Quantum thermodynamics and semi-definite optimization

TL;DR

The paper connects quantum thermodynamics with semi-definite optimization by showing that minimum energy problems with conserved non-commuting charges can be recast as SDPs through a low-temperature free-energy framework. It derives a concave dual formulation in terms of chemical potentials, enabling efficient gradient-based optimization and a natural path to hybrid quantum–classical implementations via parameterized thermal states. The authors provide first- and second-order optimization algorithms, establish convergence and polynomial sample complexity, and show how general SDPs can be reduced to energy minimization problems with scalable performance. The work offers a physical motivation for diffusion-like updates such as the matrix exponentiated gradient and matrix multiplicative weights, and it outlines concrete future directions toward quantum advantage, broader SDP classes, and more efficient subroutines for low-temperature thermalization.

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.

Figures (3)

• Figure 1: Depiction of the main idea behind solving an energy minimization problem in quantum thermodynamics, specified by a Hamiltonian $H$ and a tuple $(Q_{1}, \ldots, Q_{c})$ of conserved non-commuting charges with respective expected values $(q_{1}, \ldots, q_{c})$. The goal is to determine the minimum energy $E(\mathcal{Q},q)$ of the system. Inspired by the fact that physical systems operate at a strictly positive temperature $T>0$, we instead minimize the free energy $F_{T}(\mathcal{Q},q)$ of the system at a low temperature $T$. By employing Lagrangian duality and quantum relative entropy, we can rewrite the free energy minimization problem as the dual problem of chemical potential maximization. This latter problem is concave in the chemical potential vector $\mu$ and thus can be solved quickly by gradient ascent. The approach leads to classical and hybrid quantum--classical algorithms for energy minimization. See Section \ref{['sec:energy-min-q-thermo']} for details.
• Figure 2: Depiction of the main idea behind solving a general semi-definite optimization problem (abbreviated SDP), specified by the tuple of $d\times d$ Hermitian matrices $(C,A_{1}, \ldots, A_{c})$. The reduction from Brandao2019 allows for reducing a general SDP to an energy minimization problem, where $R \geq0$ is a guess on the trace of an optimal solution to the original SDP. The resulting algorithm is sample-efficient as long as $R$ does not grow too quickly with $d$. See Section \ref{['sec:SDP-to-energy-min']} for details.
• Figure 3: Quantum circuit that realizes an unbiased estimate of $-\frac{1}{2} \left\langle \left\{ \Phi_{\mu}(Q_{i}),Q_{j}\right\} \right\rangle _{\rho_{T}(\mu)}$. For each run of the circuit, the time $t$ is sampled independently at random from the probability density $p(t)$ in \ref{['eq:high-peak-tent-density-def']}, while $\sigma_{\vec{\ell}}$ and $\sigma_{\vec{k}}$ are sampled with probabilities $a_{i,\vec{\ell}}/\left \Vert a_i \right \Vert_1$ and $a_{j,\vec{k}}/\left \Vert a_j \right \Vert_1$, respectively. For details of the algorithm, see Appendix \ref{['app:Hessian_est']}.

Theorems & Definitions (13)

• Lemma 1: Brandao2019
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10: SGA Convergence