Tensor Network Engine¶

The core engine implements three fundamental data structures and their associated algorithms.

Data Structures¶

Matrix Product States (MPS)¶

An MPS represents a quantum/classical state as a chain of rank-3 tensors:

\[|\psi\rangle = \sum_{s_1 \ldots s_L} A^{s_1} A^{s_2} \cdots A^{s_L} |s_1 s_2 \ldots s_L\rangle\]

ontic.core.mps.MPS ¶

Matrix Product State representation.

Attributes: tensors: List of tensors A[i] with shape (χ_left, d, χ_right) L: Number of sites d: Physical dimension (assumed uniform)

Example: >>> mps = MPS.random(L=10, d=2, chi=32) >>> print(f"Norm: {mps.norm():.6f}") >>> mps.canonicalize_left_() >>> entropy = mps.entropy(bond=4)

Matrix Product Operators (MPO)¶

ontic.mpo ¶

Matrix Product Operator (MPO) framework for direct TT-core updates.

Eliminates dense-to-QTT factorization tax (6.05ms) by updating TT-cores directly. Academic validation: Oseledets (2011), Dolgov & Savostyanov (2014).

Target performance: 0.65ms physics update (5× speedup vs 3.33ms dense solver).

Quantized Tensor Train (QTT)¶

ontic.qtt ¶

ontic.qtt — QTT-specific algorithms¶

Modules in this sub-package operate directly on Tensor-Train (TT) cores without materialising full dense arrays, achieving :math:O(n r^3) or :math:O(n r^2 d) complexity where n is the number of TT-cores, r the bond dimension, and d the local mode size.

Sub-modules

sparse_direct — LU / Cholesky in TT format
rank_adaptive — Information-theoretic rank selection (AIC/BIC/MDL)
unstructured — QTT on FEM / FVM meshes via RCM + quantics
eigensolvers — Lanczos / Davidson in TT format
krylov — CG / GMRES entirely in TT
dynamic_rank — Rank adaptation during time integration
differentiable — Autograd-compatible TT operations
pde_solvers — Implicit time-steppers (backward Euler, CN, BDF-2)
qtci_v2 — Enhanced TCI with rook pivoting & error certification
time_series — Temporal signal compression via quantics mapping

Algorithms¶

Algorithm	Module	Purpose
DMRG	`ontic.algorithms.dmrg`	Variational ground state
TEBD	`ontic.algorithms.tebd`	Real/imaginary time evolution
TDVP	`ontic.algorithms.tdvp`	Time-dependent variational
Lanczos	`ontic.algorithms.lanczos`	Eigenvalue computation