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NCTSSoS.jl aims to provide a efficient tool for solving sparse noncommutative polynomial optimization problems which is based on the structured moment-SOHS hierarchy.

It is a successor to NCTSSOS.

Features

Supported Algebras

Algebra	Type	Relations	Status
Non-commutative	MonoidAlgebra	Free algebra (no relations)	✅
Projector	MonoidAlgebra	P² = P (idempotent)	✅
Unipotent	MonoidAlgebra	U² = I (involution)	✅
Pauli	TwistedGroupAlgebra	σ² = I, cyclic products	✅
Fermionic	PBWAlgebra	{aᵢ, aⱼ†} = δᵢⱼ (CAR)	✅
Bosonic	PBWAlgebra	[cᵢ, cⱼ†] = δᵢⱼ (CCR)	✅

Optimization Features

Feature	Description	Status
Moment-SOS hierarchy	SDP relaxation for polynomial optimization	✅
Correlative sparsity	Clique-based decomposition	✅
Term sparsity	Block structure exploitation	✅
Equality constraints	Linear and polynomial	✅
Inequality constraints	Localizing matrices	✅
Higher-order relaxations	Iterative refinement	✅

State Polynomial Optimization

Feature	Description	Status
Tracial optimization	tr(·) for cyclic traces	✅
State polynomials	Products of expectations ⟨A⟩⟨B⟩	✅
Maximally entangled states	Bipartite optimization	✅
Arbitrary states	General state optimization	✅
GNS construction	State reconstruction	🚧

Elimination Algorithms

Algorithm	Description
MF	Maximum fill-in
MMD	Minimum degree ordering
MaximalElimination	Maximal cliques
AsIsElimination	No reordering

Installation

NCTSSoS is a   Julia Language   package. To install NCTSSoS, please open Julia's interactive session (known as REPL) and press ] key in the REPL to use the package mode, and then type the following command:

For stable release:

pkg> add NCTSSoS 

For current master:

pkg> add NCTSSoS#master

Supporting and Citing

Much of the software in this ecosystem was developed as part of academic research. If you would like to help support it, please star the repository as such metrics may help us secure funding in the future. If you use our software as part of your research, teaching, or other activities, we would be grateful if you could cite our work. The CITATION.bib file in the root of this repository lists the relevant papers.

Contact

Jie Wang: [email protected]