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

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.

Quantum thermodynamics and semi-definite optimization

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.