Variational M-Partite Geometric Entanglement Algorithm

TL;DR

The paper introduces VMGE, a variational quantum algorithm to quantify M-partite geometric entanglement across arbitrary partitions of an N-qubit system. It defines GE as $E^{(M)}(|psi>) = 1 - max_{|phi> in S_M} |<phi|psi>|^2$ and realizes it with M disjoint parameterized circuits optimized via a two-stage Sobol-global search and BFGS refinement. Validation on analytically solvable states shows VMGE reproduces exact GE values, and applications to 1D/2D XY and XXZ spin models with unconventional partitions demonstrate detection of quantum phase transitions and scalability to larger clusters. The approach leverages symmetry reductions and is readily integrable with near-term quantum hardware frameworks like Pennylane and Qiskit, with potential synergy with neural-network quantum states. These results position VMGE as a versatile tool for multipartite entanglement quantification with implications for quantum information, condensed matter physics, and quantum field theory.

Abstract

Variational quantum algorithms have emerged as a powerful tool for harnessing the potential of near-term quantum devices to address complex challenges across quantum science and technology. Yet, the robust and scalable quantification of entanglement in many-body quantum systems remains a significant challenge, crucial for both advancing theoretical understanding and enabling practical applications. In this work, we propose a variational quantum algorithm to evaluate the $M$-partite geometric entanglement across arbitrary partitions of an $N$-qubit system into $M$ parties. By constructing tailored variational ansatz circuits for both single- and multi-qubit parties, we optimize the overlap between a target quantum state and an $M$-partite variational separable state. This method provides a flexible and scalable approach for characterizing arbitrary $M$-partite entanglement in complex quantum systems of a given dimension. The accuracy of the proposed method is assessed by reproducing known analytical results. We further demonstrate its capability to evaluate entanglement among $M$ parties for any given conventional or unconventional partitions of one- and two-dimensional spin systems, both near and at a quantum critical point. Our results establish the versatility of the variational approach in capturing different types of entanglement in various quantum systems, surpassing the capabilities of existing methods. Our approach offers a powerful methodology for advancing research in quantum information science, condensed matter physics, and quantum field theory. Additionally, we discuss its advantages, highlighting its adaptability to diverse system architectures in the context of near-term quantum devices.

Figures (6)

• Figure 1: (Color online) Variational M-partite geometric entanglement (VMGE) algorithm. For an $N$-qubit system, the algorithm involves $M$ disjoint variational circuits corresponding to a given partition of the system into $M$ parties. These circuits generate a general $M$-partite separable state, which is then compared with a given $N$-qubit target state $\ket{\psi}$. The geometric entanglement is evaluated through an optimization over the separable state space.
• Figure 2: (Color online) Variational quantum circuits: variational single-qubit ($a$) and multi-qubit ($b$) circuits used in the VMGE algorithm, as illustrated in Fig. \ref{['fig1']}. The algorithm employs universal single-qubit circuits combined with a two-qubit entangling circuits ($c$) with variational entangling power.
• Figure 3: (Color online) Validation of the VMGE algorithm in determining exact GE values. The VMGE algorithm is used to evaluate the bipartite GE in $\ket{\psi(p)}$, the tripartite GE in $\ket{W(p, \phi)}$, $\ket{GW(p, 0)}$, and $\ket{GW(p, \pi)}$, as well as the bipartite GE between subsystems $S_1 = \{\text{qubit 1, qubit 3}\}$ and $S_2 = \{\text{qubit 2, qubit 4}\}$ in the four-qubit state $\ket{BB(p)}$, all shown as functions of $p$. The resulting plots coincide precisely with the exact GE values for these states reported in Ref. Wei2003.
• Figure 4: (Color online) Results of the VMGE algorithm for evaluating 16-partite global GE in many-spin systems. Ground-state global GE for the XY (left column) and XXZ (right column) models in 1D (top row) and 2D (bottom row), obtained using the VMGE algorithm. We consider a 16-spin ring for the 1D system and a $4\times4$ square lattice for the 2D system. The results demonstrate that the VMGE algorithm successfully handles critical regions and accurately captures quantum phase transitions in all cases. We assume $J=1$ for the XY model and $J=0.25$ for XXZ model.
• Figure 5: (Color online) Results of the VMGE algorithm for evaluating GEs associated with unconventional partitions in many-spin systems. Ground-state 2- and 4-partite GEs for the XY (middle row) and XXZ (bottom row) models in 2D $4\times4$ square lattice, obtained using the VMGE algorithm. Each column correspond to GE associated to an unconventional partitions indicated by a colored square lattice (top row). The results demonstrate that the VMGE algorithm successfully handles critical regions and accurately captures quantum phase transitions in all cases. We assume $J=1$ for the XY model and $J=0.25$ for XXZ model.