量子热力学与半定优化：玻尔兹曼、费米-狄拉克及玻色-爱因斯坦框架
Semidefinite optimization as many-body thermodynamics: Boltzmann, Fermi-Dirac, and Bose-Einstein frameworks
浏览论文内容
中文总结 AI 辅助
该研究将量子热力学与半定规划（SDP）结合，为量子信息中的三类SDP约束对应不同统计的热力学正则化，提出基于热算子的混合量子-经典算法求解SDP。
中文摘要 AI 辅助
本文论证量子热力学如何为量子信息中出现的一大类半定规划（SDP）提供统一解释。三类SDP变量约束，即迹为1的密度算子、算子有界测量及无界半正定锥，分别对应与玻尔兹曼、费米-狄拉克、玻色-爱因斯坦统计相关的热力学正则化。每种情形下，熵正则化对偶问题无约束且关于化学势向量为凹函数，原问题最优解为匹配统计的热算子；对偶梯度与海森矩阵为热期望值，可实现混合量子-经典算法以求解各类SDP。
英文摘要
We study the semidefinite programs (SDPs) of quantum information whose variables are density operators, measurement operators, or unbounded positive semidefinite operators, together with their thermodynamic regularizations by the von Neumann, Fermi-Dirac, and Bose-Einstein entropies, which have been associated with Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, respectively. We find that this correspondence is exact: the Fermi-Dirac and Bose-Einstein regularized SDPs coincide with the constrained free-energy minimization of an ideal Fermi or Bose gas whose Hamiltonian and non-commuting charges are the second quantizations of the SDP data and whose one-body density matrix is the SDP variable, while the Boltzmann framework corresponds to the single-particle case. We show that the three frameworks can be unified under one framework, in which each entropy-regularized dual is, up to a term linear in the chemical potentials, the grand potential of the corresponding system; the duals are unconstrained and concave in the chemical-potential vector, with gradient and Hessian given by thermal expectation values that hybrid quantum-classical algorithms can estimate. We further identify the Fermi-Dirac and Bose-Einstein relative entropies with the Umegaki relative entropy between quasi-free states, which yields their monotonicity under gauge-covariant Gaussian channels. The result places a wide class of SDPs and their solvers within the thermodynamics of ideal quantum gases and points to extensions to interacting gases.
发表机构
- School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）
- Institute of Natural Sciences, School of Mathematical Sciences, Ministry of Education Key Laboratory in Scientific and Engineering Computing, and Global College, Shanghai Jiao Tong University（上海交通大学数学科学学院、自然研究院、教育部科学与工程计算重点实验室、全球学院）
- School of Computer Science, Cornell University（康奈尔大学计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。
