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arXiv:2205.12936 · Physical Review Applied

Advantage of Pausing: Parameter Setting for Quantum Annealers

Zoe Gonzalez Izquierdo, Shon R. Grabbe, Husni Idris, Zhihui Wang, Jeffrey Marshall, Eleanor Gilbert Rieffel

Published Nov 17, 2022• 10 citations
arXivHTMLPDF

Abstract

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 $|{J}_{F}|$, 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 $|{J}_{F}|$ 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 $|{J}_{F}|$ and embedding size. Based on these results we are able to present qualitative guidelines for parameter setting in quantum annealers.

DOI
10.1103/physrevapplied.18.054056

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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.
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 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.
Recently, many researchers are investigating quantum annealing as a solver for real-world combinatorial optimization problems. However, due to the format of problems that quantum annealing solves and the structure of the physical annealer, these problems often require additional setup prior to solving. We study how these setup steps affect performance and provide insight into the interplay among them using the job-shop scheduling problem for our evaluation. We show that the empirical probability of success is highly sensitive to problem setup and that excess variables and large embeddings reduce performance. We then show that certain problem instances are unable to be solved without the use of additional post-processing methods. Finally, we investigate the effect of pausing during the anneal. Our results show that pausing within a certain time window can improve the probability of success, which is consistent with other work. However, we also show that the performance improvement due to pausing can be masked depending on properties of the embedding, and thus, special care must be taken for embedded problems.
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.
Recent research has shown that alternative annealing schedules provide the means for improving performance in modern quantum annealing devices. One such type of schedule is forward annealing with a pause, in which there is a period of time when system evolution is paused. While the results from using this type of schedule have been promising, effectively using a pause is not a trivial task. One challenge associated with introducing a pause into the schedule is determining the point in the anneal at which the pause will start. Additionally, tuning the schedule in real-time requires a significant amount of time. A second challenge is that while a pause may increase the number of correct solutions returned from the annealer, the time-to-solution, a standard metric for measuring performance in quantum annealing, will not necessarily be improved. We propose a method for constructing annealing schedules containing a pause that avoids the costly process of determining the optimal pause location in an online manner. We also evaluate our method on the subset sum problem, a problem of practical significance, and show that our method is able to achieve a 70% reduction in time-to-solution from a standard schedule containing no pause.
Quantum annealing is a promising heuristic for combinatorial optimization, but on current hardware its performance degrades for larger and more complex problems due to noise and small energy gaps. Reverse annealing has been proposed as a refinement strategy, yet it remains unclear when it provides systematic advantages over standard forward annealing or simply increasing annealing time. We find that combining forward and reverse annealing consistently improves solution quality and efficiency across multiple problem classes. The benefits of reverse annealing increase with problem complexity and are strongest in regimes where forward annealing is increasingly limited. Moreover, reverse annealing yields larger efficiency gains than simply extending forward annealing times. We establish these results through a systematic experimental study on a D-Wave Advantage system, benchmarking reverse annealing across Max-Cut, Number Partitioning, and sparse clustering problems while varying reverse distance, pause duration, and annealing time. We identify a narrow optimal regime for reverse annealing parameters linked to the location of freeze-out points and energy-level crossings in the annealing schedule. These findings demonstrate that reverse annealing is most valuable for large, high-complexity optimization problems and is likely to gain importance as quantum annealing hardware scales toward more realistic applications.