# Quantum thermodynamics and semi-definite optimization

arXiv: https://arxiv.org/abs/2505.04514 · HTML: https://mlsanity.com/paper/2505.04514/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 91 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 22 of 22)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [A Faster Interior Point Method for Semidefinite Programming](https://mlsanity.com/paper/2009.10217/) (2020)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [Quantum SDP-Solvers: Better upper and lower bounds](https://mlsanity.com/paper/1705.01843/) (2017)
- [Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models](https://mlsanity.com/paper/1609.02542/) (2016)
- [On the Sample Complexity of Quantum Boltzmann Machine Learning](https://mlsanity.com/paper/2306.14969/) (2023)
- [A Quantum Interior Point Method for LPs and SDPs](https://mlsanity.com/paper/1808.09266/) (2018)
- [A Survey of Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics](https://mlsanity.com/paper/1908.05209/) (2019)
- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Variational Quantum Algorithms for Semidefinite Programming](https://mlsanity.com/paper/2112.08859/) (2021)
- [Quantum Interior Point Methods for Semidefinite Optimization](https://mlsanity.com/paper/2112.06025/) (2021)
- [Matrix Multiplicative Weights Updates in Quantum Zero-Sum Games: Conservation Laws & Recurrence](https://mlsanity.com/paper/2211.01681/) (2022)
- [High-Temperature Gibbs States are Unentangled and Efficiently Preparable](https://mlsanity.com/paper/2403.16850/) (2024)
- [Noisy intermediate-scale quantum algorithm for semidefinite programming](https://mlsanity.com/paper/2106.03891/) (2021)
- [Optimization over Nonnegative and Convex Polynomials With and Without Semidefinite Programming](https://mlsanity.com/paper/1806.06996/) (2018)
- [Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry](https://mlsanity.com/paper/2206.04663/) (2022)
- [Learning in Quantum Common-Interest Games and the Separability Problem](https://mlsanity.com/paper/2302.04789/) (2023)

## Built on by (most cited first, 5 of 5)

- [Fermi-Dirac thermal measurements: A framework for quantum hypothesis testing and semidefinite optimization](https://mlsanity.com/paper/2603.04061/) (2026)
- [Fermi-Dirac machines as quantizations of neurons](https://mlsanity.com/paper/2605.24386/) (2026)
- [Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems](https://mlsanity.com/paper/2508.09103/) (2025)
- [Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms](https://mlsanity.com/paper/2607.04743/) (2026)
- [Unifying quantum measurement constructions via a relative-entropy minimum change principle](https://mlsanity.com/paper/2608.04055/) (2026)

## Abstract (arXiv)

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.