Time-evolving Block Decimation - The Update of The Decomposition - Two-qubit Gates Acting On Qubits k, K+1

Two-qubit Gates Acting On Qubits k, K+1

The changes required to update the 's and the 's, following a unitary operation V on qubits k, k+1, concern only, and . They consist of a number of basic operations.

Following Vidal's original approach, can be regarded as belonging to only four subsystems:

The subspace J is spanned by the eigenvectors of the reduced density matrix :

In a similar way, the subspace K is spanned by the eigenvectors of the reduced density matrix:

The subspaces and belong to the qubits k and k + 1. Using this basis and the decomposition D, can be written as:

Using the same reasoning as for the OQG, the applying the TQG V to qubits k, k + 1 one needs only to update

, and

We can write as:

where

To find out the new decomposition, the new 's at the bond k and their corresponding Schmidt eigenvectors must be computed and expressed in terms of the 's of the decomposition D. The reduced density matrix is therefore diagonalized:

The square roots of its eigenvalues are the new 's. Expressing the eigenvectors of the diagonalized matrix in the basis : the 's are obtained as well:

From the left-hand eigenvectors,

after expressing them in the basis, the 's are: