Tensor-Network Error Analysis

Tensor-network error control requires several independent limits. A small discarded weight concerns one compression, a small eigensolver residual concerns one local optimization, and a stable energy concerns one observable. None alone proves that the full state or a long-distance correlation is accurate. This page develops the error budget shared by the algorithms in this chapter.

Required background. Canonical MPS compression supplies the meaning of a Schmidt tail. Convergence studies explains controlled parameter variation.

One truncation has a precise state error

Let $|\psi\rangle$ be normalized and $|\widetilde\psi\rangle$ the normalized retained Schmidt state after discarding weight $\epsilon$. Align their global phases so that their overlap is positive. Then

$$\|\psi-\widetilde\psi\|^2 =2-2\sqrt{1-\epsilon} =\epsilon+O(\epsilon^2).$$

For a bounded Hermitian observable $O$, the trace-distance inequality gives

$$|\langle O\rangle_\psi-\langle O\rangle_{\widetilde\psi}| \leq 2\|O\|\sqrt{\epsilon}.$$

Thus discarded probability scales as the square of a state-distance estimate. The formula requires orthonormal block bases and normalized states. It does not directly apply to arbitrary local tensor SVDs or to a noncanonical PEPS update.

Accumulating real-time truncations

Suppose the ideal sequence uses exactly unitary gates, and after gate $j$ the numerical normalized state is truncated with local norm distance $d_j=\sqrt{2-2\sqrt{1-\epsilon_j}}$. A triangle-inequality argument gives the conservative bound

$$\|\psi_{\rm exact}-\psi_{\rm truncated}\| \leq \sum_j d_j$$

for the same gate sequence. Unitary propagation does not amplify an existing norm difference. This bound can be very loose, but it explains why summing discarded weights is a diagnostic rather than a universal bound on final infidelity. Add splitting and local-exponential errors separately. Imaginary-time evolution requires different stability reasoning because normalization after a nonunitary map can amplify relative differences.

A two-error convergence experiment

For second-order TEBD at fixed final time, first compare three time steps with a generous bond cap. If successive differences have ratio near four, a nonzero quadratic time-step error is plausible. Repeat that comparison at a larger cap. Then hold the small time step fixed and inspect convergence in $D$.

For DMRG, separate local solve tolerances, sweep convergence, bond dimension, and system size. A small energy variance bounds energy spread; a bound on eigenstate fidelity additionally needs spectral separation, as explained in exact diagonalization. Independent lower-energy states and symmetry-sector comparisons address whether the result is the ground state. For PEPS, converge the contraction environment dimension at every state bond dimension. For infinite MPS, test unit-cell dependence and transfer fixed-point residuals.

A useful report contains actual observable changes under each variation, not only the parameter values used.

An observable can hide state error

Consider $|\psi\rangle=\sqrt{1-\epsilon}|00\rangle+\sqrt\epsilon|11\rangle$ and its rank-one approximation $|00\rangle$. Both give $\langle Z_1Z_2\rangle=1$, so that correlator detects no error. Their single-site expectations differ by $2\epsilon$, and their state fidelity is $1-\epsilon$. Selecting only a conserved or symmetry-fixed observable can therefore make a poor approximation look exact.

Conversely, near degeneracy the energy can be extremely accurate while the chosen linear combination has different long-distance order. Validate quantities sensitive to the intended physical conclusion.

Roundoff, symmetries, and reproducibility

Measure canonical-isometry residuals, norm discrepancies between independent contractions, Hermiticity of reduced density matrices, and conservation of declared quantum numbers. Raise arithmetic precision only when those residuals or ill-conditioning indicate a roundoff limit; it cannot repair insufficient bond dimension.

Store chain ordering, boundary conditions, symmetry sectors, initialization, bond schedule, truncation rule, local-solver tolerance, and the final observable convergence record. Tensor entries themselves can differ under gauge transformations; compare physical observables and state overlaps instead of raw tensor entries.

Exercises

If every one of $M$ truncations has discarded weight at most $\epsilon\ll1$, what conservative bound follows for the final state distance?

Solution

For unitary interleaving, it is at most $M\sqrt{2-2\sqrt{1-\epsilon}}\approx M\sqrt\epsilon$, also bounded trivially by two. The final infidelity cannot be identified with $M\epsilon$ without additional assumptions about how errors combine. This worst-case estimate motivates measuring actual convergence alongside local tails.

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.