Abstract
One of the major application areas of interest for both near-term and fault-tolerant quantum computers is the optimization of classical objective functions. In this work, we develop intuitive constructions for a large class of these algorithms based on connections to simple dynamics of quantum systems, quantum walks, and classical continuous relaxations. We focus on developing a language and tools connected with kinetic energy on a graph for understanding the physical mechanisms of success and failure to guide algorithmic improvement. This physical language, in combination with uniqueness results related to unitarity, allow us to identify some potential pitfalls from kinetic energy fundamentally opposing the goal of optimization. This is connected to effects from wavefunction confinement, phase randomization, and shadow defects lurking in the objective far away from the ideal solution. As an example, we explore the surprising deficiency of many quantum methods in solving uncoupled spin problems and how this is both predictive of performance on some more complex systems while immediately suggesting simple resolutions. Further examination of canonical problems like the Hamming ramp or bush of implications show that entanglement can be strictly detrimental to performance results from the underlying mechanism of solution in approaches like QAOA. Kinetic energy and graph Laplacian perspectives provide new insights to common initialization and optimal solutions in QAOA as well as new methods for more effective layerwise training. Connections to classical methods of continuous extensions, homotopy methods, and iterated rounding suggest new directions for research in quantum optimization. Throughout, we unveil many pitfalls and mechanisms in quantum optimization using a physical perspective, which aim to spur the development of novel quantum optimization algorithms and refinements.
Highlights
Optimization is a topic so broad reaching and powerful in its applications that the idea that it could be possibly accelerated by quantum computers has attracted incredible attention, regardless of the origins or the underlying mechanisms for speedup
While some of our results focus on the p = 1 case, where general proofs of classical algorithms are known [47], this will support our goal of understanding what happens within a differential step of a quantum optimization algorithm more generally
It is often found that a smooth increase of the potential term and a smooth decrease of the kinetic term, akin to an accelerated adiabatic path, are often close to the optimal solutions. While it has been shown in hard cases that the mechanism of solution for quantum approximate optimization algorithm (QAOA) may be diabatic [46], our results support the idea that pretraining with kinetic energy and adiabatic solutions in mind facilitates information transfer from the final point to initial stages where the diabatic transition is leveraged
Summary
Optimization is a topic so broad reaching and powerful in its applications that the idea that it could be possibly accelerated by quantum computers has attracted incredible attention, regardless of the origins or the underlying mechanisms for speedup. Using a more modern interpretation of Grover search and amplitude amplification has led to asymptotic speedups over structured classical algorithms like annealing and branch and bound [5,6,7,8,9], but the overheads for general cases remain challenging when compiled all the way to fault-tolerant gate sequences [10] This is exacerbated by the fact that real use cases have shown that it is likely the case that improved solutions are most needed on problems with more than 1000 bits, where classical heuristics break down [11,12]. We will focus on short-depth realizations to maximize understanding, these tools turn out to be powerful for better understanding how quantum optimizations based on physical dynamics proceed in general
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