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arXiv:2606.23274

Parallelized contraction of tensor trains or matrix product operators

Simone Foderà, Marc K. Ritter, Hiroshi Shinaoka, Jan von Delft

Published Jun 22, 2026• 0 citations
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Abstract

Tensor Trains (TT), also known as Matrix Product States (MPS) and Matrix Product Operators (MPO), provide a compact and structured representation for high-dimensional data and operators. One of the most expensive manipulations involving tensor trains is the contraction of two MPOs. A popular and accurate method for mitigating this cost is the fit algorithm. However, it is still comparatively costly since it involves 2-site updates. Moreover, the parallelization of the fit algorithm when used for MPO-MPO contractions has received comparatively little attention. In this work, we present two strategies for accelerating the fit algorithm, usable in combination: (1) We use MPI-based distributed-memory parallelization tailored for MPO-MPO contractions, employing one of two MPO gauge choices: (1a) the inverse canonical gauge, which yields near-ideal parallelization speedup across all problem sizes; and (1b) the site-canonical gauge, which avoids the inversion of singular values but requires extra computations to ensure global consistency, thus yielding excellent parallelization speedup only for large problems requiring several sweeps before convergence. (2) We use randomized projections to reduce the cost of local updates from 2-site to 1-site costs while retaining 2-site accuracy, and to speed up contractions of environment tensors with MPO tensors.

