Algorithms and Sum-of-Squares Certificates for Qudit Hamiltonians Over Maximally Entangles States

• URL: http://arxiv.org/abs/2410.15544v1
• Date: Mon, 21 Oct 2024 00:10:51 GMT
• Title: Algorithms and Sum-of-Squares Certificates for Qudit Hamiltonians Over Maximally Entangles States
• Authors: Zackary Jorquera, Alexandra Kolla, Steven Kordonowy, Juspreet Singh Sandhu, Stuart Wayland,
• Abstract summary: We prove monogamy of entanglement bounds by certifying the ground state energy of the Maximal Entanglement problem. We show that a simple matching-based algorithm outputs a state with energy at least $1/d$ of the ground state energy for general graphs.
• Score: 37.96754147111215
• License:
• Abstract: We introduce the Maximal Entanglement problem, a 2-local qudit Hamiltonian that we view as a quantum generalization of Unique Games and which naturally encodes the frustration present in entanglement over multiple systems. We prove monogamy of entanglement bounds by certifying the ground state energy of the Maximal Entanglement problem in terms of the maximum matching of the underlying interaction graph via low-degree sum-of-squares proofs. Algorithmically, while a random assignment achieves energy of at least $1/d^2$ times the ground state energy, we show that a simple matching-based algorithm outputs a state with energy at least $1/d$ of the ground state energy for general graphs and at least $1/d + \Theta(1/D)$ for graphs with bounded degree, $D$. Moreover, we show that this state has energy at least $1/2$ of the ground state energy on $D$-regular graphs with degree, $D \leq 5$, for any local dimension, $d$.

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