Quantum Langevin Dynamics for Optimization

TL;DR

This work develops Quantum Langevin Dynamics (QLD) as an open quantum-system approach to continuous optimization, formulating a Lindblad master equation with jump operators that encode damping and thermal effects. It provides rigorous convergence results in convex and quasar-convex settings, derives analytical behavior for quadratic potentials, and demonstrates energy dissipation via spontaneous emission. The authors introduce time-dependent variants to amplify exploration and convergence, showing empirical superiority over both quantum (QHD, QAA) and classical (SGD, NAGD) baselines across diverse nonconvex landscapes, while analyzing query complexity. The study highlights the potential of quantum noise and dissipation to enhance global optimization, outlines practical implementation strategies, and identifies theoretical gaps for future work, including extensions beyond the current approximations. Overall, the paper offers a principled framework and compelling numerical evidence that open quantum dynamics can yield competitive and, in some cases, superior optimization performance compared to classical methods and other quantum approaches.

Abstract

We initiate the study of utilizing Quantum Langevin Dynamics (QLD) to solve optimization problems, particularly those non-convex objective functions that present substantial obstacles for traditional gradient descent algorithms. Specifically, we examine the dynamics of a system coupled with an infinite heat bath. This interaction induces both random quantum noise and a deterministic damping effect to the system, which nudge the system towards a steady state that hovers near the global minimum of objective functions. We theoretically prove the convergence of QLD in convex landscapes, demonstrating that the average energy of the system can approach zero in the low temperature limit with an exponential decay rate correlated with the evolution time. Numerically, we first show the energy dissipation capability of QLD by retracing its origins to spontaneous emission. Furthermore, we conduct detailed discussion of the impact of each parameter. Finally, based on the observations when comparing QLD with classical Fokker-Plank-Smoluchowski equation, we propose a time-dependent QLD by making temperature and $\hbar$ time-dependent parameters, which can be theoretically proven to converge better than the time-independent case and also outperforms a series of state-of-the-art quantum and classical optimization algorithms in many non-convex landscapes.

Figures (28)

• Figure 1: Differences between classical dynamics (left) and quantum dynamics (right). A. Tunneling effect: An illustrative example where classical Gradient Descent algorithms will be trapped because of bad initialization, while quantum algorithms can escape easily. B. Noise: The effect of noise can be described by the system coupling with a heat bath. C. Special thing of quantum noise: The crucial difference between classical noise and quantum noise emerges when interacting with vacuum. Owing to the principles of quantum mechanics, spontaneous emission exists even in a vacuum heat bath, which indicates the fundamental origin of energy dissipation in the quantum Langevin system.
• Figure 2: Landscape of spontaneous emission: asymmetric potential (orange line), the distribution of initial state $\ket{\psi_0}$ (blue area), and the distribution of ground state $\ket{0}$ (green area).
• Figure 3: Spontaneous emission. (a) Transition probability to the ground state, i.e., $|\langle \psi(t)| 0 \rangle |^{2}$. The quadratic case is a benchmark ($V(x) = \frac{1}{2}m\Omega_0^2 x^2, \Omega_0= w$), which exhibits characteristics of spontaneous emission (blue line). When $\Omega_1 = 0.905\Omega_0$, the system maximum probability of transition is larger than $0.9$ (orange line). (b) Transition probability when increasing the number of harmonic oscillators in heat bath. Since the frequencies of oscillators are randomly chosen, the results of this experiment are not unique. We choose the best one to verify our statements.
• Figure 4: The time evolution of probability distributions for quadratic potential at mild temperature. x-coordinate represents the position $x$, and y-coordinate represents the diagonal elements of density matrix. Six different moments $t=0,~0.01,~0.15,~0.3,~0.5,~5$ with notable features are chosen. At the beginning of evolution, we can observe the oscillation of the probability distribution. After $t>0.5$, the probability distribution converges to a stable distribution gradually because of dissipation.
• Figure 5: The time evolution of expectation values for quadratic potential at mild temperature. x-coordinate represents the time $t$, and y-coordinate represents the expectation value of Hamiltonian $\left\langle {H} \right\rangle_t$, potential energy $\left\langle {V} \right\rangle_t$ and kinetic energy $\left\langle {E_k} \right\rangle_t$. The analytical convergence values $\left\langle {V} \right\rangle_f=132.3762,~\left\langle {E_k} \right\rangle_f=132.4368$ are also shown in the figure as convergence lines. We can easily observe the oscillation of the expectation values at the beginning consistent with analytical solutions Eq. (\ref{['eq:x2moment_solution']}) and Eq. (\ref{['eq:p2moment_solution']}), which originates from $\cos(2\Omega t)$ and $\sin(2\Omega t)$ factors in these equations. During the whole process, the oscillation decays gradually, which results from the damping factor $\exp(-2\eta t)$.

Theorems & Definitions (31)

• Definition 1
• Lemma 1: Generalized Ehrenfest theorem
• proof
• Lemma 2: Commutation relations
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1: Convergence of QLD
• proof