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arXiv:2005.01888 · Physical Review Applied

Why and When Pausing is Beneficial in Quantum Annealing

Huo Chen, Daniel A. Lidar

Published Jul 31, 2020• 43 citations
arXivHTMLPDF

Abstract

Recent empirical results using quantum-annealing hardware have shown that midanneal pausing has a surprisingly beneficial impact on the probability of finding the ground state for a variety of problems. A theoretical explanation of this phenomenon has thus far been lacking. Here we provide an analysis of pausing using a master-equation framework, and derive conditions for the strategy to result in a success probability enhancement. The conditions, which we identify through numerical simulations and then prove to be sufficient, require that relative to the pause duration the relaxation rate is large and decreasing right after crossing the minimum gap, small and decreasing at the end of the anneal, and is also cumulatively small over this interval, in the sense that the system does not thermally equilibrate. This establishes that the observed success probability enhancement can be attributed to incomplete quantum relaxation, i.e., is a form of beneficial nonequilibrium coupling to the environment.

DOI
10.1103/physrevapplied.14.014100

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Recent empirical results using quantum-annealing hardware have shown that midanneal pausing has a surprisingly beneficial impact on the probability of finding the ground state for a variety of problems. A theoretical explanation of this phenomenon has thus far been lacking. Here we provide an analysis of pausing using a master-equation framework, and derive conditions for the strategy to result in a success probability enhancement. The conditions, which we identify through numerical simulations and then prove to be sufficient, require that relative to the pause duration the relaxation rate is large and decreasing right after crossing the minimum gap, small and decreasing at the end of the anneal, and is also cumulatively small over this interval, in the sense that the system does not thermally equilibrate. This establishes that the observed success probability enhancement can be attributed to incomplete quantum relaxation, i.e., is a form of beneficial nonequilibrium coupling to the environment.
Prior work showed the efficacy of pausing midanneal: such a pause improved the probability of success by orders of magnitude in a class of native problem instances and improved the time to solution in a class of embedded problem instances. A physics-based picture provides qualitative suggestions for where pausing midanneal is effective, for the interplay between annealing schedule parameters and other annealing properties and parameters such as embedding size and strength of the ferromagnetic coupling ∣JF∣|{J}_{F}|∣JF​∣, and for the conditions under which pausing can improve the time to solution. Here, through demonstrations on an updated annealing architecture that has higher connectivity than previous annealers, and on multiple embedded problem classes, we are able to confirm various aspects of this picture. We demonstrate the robustness of the optimal pause parameters across platforms and problem classes, explore how to set ∣JF∣|{J}_{F}|∣JF​∣ to optimize performance in different scenarios, and provide empirical evidence that short pauses trump longer overall annealing times in time to solution. We also identify the number of different coefficients in a problem as a predictor of problem hardness, and explore its interplay with the optimal ∣JF∣|{J}_{F}|∣JF​∣ and embedding size. Based on these results we are able to present qualitative guidelines for parameter setting in quantum annealers.
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.
The probability of success of quantum annealing can be improved significantly by pausing the annealer during its dynamics, exploiting thermal relaxation in a controlled fashion. In this paper, we investigate the effect of pausing the quantum annealing of the fully connected ferromagnetic ppp-spin model. This analytically solvable model has a searchlike behavior and is often used as a benchmark for the performances of quantum annealing. We numerically show that: (i) the optimal pausing point is 60% longer than the avoided crossing time for the analyzed instance, and (ii) at the optimal pausing point, we register a 45% improvement in the probability of success with respect to a quantum annealing with no pauses of the same duration. These results are in line with those observed experimentally for less connected models with the available quantum annealers. The observed improvement for the ppp-spin model can be up to two orders of magnitude with respect to an isolated quantum dynamics of the same duration.
Annealing schedule control provides opportunities to better understand the manner and mechanisms by which putative quantum annealers operate. By appropriately modifying the annealing schedule to include a pause (keeping the Hamiltonian fixed) for a period of time, we show that it is possible to more directly probe the dissipative dynamics of the system at intermediate points along the anneal and examine thermal relaxation rates, for example, by observing the repopulation of the ground state after the minimum spectral gap. We provide a detailed comparison of experiments from a D-Wave device, simulations of the quantum adiabatic master equation, and a classical analogue of quantum annealing, spin-vector Monte Carlo, and we observe qualitative agreement, showing that the characteristic increase in success probability when pausing is not a uniquely quantum phenomena. We find that the relaxation in our system is dominated by a single timescale, which allows us to give a simple condition for when we can expect pausing to improve the time to solution, the relevant metric for classical optimization. Finally, we also explore in simulation the role of temperature whilst pausing as a means to better distinguish quantum and classical models of quantum annealers.
We study the interplay between quantum annealing parameters in embedded problems, providing both deeper insights into the physics of these devices and pragmatic recommendations to improve performance on optimization problems. We choose as our test case the class of bounded-degree minimum-spanning-tree problems. Through runs on a D-Wave quantum annealer, we demonstrate that pausing in a specific time window in the anneal provides improvement in the probability of success and in the time to solution for these problems. The time window is consistent across problem instances and its location is within the region suggested by prior theory and seen in previous results on native problems. An approach to enable gauge transformations for problems with the qubit coupling strength JJJ in an asymmetric range is presented and shown to significantly improve performance. We also confirm that the optimal pause location exhibits a shift with the magnitude of the ferromagnetic coupling, ∣JF∣|{J}_{F}|∣JF​∣, between physical qubits representing the same logical one. We extend the theoretical picture for pausing and thermalization in quantum annealing to the embedded case. This picture, along with perturbation-theory analysis and exact numerical results on small problems, confirms that the effective pause region moves earlier in the anneal as ∣JF∣|{J}_{F}|∣JF​∣ increases. It also suggests why pausing, while still providing significant benefit, has a less pronounced effect on embedded problems.
The current generation of experimental quantum annealers can shed light on the effects of noise and thermalization present in superconducting quantum circuits. Recent advances in this technology allow for direct probing of intermediate dynamics during the annealing process. By pausing the annealing over spans ranging from micro- to milliseconds and allowing the system to equilibrate, the authors identify several regimes of interest, characterized by the time scales of the dominant processes. It is shown that alternative annealing schedules inspired by physics may prove a fruitful avenue for designing more efficient solvers for optimization problems in quantum computing.