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arXiv:1906.10889 · Physical Review A

Dynamics of reverse annealing for the fully connected p -spin model

Yu Yamashiro, Masaki Ohkuwa, Hidetoshi Nishimori, Daniel A. Lidar

Published Nov 18, 2019• 76 citations
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Abstract

Reverse annealing, a variant of quantum annealing, is investigated by probing quantum transitions in a $p$-spin model and comparing it with the predictions offered by equilibrium statistical mechanics. It is shown that reverse annealing can provide an exponential speedup over conventional quantum annealing for this particular problem.

DOI
10.1103/physreva.100.052321

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Reverse annealing, a variant of quantum annealing, is investigated by probing quantum transitions in a ppp-spin model and comparing it with the predictions offered by equilibrium statistical mechanics. It is shown that reverse annealing can provide an exponential speedup over conventional quantum annealing for this particular problem.
Reverse annealing is a relatively new variant of quantum annealing in which one starts from a classical state and increases and then decreases the amplitude of the transverse field, in the hope of finding a better classical state than the initial state for a given optimization problem.We numerically study the unitary quantum dynamics of reverse annealing for the mean-field-type p-spin model and show that the results are consistent with the predictions of equilibrium statistical mechanics.In particular, we corroborate the equilibrium analysis prediction that reverse annealing provides an exponential speedup over conventional quantum annealing in terms of solving the p-spin model.This lends support to the expectation that equilibrium analyses are effective at revealing essential aspects of the dynamics of quantum annealing.We also compare the results of quantum dynamics with the corresponding classical dynamics to reveal their similarities and differences.We distinguish between two reverse annealing protocols we call adiabatic and iterated reverse annealing.We further show that iterated reverse annealing, as has been realized in the D-Wave device, is ineffective in the case of the p-spin model but note that a recently introduced protocol ("h-gain"), which implements adiabatic reverse annealing, may lead to improved performance.
Reverse annealing is a variant of quantum annealing that starts from a given classical configuration of spins (qubits). In contrast to the conventional formulation, where one starts from a uniform superposition of all possible states (classical configurations), quantum fluctuations are first increased and only then decreased. One then reads out the state as a proposed solution to the given combinatorial optimization problem. We formulate a mean-field theory of reverse annealing using the fully connected ferromagnetic ppp-spin model, with and without random longitudinal fields, and analyze it in order to understand how and when reverse annealing is effective at solving this problem. We find that the difficulty arising from the existence of a first-order quantum phase transition, which leads to an exponentially long computation time in conventional quantum annealing, is circumvented in the context of this particular problem by reverse annealing if the proximity of the initial state to the (known) solution exceeds a threshold. Even when a first-order transition is unavoidable, the difficulty is mitigated due to a smaller jump in the order parameter at a first-order transition, which implies a larger rate of quantum tunneling. Reverse annealing has not been studied analytically before, and this study paves the way toward a systematic understanding of this relatively unexplored protocol in a broader context.
In reverse quantum annealing, the initial state is an eigenstate of the final problem Hamiltonian and the transverse field is cycled rather than strictly decreased as in standard (forward) quantum annealing. We present a numerical study of the reverse quantum annealing protocol applied to the ppp-spin model (p=3p=3p=3), including pausing, in an open-system setting accounting for dephasing in the energy eigenbasis, which results in thermal relaxation. We consider both independent and collective dephasing and demonstrate that in both cases the open-system dynamics substantially enhances the performance of reverse annealing. Namely, including dephasing overcomes the failure of purely closed-system reverse annealing to converge to the ground state of the ppp-spin model. We demonstrate that pausing further improves the success probability. The collective dephasing model leads to somewhat better performance than independent dephasing. The protocol we consider corresponds closely to the one implemented in the current generation of commercial quantum annealers, and our results help to explain why recent experiments demonstrated enhanced success probabilities under reverse annealing and pausing.
We compare the performance of quantum annealing (QA, through Schr\"odinger dynamics) and simulated annealing (SA, through a classical master equation) on the ppp-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second order (p=2p=2p=2, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition ($p\ensuremath{\ge}3$, i.e., with multispin Ising interactions), in contrast, the classical annealing dynamics appears to remain stuck in the disordered phase, while we have clear evidence that QA shows a residual energy which decreases towards zero when the total annealing time $\ensuremath{\tau}$ increases, albeit in a rather slow (logarithmic) fashion. This is one of the rare examples where a limited quantum speedup, a speedup by QA over SA, has been shown to exist by direct solutions of the Schr\"odinger and master equations in combination with a nonequilibrium Landau-Zener analysis. We also analyze the imaginary-time QA dynamics of the model, finding a $1/{\ensuremath{\tau}}^{2}$ behavior for all finite values of ppp, as predicted by the adiabatic theorem of quantum mechanics. The Grover-search limit $p(\mathrm{odd})=\ensuremath{\infty}$ is also discussed.
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 ppp-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.
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) ppp-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.