Constrained optimization of sequentially generated entangled multiqubit states

We demonstrate how the matrix-product state formalism provides a flexible structure to solve the constrained optimization problem associated with the sequential generation of entangled multiqubit states under experimental restrictions. We consider a realistic scenario in which an ancillary system with a limited number of levels performs restricted sequential interactions with qubits in a row. The proposed method relies on a suitable local optimization procedure, yielding an efficient recipe for the realistic and approximate sequential generation of any entangled multiqubit state. We give paradigmatic examples that may be of interest for theoretical and experimental developments.

Figure 1

(Color online) Comparison of the variational optimization approach (solid lines) with the MPS compression technique (dotted lines). We consider the ground state of the $XXZ$ Heisenberg Hamiltonian (circles) and a randomly initialized MPS (triangles), indicating how well these MPS with bond dimension $D$ can be approximated with those of dimension $D^{\prime }<D$.

Figure 2

The contraction pattern used to calculate the cost function in Eq. (7) including the local ancilla operations ${\mathcal{U}}^{\mathcal{A}}$ and local qubit operations ${\mathcal{U}}^{\mathcal{B}}$. The initial states of the qubits are denoted by $|{\psi }_I^{\left[k\right]}⟩$.

Figure 3

(Color online) The deviation of the fidelity $1-\mathcal{F}=1-\Vert ⟨\tilde{\Psi }|\psi ⟩\Vert$ as a function of the number $n$ of qubits for the $W$ state with $D=2$ when optimizing the couplings $h_{j_{\mathcal{A}}{j}_{\mathcal{B}}}$ and the local ancilla unitaries ${\mathcal{U}}_A$, with initial qubit states all equal, $|{\psi }_I^{\left[k\right]}⟩=|0⟩$. The inset shows the case where only the couplings $h_{j_{\mathcal{A}}{j}_{\mathcal{B}}}$ are being optimized.