Conventions

Index orderCopy Anchor Link

The array layout follows the default quimb tensor indexing conventions (Gray, 2018), so results round-trip through quimb without any permutation:

import quimb.tensor as qtn

from src_method import apply

H1 = qtn.MPO_rand(6, bond_dim=4, seed=0)
H2 = qtn.MPO_rand(6, bond_dim=4, seed=1)

result = qtn.MatrixProductOperator(apply(H1.arrays, H2.arrays, chi_out=16))

• MPO tensors: bulk tensors have index order ('l', 'r', 'u', 'd'). Boundary tensors are rank 3, dropping the outer 'l' or 'r' index.
• MPS tensors: bulk tensors have index order ('l', 'r', 'u'). Boundary tensors are rank 2, dropping the outer bond index.

Here 'l' and 'r' are the left and right virtual bonds and 'u' and 'd' are the upper and lower physical legs. Keep this in mind when building or manipulating site arrays directly.

The kind of a train is inferred from the rank of its first site: rank 2 is an MPS, rank 3 an MPO. Only open-boundary trains are supported; periodic ones are rejected with an error.

Contraction orderCopy Anchor Link

src contracts a stack of trains, written in mathematical order:

state = src(U3, U2, U1, psi, chi_out=64)  # U3 U2 U1 |psi>, U1 acts first

Each contraction joins the 'd' leg of a train with the 'u' leg (or the physical leg of an MPS) of the train to its right:

An MPS may appear only first or last, and not both. apply(A, B) and compress(A) are the two- and one-train cases; apply accepts only an MPO on the left.

Bras are not conjugatedCopy Anchor Link

A leading MPS is a row vector used without conjugation: src(phi, A, B) computes $\phi^T A B$, which equals src(Bᵀ, Aᵀ, phi) with $^T$ swapping the 'u' and 'd' legs. For the physical bra $\langle\psi| A B$, conjugate first:

bra = src([t.conj() for t in psi], A, B, chi_out=64)

The result then pairs with a ket by plain contraction, with no further conjugation.