Adiabatic reverse annealing is robust to low-temperature decoherence

TL;DR

The paper investigates whether adiabatic reverse annealing (ARA) remains advantageous in the presence of low-temperature decoherence, using the solvable $p$-spin model. It develops an open-system analysis via the adiabatic master equation in the weak-coupling limit and a mean-field reduction to a self-consistent magnetization $m(t)$, showing that in the adiabatic limit the state tracks the instantaneous Boltzmann form $\rho(t)\propto e^{-\beta H_S(t)}$ and that success requires a transition-free path in the finite-temperature phase diagram. The authors identify two failure modes for open-system ARA—no transition-free path and end-state paramagnetism—and show that low but nonzero temperature can both preserve existing transition-free paths and, in a narrow range of initial guesses, create new ones, potentially enabling exponential improvements over zero temperature. The findings emphasize the nuanced role of temperature in quantum optimization protocols and suggest directions for extending beyond weak coupling and exploring temperature as a tunable resource. Overall, the work demonstrates that ARA can be robust to low-temperature decoherence under suitable conditions and provides a framework for predicting its performance via finite-temperature phase diagrams.

Abstract

Adiabatic reverse annealing (ARA) is an improvement to conventional quantum annealing (QA) that uses an initial guess at the desired ground state to circumvent problematic phase transitions. Despite encouraging results in the closed-system setting, Ref. [1] has suggested on the basis of numerical simulations that ARA may lose its advantage in the presence of decoherence. Here, we revisit this problem from a more analytical perspective. Using the $p$-spin model as a solvable example, together with the adiabatic master equation to describe the effects of the environment (valid at weak coupling), we show that ARA can in fact succeed in open systems but that the temperature of the environment plays a key role. We first demonstrate that, in the adiabatic limit, the system will follow the instantaneous equilibrium state as long as the protocol does not pass through any (finite-temperature) phase transitions. Given this, there are two distinct mechanisms by which ARA can break down at high temperature: either there are no paths that avoid transitions, or the equilibrium state itself is disordered. When the temperature is sufficiently low that neither of these occur, then ARA succeeds. Remarkably, there are even situations in which the environment benefits ARA: we find parameter values for which no transition-avoiding paths exist at zero temperature but such paths appear at non-zero temperature.

Figures (5)

• Figure 1: Success/failure of open-system ARA in the $p$-spin model ($p = 3$), as a function of the environment temperature $T$ and the fraction $c$ of spins which have the same orientation in the initial guess as in the ground state. Note that there are two failure modes for ARA: there can be no paths from $(s, \lambda) = (0, 0)$ to $(1, 1)$ which do not cross phase transitions ("no paths", shown in blue); or the equilibrium state at the end of the protocol can be in the paramagnetic phase of the model ("paramagnetic", shown in red).
• Figure 2: The magnetization $m(t)$ during the ARA protocol for different runtimes $\tau$ (along the path $s(t) = \lambda(t) = t/\tau$). The black dashed line indicates the corresponding equilibrium value of the magnetization at that point in the protocol, i.e., the average using the state $e^{-\beta H_S(t)}$, with $\beta^{-1} = 1.57$. The initial state has $c = 0.9$.
• Figure 3: The magnetization $m(t)$ during the ARA protocol at two different temperatures $T = \beta^{-1}$. Solid lines show the true dynamical evolution of $m(t)$, while the dashed lines indicate the corresponding equilibrium values at that point in the protocol, i.e., the average using the state $e^{-\beta H_S(t)}$ for the two temperatures in question. Runtime is $\tau = 50/\eta$, the path is $s(t) = \lambda(t) = t/\tau$, and $c = 0.9$. The discontinuity in the red dashed line indicates that the protocol crosses a discontinuous phase transition at that temperature.
• Figure 4: Equilibrium phase boundaries in the $s$-$\lambda$ plane for the ARA Hamiltonian $H_S(s, \lambda)$ (Eq. \ref{['eq:ARA Hamiltonian']}), at various temperatures $T$ (with $c = 0.9$). All lines indicate discontinuous phase transitions at which there is a jump in the magnetization (each boundary is identified by where the magnetization jumps by more than $0.05$ from one value of $s$ to the next).
• Figure 5: Equilibrium phase diagram for the ARA Hamiltonian $H_S(s, \lambda)$ (Eq. \ref{['eq:ARA Hamiltonian']}), with the color indicating the equilibrium value of the magnetization, at three representative temperatures (with $c = 0.9$). At $T = 0.4$ (left), ARA is successful since there are paths from $(0, 0)$ to $(1, 1)$ that do not cross discontinuous phase transitions, as evidenced by the color changing smoothly. At $T = 1.4$ (middle), ARA fails since there are no longer any such paths. At $T = 1.57$ (right), ARA fails since the final state is in the paramagnetic phase (average magnetization of zero).