---
title: double_factorized_2body (latest version)
description: API reference for qiskit_fermions.linalg.double_factorized_2body in the latest version of qiskit-fermions
source: https://eu-de.quantum.cloud.ibm.com/docs/en/api/qiskit-fermions/linalg-double-factorized-2-body
---

# double\_factorized\_2body

`double_factorized_2body(two_body_tensor, tol, max_vecs=None, cholesky=True)`

Double-factorized decomposition of a real two-body tensor.

Represents $h_{pqrs} = \sum_t \sum_{kl} Z^{(t)}_{kl} U^{(t)}_{pk} U^{(t)}_{qk} U^{(t)}_{rl} U^{(t)}_{sl}$, returning a list of $(Z, U)$ terms where each $Z$ is a real symmetric diagonal Coulomb matrix and each $U$ is a unitary orbital rotation.

When `cholesky` is `True` (the default behavior) the outer factorization uses a modified Cholesky decomposition; otherwise it uses a truncated eigendecomposition.

**Parameters**

- **two\_body\_tensor** – the real two-body tensor of shape `(norb, norb, norb, norb)`, in chemist ordering $(pq|rs)$ (i.e. the index pairs $(p,q)$ and $(r,s)$ each label a charge density, as in the reconstruction formula above).
- **tol** – the tolerance controlling the truncation error of the decomposition.
- **max\_vecs** – the maximum number of terms to retain. Defaults to `norb * (norb + 1) / 2`.
- **cholesky** – whether to use the modified Cholesky decomposition (`True`) or a truncated eigendecomposition (`False`) for the outer factorization.

**Returns**

A list of $(Z, U)$ 2-tuples, where `Z` is the diagonal Coulomb matrix and `U` is the orbital rotation.