Quantum Annealing in SK Model Employing Suzuki-Kubo-deGennes Quantum Ising Mean Field Dynamics

TL;DR

This work tackles estimating the Sherrington-Kirkpatrick model's ground-state energy by evolving continuous local magnetizations under Suzuki-Kubo-deGennes mean-field dynamics augmented with a de Gennes–Brout–Muller–Thomas reaction field. It systematically analyzes classical, quantum, and mixed annealing schemes, demonstrating an $O(N^3)$ computational cost and a universal $E^0$ scaling as $N^{-2/3}$ that yields near-ground-state energies around $-0.7623$ to $-0.7629$, close to the best known $E^0 \approx -0.763166726\ldots$. Across all schemes, the transverse field does not provide a measurable advantage within this mean-field framework, likely because the continuous magnetization variables already enable efficient exploration of the energy landscape. The results also report the absence of an Almeida-Thouless line under these dynamics, and the code is made publicly available for reproducibility. Overall, the paper presents a fast, scalable approach to approximate ground states of SK-like systems without demonstrating quantum supremacy in this setting.

Abstract

We study a quantum annealing approach for estimating the ground state energy of the Sherrington-Kirpatrick mean field spin glass model using the Suzuki-Kubo-deGennes dynamics applied for individual local magnetization components. The solutions of the coupled differential equations, in discretized state, give a fast annealing algorithm (cost $N^3$) in estimating the ground state of the model: Classical ($E^0= -0.7629 \pm 0.0002$), Quantum ($E^0=-0.7623 \pm 0.0001$) and Mixed ($E^0=-0.7626 \pm 0.0001$), all of which are to be compared with the best known estimate $E^0= -0.763166726 \dots$ . We infer that the continuous nature of the magnetization variable used in the dynamics here is the reason for reaching close to the ground state quickly and also the reason for not observing the de-Almeida-Thouless line in this approach.

Figures (7)

• Figure 1: The variations of $\Delta(\tau)=E_N^0-E^0$ are shown with annealing time ($\tau$) for quantum SK model (at $T=0$). Different system sizes are indicated. The time taken for the reaching the saturation in the energy difference scales linearly with the system size, as shown in the inset. The same order of annealing time is obtained for saturation in $\Delta$ for fixed $N$ in the classical DasPRE2025 and mixed annealing cases.
• Figure 2: The phase boundary for the SK model (cf. yamamoto) is shown that separates the $q\ne0$ (spin glass ordered) and $q=0$ (para phase) for $N=500$.
• Figure 3: Classical annealing with time variation of $T$ given by Eq. (9a) (with $\Gamma=0$, $h=0$): The lowest energy values for given system size are plotted against $N^{-2/3}$ which shows a scaling $E_{0}(N)\sim N^{-2/3}$. The ground state energy per spin ($N \rightarrow {\infty}$) is $E^0=-0.763166 \dots$. From the least-square fitting we get a ground state energy which is $E^0=-0.7629\pm 0.0002$ (considering the exponent to be 2/3). The inset shows the variation of the fluctuations $\sigma_N$ of $E^0_N$ with system size ($\sigma_N\sim N^{-3/4}$) and $\sigma_N\sim N^{-5/6}$).
• Figure 4: Quantum annealing with time variation of transverse field $\Gamma$ following Eq. (9b) ($T=0$, $h=0$): The lowest energy values for given system size are plotted against $N^{-2/3}$ which shows a scaling $E_{0}(N)\sim N^{-2/3}$ ($N$ in the range 25 to 10000). The ground state energy per spin ($N \rightarrow {\infty}$) is $E^0=-0.763166 \dots$. From the least-square fitting we get a ground state energy which is $E^0=-0.7623\pm 0.0001$ (considering the exponent to be 2/3). The inset shows the variations of the fluctuations $\sigma_N$ of $E^0_N$ with system size ($\sigma_N\sim N^{-3/4}$ and $\sigma_N\sim N^{-5/6}$).
• Figure 5: Mixed annealing (following Eqs. (9a,9b), $h=0$) starting from the critical phase boundary ($T_c=0.94$,$\Gamma_c=0.5$): The lowest energy values for given system size are plotted against $N^{-2/3}$ which shows a scaling $E_{0}(N)\sim N^{-2/3}$ ($N$ in the range 25 to 10000). The ground state energy per spin ($N \rightarrow {\infty}$) is $E^0=-0.763166 \dots$. From the least-square fitting we get a ground state energy per spin which is $E^0=-0.7626\pm 0.0001$ (considering the exponent to be 2/3). The inset shows the variations of the fluctuations $\sigma_N$ of $E^0_N$ with system size ($\sigma_N\sim N^{-3/4}$ and $\sigma_N\sim N^{-5/6}$).