Quantum thermodynamics and semi-definite optimization

2025-05-07

AI Summary

This work addresses the minimum-energy problem for quantum thermodynamic systems with noncommuting conserved quantities (e.g., particle number, charge), establishing its fundamental mathematical equivalence to semidefinite programming (SDP). Methodologically, it applies Jaynes’ maximum entropy principle to recast free-energy minimization as a concave optimization over chemical potentials and proposes a unified algorithmic framework based on gradient ascent in chemical potential space—integrating matrix multiplicative weights update (MMWU), stochastic gradient ascent, and quantum-classical hybrid strategies. Key contributions include: (i) the first rigorous physical correspondence between constrained quantum thermodynamic energy minimization and SDP; (ii) a thermodynamic interpretation of convergence and runtime guarantees for MMWU and related algorithms; and (iii) efficient low-temperature approximation of ground-state energy via free energy, significantly enhancing both SDP solution efficiency and physical interpretability.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Security and Privacy: Large-scale security measurements

Abstract

In quantum thermodynamics, a system is described by a Hamiltonian and a list of non-commuting charges representing conserved quantities like particle number or electric charge, and an important goal is to determine the system's minimum energy in the presence of these conserved charges. In optimization theory, a semi-definite program (SDP) involves a linear objective function optimized over the cone of positive semi-definite operators intersected with an affine space. These problems arise from differing motivations in the physics and optimization communities and are phrased using very different terminology, yet they are essentially identical mathematically. By adopting Jaynes' mindset motivated by quantum thermodynamics, we observe that minimizing free energy in the aforementioned thermodynamics problem, instead of energy, leads to an elegant solution in terms of a dual chemical potential maximization problem that is concave in the chemical potential parameters. As such, one can employ standard (stochastic) gradient ascent methods to find the optimal values of these parameters, and these methods are guaranteed to converge quickly. At low temperature, the minimum free energy provides an excellent approximation for the minimum energy. We then show how this Jaynes-inspired gradient-ascent approach can be used in both first- and second-order classical and hybrid quantum-classical algorithms for minimizing energy, and equivalently, how it can be used for solving SDPs, with guarantees on the runtimes of the algorithms. The approach discussed here is well grounded in quantum thermodynamics and, as such, provides physical motivation underpinning why algorithms published fifty years after Jaynes' seminal work, including the matrix multiplicative weights update method, the matrix exponentiated gradient update method, and their quantum algorithmic generalizations, perform well at solving SDPs.

Problem

• Determine minimum energy in quantum systems with conserved charges
• Link quantum thermodynamics and semi-definite optimization mathematically
• Develop efficient algorithms for energy minimization and SDP solving

Innovation

• Minimizing free energy via dual chemical potential maximization
• Employing gradient ascent for optimal parameter convergence
• Applying quantum-classical algorithms for efficient SDP solutions