【日時/Date】
2024年4月10日（水）10時30分～12時 /Apr. 10, 10:30 - 12:00 (JST)

【発表者/Speaker】
高橋 惇 氏（ニューメキシコ大学）

【タイトル/Title】
“A Semidefinite Programming approach for Quantum Heisenberg Ground States"

【概要/Abstract】
The Heisenberg model plays a central role in understanding quantum materials, realizing numerous quantum phases as ground states. However, the task of obtaining the ground state of a given Heisenberg-type Hamiltonian is in general very hard. While quantum Monte Carlo methods and tensor network type methods can be extremely powerful in many cases, they are not a single panacea for the problem; as a matter of fact, obtaining the ground state of the Heisenberg model in general is known to be QMA-complete (quantum analogue of NP-complete). 
Here, we focus on the fact that for the (antiferromagnetic) Ising model, computational complexity theory provides strong evidence for the Goemans-Williamson algorithm, a semidefinite programming (SDP) based approach, to be *optimal* in the approximation ratio sense. As the Heisenberg model could be regarded as a natural quantum extension of the Ising model, we can ask the question "what would be the similarly optimal algorithm for the Heisenberg model?". This problem has triggered a number of recent research in the quantum computation community known as the "quantum Max Cut problem" [1]. 
Focusing on the SU(2) symmetry and its algebraic structure, we construct an SDP algorithm that is both theoretically most natural and practically implementable for the first time [2]. We prove that it could be regarded as a first-order of a systematically improving sequence of approximations (properly converging Lasserre/NPA-hierarchy in mathematics terminology), as well as exactness and inexactness for some natural families of Heisenberg models.
In the talk I will start from the classical Goemans-Williamson result and motivate our approach before introducing the algorithm. I will also show its potential use for understanding ground states of actual condensed matter stems as well as how it connects to "frustration-free" models known in the context of exact solvability, as well as sum-of-squares proof systems, which is the dual to our SDP. If we have time, I would also like to discuss its connection to other approaches for solving the Heisenberg ground state problem and its computational complexity, when restricted to the bipartite case [3]. 
[1] Sevag Gharibian, ACM SIGACT News, 54 (4) 54-91 (2024). 
[2] Jun Takahashi, Chaithanya Rayudu, Cunlu Zhou, Robbie King, Kevin Thompson, and Ojas Parekh, arXiv:2307.15688
[3] Jun Takahashi, Sam Slezak, and Elizbeth Crosson, in preparation; 日本物理学会2023春季

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