Low depth mechanisms for quantum optimization

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.

Power of data in quantum machine learning

“…This could include finding better optima in a training landscape or finding optima with fewer queries. However, without more structure known in the problem, the advantage along these lines may be limited to quadratic or small polynomial speedups [8,9].…”

Quantum Optimization for Training Quantum Neural Networks

“…• Adaptive Mixers. These exploit hidden structure in QNN optimisation problems, hence can achieve short-depth circuit [56].…”

“…Adopting "AC-QAOA" could exploit hidden structure in QNN optimisation problem and dramatically shorten the depth of QAOA layers while significantly improving the quality of the solution [56].…”

Analyzing the Performance of Variational Quantum Factoring on a Superconducting Quantum Processor

“…In order to optimize the QAOA circuit to find γ opt and β opt we employ a layer-by-layer approach [19]. Although there are alternative strategies for training QAOA circuits [20,21], this approach has been shown to require a grid resolution that scales polynomially with respect to the number of qubits required [9]. This approach has two phases.…”