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

TL;DR

The paper develops and benchmarks a framework for constrained free-energy minimization in quantum thermodynamics with non-commuting charges, using dual chemical-potential optimization and parameterized non-Abelian thermal states. It reframes LMPW classical and HQC algorithms as tools for designing ground and thermal states of controllable Hamiltonians, and extends the formalism to stabilizer thermodynamic systems that map stabilizer codes to thermodynamic observables. The authors demonstrate that stabilizer codes can be leveraged for encoding quantum information by preparing low-temperature thermal states of stabilizer thermodynamic systems, and they introduce warm-start strategies to accelerate encoding. Through extensive numerical experiments on 1D/2D Heisenberg models and several stabilizer codes, they compare classical and HQC variants, show the advantages of second-order information, and discuss practical considerations for thermal-state preparation and scalability to larger systems.

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.

Figures (12)

• Figure 1: Depiction of one- and two-dimensional Heisenberg models. (Top) Qubits of the one-dimensional model are arranged on a line with edges placed between nearest neighbors (black lines), and edges additionally placed between next-to-nearest neighbors (blue curves). (Bottom) Qubits of the two-dimensional model are arranged on a square lattice, with nearest neighbor and next-to-nearest neighbor interactions shown in black and blue lines, respectively.
• Figure 2: The figure depicts the logarithm of the error metric in \ref{['eq:error-metric']} versus the logarithm of the number of iterations, for the task of constrained energy minimization for the two-dimensional, four-qubit quantum Heisenberg model with nearest- and next-to-nearest-neighbor interactions and constraints on the total magnetizations in the $x$, $y$, and $z$ directions set to be 1, 0, and 1, respectively. All of the algorithms converge, but the HQC algorithms, shown as the average over five independent runs with shaded regions denoting one standard deviation, require more iterations to converge due to sampling noise inherent in them.
• Figure 3: The figure depicts the logarithm of the error metric in \ref{['eq:error-metric']} versus the number of iterations, for all of the LMPW algorithms (1st- and 2nd-order, and classical and HQC) for the task of constrained energy minimization for a stabilizer thermodynamic system formed from the perfect five-qubit error-correcting code. The constraints on the logical operators $\overline{X}$, $\overline{Y}$, and $\overline{Z}$ were set to $0.2$, 0, and $0.5$, respectively. All of the algorithms converge, but the HQC algorithms, shown as the average over five independent runs with shaded regions denoting one standard deviation, require more iterations to converge due to sampling noise inherent in them. For this simulation, we did not warm-start the algorithm according to the recipe from Section \ref{['sec:warm-start']}, but we instead started with $\mu_x = 1$, $\mu_y = 1$, and $\mu_z = 1$.
• Figure 4: The figure depicts the logarithm of the error metric in \ref{['eq:error-metric']} versus the number of iterations, for the task of constrained energy minimization for the one-dimensional quantum Heisenberg model with nearest- and next-to-nearest-neighbor interactions and constraints on the total magnetizations in the $x$, $y$, and $z$ directions set to be 1, 0, and 1, respectively. The LMPW classical algorithm converges in all cases (3, 4, 5, and 6 qubits), with larger systems taking more iterations to converge.
• Figure 5: The figure depicts the average of the logarithm of the error metric in \ref{['eq:error-metric']} over five independent runs, plotted against the number of iterations, with shaded regions indicating one standard deviation. The task is constrained energy minimization for the one-dimensional quantum Heisenberg model with nearest- and next-to-nearest-neighbor interactions, and constraints on the total magnetizations in the $x$, $y$, and $z$ directions set to be 1, 0, and 1, respectively. The LMPW HQC algorithm converges in all cases (3, 4, 5, and 6 qubits), with larger systems taking more iterations to converge.

Theorems & Definitions (23)

• Definition 5: Stabilizer thermodynamic system
• Example 6
• Theorem 7
• Theorem 8
• Proposition 9
• proof
• Corollary 10
• proof
• Proposition 11
• proof