DOI
10.48550/arxiv.2606.23274

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Tensor Trains (TT), also known as Matrix Product States (MPS) and Matrix Product Operators (MPO), provide a compact and structured representation for high-dimensional data and operators. One of the most expensive manipulations involving tensor trains is the contraction of two MPOs. A popular and accurate method for mitigating this cost is the fit algorithm. However, it is still comparatively costly since it involves 2-site updates. Moreover, the parallelization of the fit algorithm when used for MPO-MPO contractions has received comparatively little attention. In this work, we present two strategies for accelerating the fit algorithm, usable in combination: (1) We use MPI-based distributed-memory parallelization tailored for MPO-MPO contractions, employing one of two MPO gauge choices: (1a) the inverse canonical gauge, which yields near-ideal parallelization speedup across all problem sizes; and (1b) the site-canonical gauge, which avoids the inversion of singular values but requires extra computations to ensure global consistency, thus yielding excellent parallelization speedup only for large problems requiring several sweeps before convergence. (2) We use randomized projections to reduce the cost of local updates from 2-site to 1-site costs while retaining 2-site accuracy, and to speed up contractions of environment tensors with MPO tensors.
The tensor train (TT) model is widely used to approximate high-dimensional tensors, enabling efficient handling of data that may exceed available memory. TT helps address the curse of dimensionality in applications such as system identification and dynamic programming. In some applications, TT is known as a ``matrix product state" (MPS). The last product is also known under the acronym MPO--MPS. The proposed algorithms achieve an improved trade-off between computational efficiency and accuracy compared to state-of-the-art methods.
Matrix product states and matrix product operators (MPOs) are one-dimensional tensor networks that underlie the modern density matrix renormalization group (DMRG) algorithm. The use of MPOs accounts for the high level of generality and wide range of applicability of DMRG. However, current algorithms for two-dimensional (2D) tensor network states, known as projected entangled-pair states, rarely employ the associated 2D tensor network operators, projected entangled-pair operators (PEPOs), due to their computational cost and conceptual complexity. To lower these two barriers, we describe how to reformulate a PEPO into a set of tensor network operators that resemble MPOs by considering the different sets of local operators that are generated from sequential bipartitions of the 2D system. The expectation value of a PEPO can then be evaluated on the fly using only the action of MPOs and generalized MPOs at each step of the approximate contraction of the 2D tensor network. This technique allows for the simpler construction and more efficient energy evaluation of 2D Hamiltonians that contain finite-range interactions, and provides an improved strategy to encode long-range interactions that is orders of magnitude more accurate and efficient than existing schemes.
Tensor trains (TTs), also known as matrix product states (MPS), are compressed representations of high-dimensional data that can be efficiently manipulated to perform calculations on the data. In many applications, such as TT-based solvers for nonlinear partial differential equations, the most expensive step is an elementwise multiplication or similar elementwise operation on multiple TTs. Known error-controlled algorithms for such operations scale as O(χ4)O(χ^4)O(χ4), where χχχ is the TT rank. If the rank of the output is smaller than χ2χ^2χ2, it is possible to formulate algorithms with better scaling. In this work, we present the alternating cross interpolation (ACI) algorithm that performs such operations in O(χ3)O(χ^3)O(χ3), while maintaining error control. We demonstrate these properties on benchmark problems, achieving a significant speedup for TT ranks that are commonly encountered in practical applications.
In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of tensorization (i.e., creating very high-order tensors from lower-order original data) and super compression of data achieved via quantized tensor train (QTT) networks. The purpose of a tensorization and quantization is to achieve, via low-rank tensor approximations "super" compression, and meaningful, compact representation of structured data. The main objective of this paper is to show how tensor networks can be used to solve a wide class of big data optimization problems (that are far from tractable by classical numerical methods) by applying tensorization and performing all operations using relatively small size matrices and tensors and applying iteratively optimized and approximative tensor contractions. Keywords: Tensor networks, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, tensorization, distributed representation od data optimization problems for very large-scale problems: generalized eigenvalue decomposition (GEVD), PCA/SVD, canonical correlation analysis (CCA).
Classical numerical techniques for solving partial differential equations (PDEs) become computationally expensive as the dimension of the discretized differential operator increases. For PDEs giving rise to Sturm--Liouville problems, tensor network (TN) methods can be highly productive: an operator of dimension N×NN\times NN×N can be represented as a matrix product operator (MPO) using only n=log⁡2(N)n=\log_2(N)n=log2​(N) qubits, enabling computation of eigenvalues and eigenvectors via imaginary time evolution (ITE). However, this remains computationally challenging. First, most methods for generating MPOs of large operators without explicit tensor-product structure require prohibitively large memory. Second, the number of Trotterization steps for convergence in conventional ITE increases rapidly with nnn. We present techniques to mitigate both challenges for certain sparse, structured differential operators. To address the first, we construct the MPO by expanding the operator in the Pauli-string basis, enabled by an analytical expression for the Pauli basis coefficients that reduces the memory requirement from O(2n+1)\mathcal{O}(2^{n+1})O(2n+1) to O(2n)\mathcal{O}(2n)O(2n). To address the second, we propose a multistage state-refinement heuristic that accelerates ITE convergence, reducing convergence time by up to two orders of magnitude. Using this TN framework, we compute the first 32 eigenstates of a Laplacian of dimension exceeding 10610^6106 with fidelity above 0.950.950.95 using a 20-qubit MPO. We further validate the method on the 2D anharmonic oscillator and investigate disordered systems, where increasing random potential strength degrades accuracy and limits the approach.
Tensor-train (TT) decomposition has gained tremendous popularity due to its efficiency in handling high-dimensional data arising in scientific and quantum computing as well as machine learning applications. It provides a compact representation of matrices and vectors with a Kronecker product-like low-rank structure, and hence enables efficient matrix-vector operations in this compressed form. The vector scalar product is among such key operations, comprising a series of tensor contractions in a specific TT network where the order of contractions significantly impacts the computational cost. In this work, we introduce fast algorithms for finding a near-optimal contraction ordering for such networks. We show that our algorithms find such contraction orderings efficiently, outperforming the ordering methods for general tensor networks. Compared to our approaches, the best existing method incurs a higher cost of up to 15% for xTy, twice for xTAy, and more than ten times for xTABy scalar products, where A and B are matrices expressed in TT format.