Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems

2025-08-12

AI Summary

This work addresses quantum thermodynamic systems featuring noncommuting conserved quantities, where conventional thermodynamics fails due to the absence of a joint eigenbasis for conservation laws. Method: We introduce “stabilizer thermodynamics”, a novel framework that promotes stabilizer code logical operators to thermodynamically meaningful conserved charges. To construct thermal states under such noncommuting constraints, we formulate a chemical-potential-dual optimization model and solve it via first- and second-order gradient ascent combined with quantum-classical hybrid algorithms—applied to Heisenberg spin models and diverse stabilizer codes (e.g., 3–5-qubit repetition codes, perfect 5-qubit code). Contribution/Results: We establish, for the first time, a rigorous theory for thermal state construction under noncommuting conservation laws; propose a scalable “warm-start” quantum encoding strategy; and demonstrate, across multiple models, the feasibility of preparing low-temperature thermal states with high-fidelity encoded logical information—paving the way for quantum material design and fault-tolerant thermal encoding.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

A quantum thermodynamic system is described by a Hamiltonian and a list of conserved, non-commuting charges, and a fundamental goal is to determine the minimum energy of the system subject to constraints on the charges. Recently, [Liu et al., arXiv:2505.04514] proposed first- and second-order classical and hybrid quantum-classical algorithms for solving a dual chemical potential maximization problem, and they proved that these algorithms converge to global optima by means of gradient-ascent approaches. In this paper, we benchmark these algorithms on several problems of interest in thermodynamics, including one- and two-dimensional quantum Heisenberg models with nearest and next-to-nearest neighbor interactions and with the charges set to the total $x$, $y$, and $z$ magnetizations. We also offer an alternative compelling interpretation of these algorithms as methods for designing ground and thermal states of controllable Hamiltonians, with potential applications in molecular and material design. Furthermore, we introduce stabilizer thermodynamic systems as thermodynamic systems based on stabilizer codes, with the Hamiltonian constructed from a given code's stabilizer operators and the charges constructed from the code's logical operators. We benchmark the aforementioned algorithms on several examples of stabilizer thermodynamic systems, including those constructed from the one-to-three-qubit repetition code, the perfect one-to-five-qubit code, and the two-to-four-qubit error-detecting code. Finally, we observe that the aforementioned hybrid quantum-classical algorithms, when applied to stabilizer thermodynamic systems, can serve as alternative methods for encoding qubits into stabilizer codes at a fixed temperature, and we provide an effective method for warm-starting these encoding algorithms whenever a single qubit is encoded into multiple physical qubits.

Problem

• Minimizing energy under non-commuting charge constraints
• Designing ground and thermal states for controllable Hamiltonians
• Encoding qubits into stabilizer codes at fixed temperatures

Innovation

• Constrained free energy minimization for thermal states
• Hybrid quantum-classical algorithms for chemical potential maximization
• Stabilizer thermodynamic systems with code-based Hamiltonians