Introduction to matrix-product states and tensor networks

Abstract: 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.

• The paper provides a practical, code-supported introduction to MPS, MPOs, DMRG, TEBD, TDVP, PEPS, and tensor-network methods for mixed states and Lindblad dynamics.
• Tensor networks efficiently represent area-law states, while entanglement growth, criticality, hard PEPS contractions, and positivity issues limit accuracy and simulation times.
• The notes connect theory with Julia implementations and benchmarks, including exact free-fermion results, thermal states, dissipative chains, quantum circuits, and symmetry-protected phases.

These lecture notes by Grégoire Misguich provide a pedagogical introduction to tensor-network (TN) methods in quantum many-body physics, written for the 9th Les Houches Summer School on Computational Physics (Open Quantum Systems). The notes emphasize matrix-product states (MPS) and their associated algorithms, extend to projected entangled-pair states (PEPS), and devote substantial material to tensor-network representations of mixed states, quantum channels, and Lindblad dynamics. A distinctive feature is the systematic inclusion of short Julia code examples based on ITensor, ITensorMPS, and the TensorMixedStates library, which ground the formalism in executable implementations.

Scope and motivation

The notes position TNs as compressed representations of many-body states whose efficiency is governed by entanglement structure rather than Hilbert-space dimension per se. States obeying area-law entanglement—typically low-energy states of local Hamiltonians in one and two dimensions—can be represented with resources growing in a controlled way with system size, whereas volume-law states generically cannot be captured without exponentially large bond dimensions. The author is explicit that TNs are not a universal remedy: long-time dynamics, critical systems, and generic contractions in $d>1$ remain computationally hard.

Beyond lattice simulation, the notes survey several application domains: quantum chemistry (where MPS sites are molecular orbitals), infinite and continuous systems, classification of gapped symmetry-protected phases via projective representations on virtual indices, classical statistical mechanics (including the TASEP steady state as an MPS and transfer-matrix DMRG for 2d classical models), simulation of quantum circuits, and tensor cross interpolation (TCI) with quantics representations of multivariate functions. On the circuit-simulation side, the notes note that TN calculations have raised the performance bar that quantum processors must clear to demonstrate quantum advantage.

Matrix-product states: structure and canonical forms

An open-boundary MPS of bond dimension $\chi$ writes a state as a product of $\chi\times\chi$ matrices carrying physical indices, with boundary tensors of compatible shapes. The notes develop the gauge redundancy $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$ and state the fundamental theorem of MPS: two injective translationally invariant MPS representing the same state are related by such a gauge transformation up to a phase—a result underpinning the classification of 1d symmetry-protected topological phases through the projective representations $V_gV_h=\omega(g,h)V_{gh}$ carried on virtual indices.

Canonical forms are constructed by successive QR decompositions, either from a full vector representation or directly on an existing MPS. The mixed-canonical form, with left-orthonormal tensors, right-orthonormal tensors, and an orthogonality center at site $k$, reduces local expectation values to contractions costing $\mathcal{O}(d^2\chi^2)$ and yields the Schmidt decomposition across any bond from the SVD of the center tensor alone. The notes quantify the cost of exactness: converting a generic state yields bond dimensions $\chi_i=d^{\min(i,N-i)}$, confirmed numerically for a random spin-$\frac{1}{2}$ chain where the printed bond dimensions are $(2,4,8,16,32,64,128,64,32,16,8,4,2)$.

The transfer-matrix analysis establishes a structural limitation: for a translationally invariant MPS with finite $\chi$0 and non-degenerate dominant eigenvalue $\chi$1, connected correlations decay exponentially with correlation length $\chi$2. The GHZ state is cited as the counterexample where eigenvalue degeneracy produces long-range order.

Operators and algorithms

Matrix product operators are treated through the finite-state automaton picture: for a Hamiltonian with $\chi$3 terms crossing a given bond, the MPO bond dimension is $\chi$4, with explicit constructions given for onsite, transverse-field Ising ($\chi$5), and Heisenberg ($\chi$6) Hamiltonians. Crucially, local Hamiltonians admit exact MPO representations with system-size-independent bond dimension, in contrast to states, which generally require truncation.

DMRG is presented as variational optimization within the MPS manifold: sweeping through the chain, optimizing one or two site tensors at a time via the generalized eigenvalue problem $\chi$7, which collapses to a standard eigenproblem when the orthogonality center sits on the optimized bond. Iterative eigensolvers (Lanczos) avoid forming the $\chi$8 effective Hamiltonian explicitly; the notes report that state-of-the-art DMRG reaches $\chi$9.

Time evolution covers three families. TEBD applies Trotter-Suzuki decomposed gates with SVD-based truncation, restricted to nearest-neighbor Hamiltonians. TDVP projects the Schrödinger equation onto the tangent space of the fixed-$\chi\times\chi$0 manifold, guaranteeing energy conservation, with a two-site variant allowing dynamic bond-dimension growth. The W$\chi\times\chi$1 and W$\chi\times\chi$2 MPO methods of Zaletel et al. approximate $\chi\times\chi$3 with error $\chi\times\chi$4 while retaining compact MPO form; composing complex time steps $\chi\times\chi$5 reduces the error to $\chi\times\chi$6, and higher-order schemes reach $\chi\times\chi$7. W$\chi\times\chi$8 additionally exponentiates onsite terms exactly, demonstrated numerically by exact evolution under a uniform transverse field with a large single step. All methods confront the entropy barrier: entanglement growth forces bond-dimension growth, limiting accessible times—an active research problem the notes flag as unresolved.

