Matrix-product entanglement characterizing the optimality of state-preparation quantum circuits

Abstract

Multipartite entanglement offers a powerful framework for understanding the complex collective phenomena in quantum many-body systems that are often beyond the description of conventional bipartite entanglement measures. Here, we propose a class of multipartite entanglement measures that incorporate the matrix product state (MPS) representation, enabling the characterization of the optimality of quantum circuits for state preparation. These measures are defined as the minimal distances from a target state to the manifolds of MPSs with specified virtual bond dimensions $χ$, and thus are dubbed as $χ$-specified matrix product entanglement ($χ$-MPE). We demonstrate superlinear, linear, and sublinear scaling behaviors of $χ$-MPE with respect to the negative logarithmic fidelity $F$ in state preparation, which correspond to excessive, optimal, and insufficient circuit depth $D$ for preparing $χ$-virtual-dimensional MPSs, respectively. Specifically, a linearly-growing $χ$-MPE with $F$ suggests $\mathcal{H}_χ \simeq \mathcal{H}_{D}$, where $\mathcal{H}_χ$ denotes the manifold of the $χ$-virtual-dimensional MPSs and $\mathcal{H}_{D}$ denotes that of the states accessible by the $D$-layer circuits. We provide an exact proof that $\mathcal{H}_{χ=2} \equiv \mathcal{H}_{D=1}$. Our results establish tensor networks as a powerful and general tool for developing parametrized measures of multipartite entanglement. The matrix product form adopted in $χ$-MPE can be readily extended to other tensor network ansätze, whose scaling behaviors are expected to assess the optimality of quantum circuit in preparing the corresponding tensor network states.

Figures (4)

• Figure 1: (Color online) Illustration of $\chi$-MPE. The circles indicate different manifolds. The dash circles indicate the manifolds formed by the MPSs with specified virtual bond dimension $\chi$. The solid lines indicate the $\chi$-MPE as the minimal distances from the target state $|\Psi\rangle$ to the corresponding manifolds. For $\chi=1$, the MPSs are product states, and the $1$-MPE reduces to GE.
• Figure 2: (Color online) (a) The illustration of the equivalence between the MPSs with $\chi=2$ and the quantum states prepared by single-layer VQCs from the product state $|0\cdots0\rangle$. (b) Our numerical results suggest the MPSs with $\chi=4$ can be efficiently prepared using a VQC with 3 layers.
• Figure 3: (Color online) The $\chi$-MPE $E_{\chi}$ [Eq. (\ref{['eq_TTGE']})] versus the NLF $F$ [Eq. (\ref{['eq_NLF']})] for (a) $D=1$, (b) $D=2$, and (c) $D=3$, with $D$ the number of layers of the VQC. We take the $N$-qubit generalized RPSs as the target states with $N=12$. The insets show the coefficient of determination ($R^2$) that characterizes the linearity of the relation between $E_{\chi}$ and $F$ for different values of $\chi$.
• Figure 4: (Color online) The scaling diagram demonstrating three regions, where the $\chi$-MPE exhibits superlinear, linear, and sublinear behaviors versus the NLF with $D$-layer VQCs, respectively. Note we consider the scaling to be linear with the coefficient of determination $R^2> 0.999$. The black lines are drawn as curves using cubic spline interpolation between integer-valued data points, with the intention solely for an artistic illustration. The red dashed line shows the theoretical upper bounds $\chi=2^{D}$SM.