ORQVIZ: Visualizing High-Dimensional Landscapes in Variational Quantum Algorithms

Abstract

Variational Quantum Algorithms (VQAs) are promising candidates for finding practical applications of near to mid-term quantum computers. There has been an increasing effort to study the intricacies of VQAs, such as the presence or absence of barren plateaus and the design of good quantum circuit ansätze. Many of these studies can be linked to the loss landscape that is optimized as part of the algorithm, and there is high demand for quality software tools for flexibly studying these loss landscapes. In our work, we collect a variety of techniques that have been used to visualize the training of deep artificial neural networks and apply them to visualize the high-dimensional loss landscapes of VQAs. We review and apply the techniques to three types of VQAs: the Quantum Approximate Optimization Algorithm, the Quantum Circuit Born Machine, and the Variational Quantum Eigensolver. Additionally, we investigate the impact of noise due to finite sampling in the estimation of loss functions. For each case, we demonstrate how our visualization techniques can verify observations from past studies and provide new insights. This work is accompanied by the release of the open-source Python package orqviz, which provides code to compute and flexibly plot 1D and 2D scans, Principal Component Analysis scans, Hessians, and the Nudged Elastic Band algorithm. orqviz enables flexible visual analysis of high-dimensional VQA landscapes and can be found at: github.com/zapatacomputing/orqviz.

Quantum circuit architecture search on a superconducting processor

“…After dimension reduction, we choose the obtained first two principal components that explain most of variance as the landscape spanning vector. Refer to [71] and Appendix C for details. For HAA and HEA, the objective function is governed by both the 0-th component (98.75% of variance for HAA, 98.55% of variance for HEA) and 1-th component (1.24% of variance for HAA, 1.45% of variance for HEA).…”

“…Simultaneously, the projection vector e i of each component indicates the contribution of each parameter to this component, implying how many parameters determine the value of objective function. Refer to [71] for details.…”

Neural annealing and visualization of autoregressive neural networks in the Newman-Moore model

“…Note that the filter-wise normalization technique of [24] is not implemented here, given that the RNN ansatz used is not scale-invariant due to the presence of ELU activation functions. A principal component analysis approach, similar to the one used in the visualization of variational quantum circuits landscapes could also be implemented [26]. However, we found the previously described visualization method sufficient to interpret our results.…”

“…In particular, it was used to understand why skip-connections in residual neural networks are generalizing better than vanilla convolutional neural networks. Nowadays, in the quantum computing community, it is more and more employed to benchmark different quantum circuits architectures over a variety of tasks such as quantum optimization or quantum machine learning [25,26]. This paper uses it to study trainability in the neural annealing paradigm.…”

Neural Annealing and Visualization of Autoregressive Neural Networks in the Newman–Moore Model

“…In particular, it was used to understand why skip connections in residual neural networks are generalizing better than vanilla convolutional neural networks. Nowadays, in the quantum computing community, it is more and more employed to benchmark different quantum circuit architectures over a variety of tasks, such as quantum optimization or quantum machine learning [25,26]. This paper uses it to study trainability in the neural annealing paradigm.…”