Tensor-Network Algorithms

Tensor-network methods trade the full Hilbert-space array for a structured factorization. They are useful when the relevant state, operator, or requested contraction has manageable internal dimensions. This chapter develops the numerical algorithms and their error controls; the physical entanglement structure is treated in Many-Body.

Enter this chapter

Begin with tensor-network representations and contraction costs, then implement matrix product states and matrix product operators. These pages establish index ordering, canonical gauges, environment contractions, and a short-chain benchmark before introducing an optimizer.

The central resource question is entanglement versus bond dimension. Across a chain cut an MPS of dimension $D$ has Schmidt rank at most $D$, but the required approximation dimension is determined by the spectral tail, not merely by entropy.

Choose an algorithm from the requested output

For a finite-chain ground state, use DMRG: local variational eigensolves and sweeps with independent bond-dimension and size convergence. For real-time dynamics of local Hamiltonians, TEBD uses gates and truncation. TDVP projects the dynamics onto an MPS manifold and is compatible with general MPO Hamiltonians.

For bulk one-dimensional observables, infinite MPS uses transfer fixed points and an explicit unit cell. For two-dimensional states, PEPS preserves lattice connectivity but introduces approximate environment contractions. For a hierarchy of length scales, MERA uses isometries, disentanglers, and causal cones.

For a circuit amplitude, expectation, or output sample, use tensor-network circuit simulation to formulate the correct network before optimizing its contraction. A single amplitude and an entire probability distribution have different costs.

A benchmark ladder that diagnoses errors

Start with a product state, where $D=1$ is exact. Next use a two-spin singlet: it requires $D=2$, has Heisenberg energy $-3J/4$, and exposes a restricted product-state approximation. Finally evolve $|00\rangle$ with $JX_1X_2$, for which

$$|\psi(t)\rangle =\cos(Jt/\hbar)|00\rangle-i\sin(Jt/\hbar)|11\rangle.$$

A two-site gate has no Hamiltonian splitting error, so discrepancies isolate contraction, phase, or truncation mistakes. Add a noncommuting three-site Hamiltonian only after this check passes; that additional problem tests the time integrator.

Compare short-chain observables with exact diagonalization before using a network at inaccessible sizes. The reference must use the same spin normalization, boundary conditions, and symmetry sector.

Separate the sources of error

The error-analysis page distinguishes state truncation, local solver accuracy, integration step, environment approximation, finite size, and unit-cell restrictions. Report observable changes under each relevant control parameter.

A calculation can conserve norm while losing entanglement, conserve energy while following the wrong projected trajectory, or minimize energy inside the wrong symmetry sector. The chapter’s worked examples make these failures measurable. A useful final result includes an error budget and the largest distance or time scale over which the requested observable is converged.

References

• Haegeman, Jutho, Christian Lubich, Ivan Oseledets, Bart Vandereycken, and Frank Verstraete, “Unifying time evolution and optimization with matrix product states,” Physical Review B 94, 165116 (2016), arXiv:1408.5056 — tangent-space projector splitting.
• Orús, Román, “A practical introduction to tensor networks: Matrix product states and projected entangled pair states,” Annals of Physics 349, 117–158 (2014), arXiv:1306.2164 — representations and approximate contractions.
• Schollwöck, Ulrich, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011), arXiv:1008.3477 — canonical gauges, contractions, and finite-chain algorithms.