Projected entangled-pair states

The notes justify PEPS quantitatively: mapping a 2d area-law state onto a snake-ordered MPS requires $\chi\times\chi$9, exponentially in the system width, whereas a PEPS cuts only boundary-proportional bonds, giving $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$0. A numerical example on a $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$1 free-fermion lattice illustrates the practical consequence: at $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$2, DMRG achieves relative energy error $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$3 but relative corner-to-corner density-correlation error $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$4—energies converge far faster than long-range correlations.

Two canonical PEPS examples are worked out with bond dimension $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$5: a critical state built from the classical 2d Ising Boltzmann weights, whose correlations decay algebraically at $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$6 (demonstrating that finite-$A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$7 PEPS evade the MPS exponential-decay constraint), and Kitaev's toric code, showing that finite-bond-dimension PEPS capture topological order. Higher-dimensional AKLT generalizations and RVB states (from $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$8) round out the examples.

The central difficulty is contraction: exact PEPS contraction is computationally hard (#P-hard in general), so expectation values require approximate schemes—boundary MPS, corner transfer matrix, Monte Carlo sampling, tensor renormalization group, or belief propagation—with polynomial costs of degree typically six or more in $A^{[i]\,s}\to X_{i-1}A^{[i]\,s}X_i^{-1}$9. This sharply limits accessible bond dimensions relative to MPS. The boundary-MPS method is sketched in detail, with truncation of the environment bond dimension $V_gV_h=\omega(g,h)V_{gh}$0 controlling the approximation.

Mixed states, channels, and Lindblad dynamics

The final section treats open systems via two routes. Purification represents $V_gV_h=\omega(g,h)V_{gh}$1 as the thermofield-double state, prepared by imaginary-time evolution from maximally entangled infinite-temperature Bell pairs; a code example validates TDVP-prepared thermal energies of the XX chain against the exact free-fermion expression $V_gV_h=\omega(g,h)V_{gh}$2.

Vectorization maps density matrices to super-kets $V_gV_h=\omega(g,h)V_{gh}$3 in a doubled Hilbert space, so that $V_gV_h=\omega(g,h)V_{gh}$4 and Kraus channels become MPOs acting on MPS. The operator-space entanglement entropy (OSEE) then plays the role that entanglement entropy plays for pure states, controlling the required bond dimension; a dissipative XX-chain example shows OSEE rising under Hamiltonian-driven correlation buildup and falling as dissipation drives the system toward the fully mixed product state. The Lindbladian itself becomes an MPO with bond dimension $V_gV_h=\omega(g,h)V_{gh}$5 for the commutator part, and Lindblad evolution reduces to MPS time evolution in operator space, amenable to TDVP or W-methods. A noisy random-circuit example computes the linear XEB estimator $V_gV_h=\omega(g,h)V_{gh}$6, showing saturation near 1 for ideal circuits approaching Haar-random behavior and exponential decay with depth for depolarizing noise at $V_gV_h=\omega(g,h)V_{gh}$7—consistent with the expected $V_gV_h=\omega(g,h)V_{gh}$8 scaling.

Limitations and open questions

The notes are candid about boundaries of applicability. Vectorized MPS representations do not enforce positive semidefiniteness after truncation; positivity-guaranteeing MPDO constructions exist but are numerically more expensive, so practitioners monitor trace, Hermiticity, and bond-dimension convergence instead. Variational steady-state search via DMRG on $V_gV_h=\omega(g,h)V_{gh}$9 may converge slowly when the spectral gap above zero is small, a situation the author reports as "not uncommon" in simple spin-chain models and expected to worsen if the gap closes in the thermodynamic limit. The entropy barrier for real-time dynamics remains the field's central unsolved algorithmic problem, and exact PEPS contraction hardness caps achievable accuracy in $k$0. Periodic MPS, fermionic TNs beyond Jordan-Wigner in higher dimensions, and non-Markovian process tensors are mentioned but deliberately excluded from the detailed treatment.

Conclusion

These lecture notes offer a coherent, self-contained path from elementary tensor notation to research-level tools for open quantum systems, distinguished by the integration of runnable Julia examples at every stage. Their principal contributions are pedagogical: explicit derivations of canonical forms, MPO automaton constructions, the W$k$1/W$k$2 expansions, and vectorized Lindblad dynamics, each tied to verifiable numerical benchmarks against exact free-fermion or known-analytic results. For researchers entering the field—or those seeking a compact treatment of mixed-state TN methods—the notes serve as both a tutorial and a practical reference, while honestly delineating the entanglement-growth and contraction-hardness barriers that continue to define the limits of tensor-network computation.