Matrix Product Operators

A matrix product operator (MPO) factorizes an operator into local input-output tensors. It is useful when a Hamiltonian contains many terms but only a small number of patterns connecting them. The operator bond dimension measures the complexity of those patterns; it need not equal the MPS bond dimension of a state acted on by the operator.

Required background. MPS implementation supplies virtual-index conventions, and sparse many-body Hamiltonians supplies an independent small-system reference.

Operator strings as virtual transitions

Use a tensor $W^{[j]\,s't}_{ab}$ with virtual indices $a,b$ and physical output/input indices $s',t$. Contracting virtual indices produces $\langle s'_1\cdots s'_N|H|t_1\cdots t_N\rangle$. Physical index pairs must never be interchanged silently.

For an open Ising chain,

$$H=-J\sum_{j=1}^{N-1}Z_jZ_{j+1}-h\sum_{j=1}^{N}X_j,$$

a bond-dimension-three bulk MPO is the operator-valued matrix

$$W= \begin{pmatrix} I&0&0\\ Z&0&0\\ -hX&-JZ&I \end{pmatrix}, \qquad v_L=(0,0,1),\quad v_R=(1,0,0)^T.$$

The scalar operator $v_LW^{[1]}\cdots W^{[N]}v_R$ has three kinds of virtual path: wait in state three, start a term, then finish in state one and propagate identities. A transition $3\to1$ inserts $-hX$; $3\to2\to1$ inserts $-JZZ$. The zero at $2\to2$ prevents unwanted longer strings.

Contracting an expectation without constructing H

Let $L_{a\alpha\alpha'}$ be an environment carrying one MPO bond and two MPS bonds. Advancing one site contracts $L$, the ket tensor, $W$, and the conjugate bra tensor. The final scalar divided by the contracted state norm is the expectation. If the MPS bond is $D$ and MPO bond is $w$, common dense contraction orders cost terms of order $Nd^2wD^3$ and $Nd^2w^2D^2$; sparsity of $W$ can reduce this substantially.

Applying an MPO to an MPS combines virtual indices and produces bonds no larger than $wD$. Compress this result with a canonical SVD or a variational fit. The multiplication is exact before compression; the fit tolerance is an additional error control, especially in repeated time evolution.

A two-site construction test

For $N=2$, the path sum must yield

$$H=-JZ\otimes Z-h(X\otimes I+I\otimes X).$$

At $h=0$, the ordered $Z$-basis diagonal is $(-J,J,J,-J)$. At $J=0$, the two-site ground energy is $-2|h|$. These tests expose virtual-boundary mistakes and incorrect coefficients. They cannot detect every physical-axis transposition: this particular Hamiltonian is real and symmetric. Also compare a complex off-diagonal operator, such as $Y_1$, with its dense matrix to test conjugation and input-output conventions.

Compression and fermionic strings

Several operator paths can be combined or compressed by treating each physical input-output pair as one index and applying tensor-train compression. Frobenius compression controls the operator Hilbert–Schmidt norm in the chosen basis; it does not automatically certify a small relative error for every low-energy expectation.

Fermionic operators require the chosen parity convention. A Jordan–Wigner hopping term includes a parity string between its endpoints; dropping it changes the operator, even if the resulting MPO remains Hermitian. Use fermionic basis construction for the sign convention and verify the MPO within multiple particle-number sectors.

Exercises

Modify the displayed MPO for an inhomogeneous chain with field $h_j$ and coupling $J_j$ on bond $(j,j+1)$.

Solution

Replace $-hX$ at site $j$ by $-h_jX$ and the transition $3\to2$ by $-J_jZ$. The next site’s $2\to1$ remains $Z$. The last site’s $J_N$ is unused because the right boundary cannot end in virtual state two; it may be set to zero. This gives each bond coupling exactly once.

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.