Yuzuha Protocol

The Yuzuha Protocol is a formal specification for recoupling-coefficient libraries in tensor network applications. It defines the abstract types, self-consistency requirements, and function interfaces that any conforming implementation must satisfy.

Motivation

Yuzuha currently implements recoupling theory for the SU(2) symmetry group. However, the interface it provides is intentionally designed to generalize to other compact Lie groups, particularly SU(N) for \(N > 2\). The Yuzuha Protocol makes this generalization explicit by separating the abstract interface from the concrete SU(2) implementation.

A library that conforms to the Yuzuha Protocol:

1. Provides the required abstract types (Type System)
2. Implements the required symbol functions (Symbol Interface)
3. Implements the required duality functions (Duality Interface)
4. Adopts a self-consistent internal convention — the SU(2) Conventions section documents one valid choice used by the reference implementation
5. Satisfies the caching requirements if caching is implemented (Data Caching)

The current yuzuha Python package is the reference implementation of this protocol for SU(2).

Scope

The protocol covers:

• The representation label type — how irreducible representations are identified
• The edge type — how labeled, directed edges are represented
• The specification type — how a full fusion tree is described
• The contraction type — how edge-pair contractions are specified
• The canonical basis function — normalization and fusion-tree conventions
• The X-symbol function — its required mathematical properties and API shape
• The R-symbol function — its required unitarity and composition properties
• The FS phase function — its required properties and relationship to arrow reversal
• The conjugate spec function — its required properties and relationship to duality
• The caching layer — determinism, thread-safety, and key-space requirements

The protocol does not prescribe:

• Implementation language or performance characteristics
• Internal algorithms (Racah formula, Wigner 3j/6j, etc.)
• Specific phase choices, fusion-tree coupling order, or OM-index labeling conventions — any self-consistent choice is valid; the SU(2) Conventions section documents the choices made by the reference implementation
• The file format of any cache
• How floating-point precision is handled beyond the requirement that all outputs are real-valued

Conformance

A library conforms to the Yuzuha Protocol if:

1. All required types are present and satisfy the interface constraints in Type System
2. All outputs satisfy the mathematical identities specified in Symbol Interface and Duality Interface
3. All error conditions specified in Symbol Interface and Duality Interface are raised with appropriate exceptions
4. If caching is implemented, it satisfies Data Caching

Conformance may be verified using the protocol's reference test suite (to be published separately).

Terminology

The following terms are used throughout this document with their precise meanings:

Term	Definition
Representation label	An element of the set of irreducible representations of the symmetry group
Edge	A pair (representation label, direction) identifying one external edge of a tensor
Spec	A complete description of a CG tensor: edges and internal representations
Internal representation	An intermediate representation appearing in a fusion tree
OM dimension	Number of valid internal representation paths for a given spec
OM index	Integer index \(\mu\) labeling one such path
Canonical basis	The chosen orthonormal basis for the space of CG tensors with a given spec
X-symbol	Recoupling coefficient tensor for contracting two CG tensors
R-symbol	Transformation matrix for permuting the edges of a CG tensor

Document Structure

Page	Contents
Type System	Required types: RepLabel, Direction, Edge, Spec, Contraction
Symbol Interface	canonical_basis, compute_x_symbol, compute_r_symbol — signatures and properties
Duality Interface	fs_phase, compute_conjugate — Frobenius-Schur phase and spec dualisation
SU(2) Conventions	One valid self-consistent convention: phase choices, fusion-tree form, metric
Data Caching	Key spaces, determinism, thread-safety

References

[1] A. Weichselbaum, Non-abelian symmetries in tensor networks: A quantum symmetry space approach, Ann. Phys. 327, 2972 (2012).

[2] A. Weichselbaum, X-symbols for non-Abelian symmetries in tensor networks, Phys. Rev. Res. 2, 023385 (2020).

[3] S. Singh and G. Vidal, Tensor network states and algorithms in the presence of a global SU(2) symmetry, Phys. Rev. B 86, 195114 (2012).

[4] P. Schmoll, S. Singh, M. Rizzi, and R. Orús, A programming guide for tensor networks with global SU(2) symmetry, Annals of Physics 419, 168232 (2020).