Simulated outperforms quantum reverse annealing in mean-field models

TL;DR

The paper interrogates whether Adiabatic Reverse Annealing (ARA) offers a quantum advantage by introducing a classical analogue, Simulated Reverse Annealing (SRA), and comparing their performance in a solvable mean-field $p$-spin model with a two-pattern Hopfield extension. Using a two-order-parameter mean-field analysis, the authors map the free-energy landscape and identify how fluctuations can merge local minima to create barrier-free paths from a marked state to the true ground state. Dynamics, via path-integral formalisms for both ARA and SRA, show that when a transition-free path exists, both protocols reach the ground state in $O(1)$ time, with SRA typically achieving it faster; crucially, there are parameter regimes where SRA succeeds while ARA fails. Overall, the study argues that SRA unambiguously outperforms ARA in the mean-field models considered, implying that quantum fluctuations are not essential to the mechanism and that classical fluctuation-driven approaches can mimic or exceed ARA's performance in these contexts, while outlining future directions to test these ideas on harder problems and in finite-temperature settings.

Abstract

Adiabatic reverse annealing (ARA) has been proposed as an improvement to conventional quantum annealing for solving optimization problems, in which one takes advantage of an initial guess at the solution to suppress problematic phase transitions. Here we interpret the performance of ARA through its effects on the free energy landscape, and use the intuition gained to introduce a classical analogue to ARA termed ``simulated reverse annealing'' (SRA). This makes it more difficult to claim that ARA provides a quantum advantage in solving a given problem, as not only must ARA succeed but the corresponding SRA must fail. As a solvable example, we analyze how both protocols behave in the infinite-range (non-disordered) $p$-spin model. Through both the thermodynamic phase diagrams and explicit dynamical behavior, we establish that the quantum algorithm has no advantage over its classical counterpart: SRA succeeds not only in every case where ARA does but even in a narrow range of parameters where ARA fails.

Figures (9)

• Figure 1: Comparison of the equilibrium phase diagram for ARA (left) and our classical analogue SRA (right), specifically for the target Hamiltonian $H_0$ given by Eq. \ref{['eq:Hopfield_model']}, using parameter values $p = 3$, $\alpha = 0.5$, $x = 0.2$. The color indicates the equilibrium value of the magnetization $m$, with the ground state of $H_0$ corresponding to $m = 1$. Black lines indicate discontinuous phase transitions (estimated somewhat crudely as where $m$ changes by more than $0.05$ from one pixel to the next). Note that both ARA and SRA succeed for these parameter values --- there are paths from $s = 0$ to $s = 1$ that avoid phase transitions.
• Figure 2: Comparison of the phase diagram for ARA (left) and SRA (right), using parameter values $p = 5$, $\alpha = 0.9$, $x = 0.2$ (for $H_0$ given by Eq. \ref{['eq:Hopfield_model']}). The color indicates the equilibrium value of the magnetization $m$. Black lines indicate discontinuous phase transitions (again identified as where $m$ changes by more than $0.05$ from one pixel to the next). Here, in contrast to Fig. \ref{['fig:phase_diagrams_example_success']}, both ARA and SRA fail --- there are no paths from $s = 0$ to $s = 1$ that do not cross a phase transition.
• Figure 3: Contour plot of the free energy landscape $\Phi(m_u, m_d)$ for $p = 5$, $\alpha = 0.9$, $x = 0.2$, at $s = 0.4$ and $\lambda = 0.88$. Only contours for $\Phi \leq -0.48$ are shown --- the white regions correspond to $\Phi(m_u, m_d) > -0.48$.
• Figure 4: Evolution of the free energy landscape $\Phi'(m_d)$ along the straight line from $(s, \lambda) = (0, 0)$ to $(1, 0)$ (same parameters as in Fig. \ref{['fig:phase_diagrams_example_success']}) --- blue to red corresponds to increasing $s$ from 0 to 1. The $y$-axis of each curve is shifted so that the minimum always has a value of zero.
• Figure 5: Evolution of the free energy landscape $\Phi'(m_d)$ along a three-stage path that avoids phase transitions (same parameters as in Fig. \ref{['fig:phase_diagrams_example_success']}). Top panel shows the first stage from $(s, \lambda) = (0, 0)$ to $(0.2, 0.7)$. Middle panel shows the second stage from $(0.2, 0.7)$ to $(0.6, 0.7)$. Bottom panel shows the third stage from $(0.6, 0.7)$ to $(1, 0)$. In each, blue to red corresponds to moving forward along the path (increasing $s$), and the $y$-axis of each curve is shifted so that the minimum always has a value of zero.