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arXiv:2509.03699 · SciPost Physics Codebases

The ITransverse.jl library for transverse tensor network contractions

Stefano Carignano

Published Jan 9, 2026• 1 citation
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Abstract

Transverse contraction methods are extremely promising tools for the efficient contraction of tensor networks associated with the time evolution of quantum many-body systems, allowing in some cases to circumvent the entanglement barrier that would normally prevent the study of quantum dynamics with classical resources. We present here the ITransverse.jl package, written in Julia and based on ITensors.jl, containing several of these high-level algorithms, including novel prescriptions for efficient truncations of temporal matrix product states.

DOI
10.21468/scipostphyscodeb.63

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Transverse contraction methods are extremely promising tools for the efficient contraction of tensor networks associated with the time evolution of quantum many-body systems, allowing in some cases to circumvent the entanglement barrier that would normally prevent the study of quantum dynamics with classical resources. We present here the ITransverse.jl package, written in Julia and based on ITensors.jl, containing several of these high-level algorithms, including novel prescriptions for efficient truncations of temporal matrix product states.
Transverse contraction methods are extremely promising tools for the efficient contraction of tensor networks associated with the time evolution of quantum many-body systems, allowing in some cases to circumvent the entanglement barrier that would normally prevent the study of quantum dynamics with classical resources. We present here the ITransverse.jl package, written in Julia and based on ITensors.jl, containing several of these high-level algorithms, including novel prescriptions for efficient truncations of temporal matrix product states.
Tensor Networks are graph representations of summation expressions in which vertices represent tensors and edges represent tensor indices or vector spaces. In this work, we present EinExprs.jl, a Julia package for contraction path optimization that offers state-of-art optimizers. We propose a representation of the contraction path of a Tensor Network based on symbolic expressions. Using this package the user may choose among a collection of different methods such as Greedy algorithms, or an approach based on the hypergraph partitioning problem. We benchmark this library with examples obtained from the simulation of Random Quantum Circuits (RQC), a well known example where Tensor Networks provide state-of-the-art methods.
We present the Julia package PauliStrings.jl for quantum many-body simulations, which performs fast operations on the Pauli group by encoding Pauli strings in binary. All of the Pauli string algebra is encoded into low-level logic operations on integers, and is made efficient by various truncation methods which allow for systematic extrapolation of the results. We illustrate the effectiveness of our package by (i) performing Heisenberg time evolution through direct numerical integration and (ii) by constructing a Liouvillian Krylov space. We benchmark the results against tensor network methods, and we find our package performs favorably. In addition, we show that this representation allows for easy encoding of any geometry. We present results for chaotic and integrable spin systems in 1D as well as some examples in 2D. Currently, the main limitations are the inefficiency of representing non-trivial pure states (or other low-rank operators), as well as the need to introduce dissipation to probe long-time dynamics.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.
We introduce TensorMixedStates, a Julia library built on top of ITensor which allows the simulation of quantum systems in the presence of dissipation using matrix product states (MPS). It offers three key features: i) it implements the MPS representation for mixed states along with associated operations, in particular the time evolution according to a Lindblad equation or discrete time evolution using non-unitary gates (quantum channels), ii) it is based on ITensor, which has proven its effectiveness and which gives access to efficient low-level tensor manipulation as well as state-of-the-art algorithms (like DMRG or TDVP), finally iii) it presents a user-friendly interface allowing users to write sophisticated simulations for pure and mixed quantum states in a few lines of code.
The transverse folding algorithm [M. C. Ba\~nuls et al., Phys. Rev. Lett. 102, 240603 (2009)] is a tensor network method to compute time-dependent local observables in out-of-equilibrium quantum spin chains that can overcome the limitations of matrix product states when entanglement grows slower in the time than in the space direction. We present a contraction strategy that makes use of the exact light cone structure of the tensor network representing the observables. The strategy can be combined with the hybrid truncation proposed for global quenches by Hastings and Mahajan Phys. Rev. A 91, 032306 (2015), which significantly improves the efficiency of the method. We demonstrate the performance of this transverse light cone contraction also for transport coefficients, and discuss how it can be extended to other dynamical quantities.