The OQGs are affecting only the qubit they are acting upon, the update of the state | ψ ψ ⟩ ⟩ after a unitary operator at qubit k does not modify the Schmidt eigenvalues or vectors on the left, consequently the Γ Γ [ k − − 1 ] 's, or on the right, hence the Γ Γ [ k + 1 ] 's. The only Γ Γ 's that will be updated are the Γ Γ [ k ] 's (requiring only at most O ( M 2 ⋅ ⋅ χ χ 2 ) operations), as