GitHub

TensorTrainNumerics.jl

TensorTrainNumerics.jl is a Julia package for numerical computation with tensors in the tensor-train (TT) format [1, 2] and its quantics variant (QTT) [3, 4]. A TT representation stores a tensor with d indices of size n using O(d n r²) numbers, where r is the TT rank, instead of nᵈ. In the QTT format a vector of length 2^d becomes a TT with d binary sites, so smooth functions on grids with billions of points can be stored and manipulated with a few kilobytes of memory.

The package is aimed at scientific computing: solving high-dimensional or very finely resolved linear systems, eigenvalue problems, and evolution equations, and approximating functions from a limited number of samples. Many tensor-network packages already exist (see Resources); this one collects TT solvers, QTT operator constructions, time integrators, and cross interpolation behind one set of types.

Features

  • TT and QTT data types. TTVector and TTOperator, with multidimensional QTT wrappers QTTVector and QTTOperator that support serial and interleaved bit orderings.
  • Decomposition and rank control. TT-SVD decomposition of dense tensors, left/right canonical forms, TT rounding with a relative tolerance or a rank cap, and entanglement entropies across every bond.
  • Arithmetic. Addition, scalar and operator products, inner products and norms, Hadamard (elementwise) and Kronecker products, and diagonal operators built from vectors [5]. The Hadamard product is also available by tensor train multiplication [6], which truncates while it multiplies.
  • QTT functions and operators. Low-rank QTT constructions of polynomials, trigonometric functions, exponentials, and Chebyshev polynomials; finite- difference Laplacians with Dirichlet, Neumann, and periodic boundary conditions and an explicit inverse Laplacian [7]; gradient and shift operators; prolongation operators between grids; and the discrete Fourier transform [8–11].
  • Spin-chain Hamiltonians. Ising, XY, XXZ, XXX, and XYZ Hamiltonians as low-rank TT operators.
  • Linear and eigenvalue solvers. ALS and MALS [12], one- and two-site DMRG [13], AMEn [14, 15], and Krylov methods from KrylovKit.jl with rank truncation, all through linear_solve and eigen_solve.
  • Nonlinear problems. A penalty-method ALS solver and multigrid renormalization [16] for Gross–Pitaevskii-type ground states.
  • Time evolution. Explicit Euler, fourth-order Runge–Kutta, implicit Euler, and Crank–Nicolson steppers that accept any of the linear solvers, and one- and two-site TDVP in real and imaginary time [17, 18], all through time_evolve.
  • Cross interpolation. MaxVol, DMRG-cross, and greedy TT-cross algorithms that build a TT from function evaluations, and high-dimensional quadrature built on them [19–21].
  • Interoperability. TTVector implements the VectorInterface.jl interface, so it can be used directly with KrylovKit.jl and OptimKit.jl. Package extensions add reverse-mode differentiation rules for ChainRulesCore.jl (and hence Zygote.jl), Riemannian optimization with Manopt.jl, and conversion from TensorCrossInterpolation.jl.

Installation

using Pkg
Pkg.add("TensorTrainNumerics")

Where to go next