Abstract
There are well-developed theoretical tools to analyze how quantum dynamics can solve computational problems by varying Hamiltonian parameters slowly, near the adiabatic limit. On the other hand, there are relatively few tools to understand the opposite limit of rapid quenches, as used in quantum annealing and (in the limit of infinitely rapid quenches) in quantum walks. In this paper, we develop several tools that are applicable in the rapid-quench regime. Firstly, we analyze the energy expectation value of different elements of the Hamiltonian. From this, we show that monotonic quenches, where the strength of the problem Hamiltonian is consistently increased relative to fluctuation (driver) terms, will yield a better result on average than random guessing. Secondly, we develop methods to determine whether dynamics will occur locally under rapid-quench Hamiltonians and identify cases where a rapid quench will lead to a substantially improved solution. In particular, we find that a technique we refer to as “preannealing” can significantly improve the performance of quantum walks. We also show how these tools can provide efficient heuristic estimates for Hamiltonian parameters, a key requirement for practical application of quantum annealing.
5 More- Received 9 September 2020
- Accepted 21 January 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.010338
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
One important practical area where quantum computing is poised to provide faster and better solutions is combinatorial optimization, solving problems such as scheduling deliveries efficiently. Quantum hardware designed for optimization uses a continuous-time process that seeks out lower-energy states that encode good solutions. Most of the theoretical results in this area focus on slow evolutions that do not work well on current imperfect quantum hardware. In this work, we provide tools for rapid quenches, fast evolutions that are suitable for current hardware. Each run is less likely to find a good solution, but it is cost-effective to do many repeats and take the best solution overall.
Several types of fast evolutions are known, including quantum annealing, where the cold environment of the quantum hardware helps to find low-energy states, and quantum walks, in which the quantum hardware is run for a short time without variation of the controls from their initial settings. We show that if the controls are varied monotonically, on average these fast evolutions will find better solutions. We then show that local information about the evolution is enough to predict when the solutions will be substantially better than guessing. Combining both results provides a way to efficiently specify the control settings to achieve these gains.
The quantum computers with most qubits currently (around 2000) are built specifically for solving optimization problems. Our tools enable this hardware to be exploited more effectively and facilitate the design of better optimization algorithms for solving practical problems.