Trace norms are fundamental to quantum information theory, yet in many-body systems their evaluation remains a major computational bottleneck, as it generally requires diagonalizing exponentially large operators. Here, we overcome this bottleneck by introducing a controlled tensor-network algorithm for estimating the trace norm of matrix product operators without full diagonalization. The key idea is to combine Zolotarev's rational approximation to the sign function with a variational formulation solved using a density-matrix-renormalization-group-like algorithm. The resulting approximation is systematically improvable, with its accuracy controlled by the rational approximation parameters and the spectral weight near zero. Beyond the reach of exact diagonalization, we demonstrate controlled trace-norm calculations for entanglement negativity, quantum fidelity and quantum Fisher information, achieving substantially improved accuracy over polynomial-based Lanczos approaches. Our results establish trace-norm-based quantities as practical tensor-network observables, opening a route toward tensor-network studies of quantum information in mixed states.
Trace norms are fundamental to quantum information theory, yet in many-body systems their evaluation remains a major computational bottleneck, as it generally requires diagonalizing exponentially large operators. Here, we overcome this bottleneck by introducing a controlled tensor-network algorithm for estimating the trace norm of matrix product operators without full diagonalization. The key idea is to combine Zolotarev's rational approximation to the sign function with a variational formulation solved using a density-matrix-renormalization-group-like algorithm. The resulting approximation is systematically improvable, with its accuracy controlled by the rational approximation parameters and the spectral weight near zero. Beyond the reach of exact diagonalization, we demonstrate controlled trace-norm calculations for entanglement negativity, quantum fidelity and quantum Fisher information, achieving substantially improved accuracy over polynomial-based Lanczos approaches. Our results establish trace-norm-based quantities as practical tensor-network observables, opening a route toward tensor-network studies of quantum information in mixed states.
Building on a previously introduced block Lanczos method, we demonstrate how to approximate any operator function of the form Trf(A) when the argument A is given as a Hermitian matrix product operator (MPO). This gives access to quantities that, depending on the full spectrum, are difficult to access for standard tensor network techniques, such as the von Neumann entropy and the trace norm of an MPO. We present a modified, more efficient strategy for computing thermal properties of short- or long-range Hamiltonians, and we illustrate the performance of the method with numerical results for the thermal equilibrium states of the Lipkin-Meshkov-Glick and Ising Hamiltonians.
Abstract Tensor network methods as presented in our open source matrix product states software have opened up the possibility to study many-body quantum physics in one and quasi-one-dimensional systems in an easily accessible package similar to density functional theory codes but for strongly correlated dynamics. Here, we address methods which allow one to capture the full entanglement without truncation of the Hilbert space. Such methods are suitable for validation of and comparisons to tensor network algorithms, but especially useful in the case of new kinds of quantum states with high entanglement violating the truncation in tensor networks. Quantum cellular automata are one example for such a system, characterized by tunable complexity, entanglement, and a large spread over the Hilbert space. Beyond the evolution of pure states as a closed system, we adapt the techniques for open quantum systems simulated via the Lindblad master equation. We present three algorithms for solving closed-system many-body time evolution without truncation of the Hilbert space. Exact diagonalization methods have the advantage that they not only keep the full entanglement but also require no approximations to the propagator. Seeking the limits of a maximal number of qubits on a single core, we use Trotter decompositions or Krylov approximation to the exponential of the Hamiltonian. All three methods are also implemented for open systems represented via the Lindblad master equation built from local channels. We show their convergence parameters and focus on efficient schemes for their implementations including Abelian symmetries, e.g. symmetry used for number conservation in the Bose–Hubbard model or discrete symmetries in the quantum Ising model. We present the thermalization timescale in the long-range quantum Ising model as a key example of how exact diagonalization contributes to novel physics.
Open quantum many-body systems play an important role in quantum optics and condensed matter physics, and capture phenomena like transport, the interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate one-dimensional open quantum many-body dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is preserved at all times. Moreover, the approximation error is controlled with respect to the trace norm. Hence, this scheme overcomes various obstacles of the known numerical open-system evolution schemes. To exemplify the functioning of the approach, we study both stationary states and transient dissipative behavior, for various open quantum systems ranging from few to many bodies.
We propose a tensor-network (TN) approach for solving classical optimization problems that is inspired by spectral filtering and sampling on quantum states. We first shift and scale an Ising Hamiltonian of the cost function so that all eigenvalues become non-negative and the ground states correspond to the the largest eigenvalues, which are then amplified by power iteration. We represent the transformed Hamiltonian as a matrix product operator (MPO) and form an immense power of this object via truncated MPO-MPO contractions, embedding the resulting operator into a matrix product state for sampling in the computational basis. In contrast to the density-matrix renormalization group, our approach provides a straightforward route to systematic improvement by increasing the bond dimension and is better at avoiding local minima. We also study the performance of this power method in the context of a higher-order Ising Hamiltonian on a heavy-hexagonal lattice, making a comparison with simulated annealing. These results highlight the potential of quantum-inspired algorithms for solving optimization problems and provide a baseline for assessing and developing quantum algorithms.
Understanding quantum materials at meso and even macroscopic scales requires tight-binding calculations on system sizes where explicit matrix representations become prohibitively costly. This represents a major bottleneck to rationalize phenomena in moiré and super-moiré heterostructures and quasicrystals. Here, we present a unified tensor-network methodology to solve tight-binding problems at exceptionally large scales, by mapping a system of N=2L sites onto a many-body problem of L pseudospin sites, which is subsequently solved with tensor network algorithms. For Hamiltonians with compressible real-space structure, the tensor network bond dimension remains modest, typically of order a few tens, independent of N. Tensor network representations of arbitrary hopping functions including long-range, spatially modulated, and twisted-layer couplings are built with quantics tensor cross interpolation, and all physical observables are evaluated entirely with tensor network algebra without explicit matrix storage or diagonalization. We demonstrate applications to spectral functions, momentum-space spectra via the tensor-network quantum Fourier transform, real-space topological invariants, real-time dynamics, correlation induced symmetry breaking with self-consistent mean-field calculations, non-Hermitian phenomena, and excitonic many-body physics. Our methodology enables routinely solving systems with billions of sites, by leveraging the tensor network compressibility of real-space structures, and establishing a flexible framework to study quantum matter at ultra-large length scales. The methodology is implemented in the open-source Julia package TensorBinding.
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.