Matrix Product States

A matrix product state (MPS) stores a chain wavefunction as a product of site tensors. A reliable implementation must maintain both physical index order and a useful gauge: canonical gauges make the norm metric local and give singular values a direct truncation meaning. The many-body introduction explains the physical representation; here we construct, contract, and compress its arrays.

Required background. Schmidt decomposition and tensor products in code supply SVD and reshape conventions.

Successive SVD construction

Take an open chain with $C_{s_1\ldots s_N}$, $s_j=0,\ldots,d_j-1$. Reshape the first coefficient array into a matrix with row $s_1$ and column $(s_2,\ldots,s_N)$, without permuting physical axes, and compute $C=U\Sigma V^\dagger$. Store $A^{[1]s_1}_{1\alpha}=U_{s_1\alpha}$. Reshape $\Sigma V^\dagger$ into rows $(\alpha,s_2)$ and the remaining physical columns, then repeat. The last residual becomes the final tensor.

At site $j$ the array shape is $(D_{j-1},d_j,D_j)$. A reconstruction test must compare coefficients in the same lexicographic order as the input. An SVD can return different phases and bases in degenerate singular spaces; reconstructed states and observables, rather than individual tensor entries, are the invariants.

Canonical gauges and the center tensor

The transformation $A^{[j]s}\mapsto A^{[j]s}X$, $A^{[j+1]s}\mapsto X^{-1}A^{[j+1]s}$ leaves the state invariant. Avoid explicitly forming poorly conditioned inverses. A QR sweep moves the gauge center stably: reshape the site as $(D_{j-1}d_j)\times D_j$, factor it as $QR$, store $Q$, and multiply $R$ into its neighbor.

Left-canonical tensors satisfy

$$\sum_s (A^{[j]s})^\dagger A^{[j]s}=I,$$

while right-canonical tensors obey $\sum_s A^{[j]s}(A^{[j]s})^\dagger=I$. With left and right blocks orthonormal, the Frobenius norm of the remaining center tensor equals the wavefunction norm. Outside this gauge, a local tensor norm has no such interpretation.

Compression and a two-qubit example

Across a canonical bond, retain the largest $\chi$ singular values $\lambda_a$ and define

$$\epsilon=\sum_{a>\chi}\lambda_a^2, \qquad \lvert\widetilde\psi\rangle= \frac{\sum_{a\leq\chi}\lambda_a|L_aR_a\rangle} {\sqrt{1-\epsilon}}.$$

For an initially normalized state, $|\langle\psi|\widetilde\psi\rangle|^2=1-\epsilon$. The discarded weight is a squared norm loss, not the norm distance itself.

For $|\psi\rangle=\sqrt{0.9}|00\rangle+\sqrt{0.1}|11\rangle$, compression to $\chi=1$ gives $|00\rangle$, fidelity $0.9$, and norm distance $\sqrt{2-2\sqrt{0.9}}$. The discarded component has amplitude $\sqrt{0.1}$, which is much larger than $0.1$. This distinction matters when estimating observable errors.

Sequential truncations are each optimal for the current cut but do not generally yield the globally optimal fixed-bond MPS. A later variational compression can improve the result. See error analysis before summing discarded weights.

Implementation checks and boundaries

Check left/right isometry residuals, the independently contracted norm, reconstruction on short chains, and expectations before and after moving the gauge center. Test complex tensors so that an accidental transpose in place of a conjugate transpose cannot pass unnoticed.

Periodic MPS replace boundary vectors by a trace, but the finite open-chain canonical identities do not transfer unchanged. Cutting a ring changes the effective entanglement crossing that cut; periodic algorithms can therefore need larger bonds and more expensive environments.

Exercises

Why can one not read the Schmidt spectrum from an arbitrary site’s SVD?

Solution

The remaining tensor network supplies nontrivial left and right Gram matrices. A plain Euclidean SVD of the local array ignores those metrics. Bring both surrounding blocks into orthonormal gauges first; the center-bond SVD then computes the physical Schmidt coefficients. A gauge transformation with a badly scaled $X$ can change arbitrary local singular values without changing the state.

References

• 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.