

arXiv:2310.00762Quantum Information ProcessingFeb 29, 20240 citations

Scattering in a three-dimensional fuzzy space

Johannes N. Kriel, H. W. Groenewald, Frederik G. Scholtz

We develop scattering theory in a noncommutative space defined by an $su(2)$ coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spatial noncommutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of noncommutativity on the well's spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well depths the potential effectively becomes repulsive and that the cross section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle's wavelength inside the well becomes comparable to the noncommutative length scale.We develop scattering theory in a noncommutative space defined by an $su(2)$ coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spatial noncommutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of noncommutativity on the well's spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well depths the potential effectively becomes repulsive and that the cross section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle's wavelength inside the well becomes comparable to the noncommutative length scale.

arXiv:1612.01306Physical review. D/Physical review. D.Jan 4, 201710 citations

We develop scattering theory in a noncommutative space defined by an

su(2)

coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spatial noncommutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of noncommutativity on the well's spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well depths the potential effectively becomes repulsive and that the cross section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle's wavelength inside the well becomes comparable to the noncommutative length scale.