[Paper Review] Pegasus: The second connectivity graph for large-scale quantum annealing hardware

Nike Dattani, Szilárd Szalay+1 more|arXiv (Cornell University)|Jan 22, 2019

Quantum Computing Algorithms and Architecture2 references66 citations

TL;DR

The paper introduces Pegasus, a second connectivity graph for quantum annealers that dramatically increases qubit connectivity beyond Chimera and provides an algorithm, visualizations, and open-source tools for generating and exploring Pegasus graphs.

ABSTRACT

Pegasus is a graph which offers substantially increased connectivity between the qubits of quantum annealing hardware compared to the graph Chimera. It is the first fundamental change in the connectivity graph of quantum annealers built by D-Wave since Chimera was introduced in 2009 and then used in 2011 for D-Wave's first commercial quantum annealer. In this article we describe an algorithm which defines the connectivity of Pegasus and we provide what we believe to be the best way to graphically visualize Pegasus in order to see which qubits couple to each other. As supplemental material, we provide a wide variety of different visualizations of Pegasus which expose different properties of the graph in different ways. We provide an open source code for generating the many depictions of Pegasus that we show.

Motivation & Objective

• Motivate and define a higher-connectivity qubit graph to outperform Chimera for large-scale quantum annealing.
• Describe an explicit algorithm to generate Pegasus connectivity from layered Chimera graphs.
• Provide visualization techniques and open-source code to illustrate Pegasus properties.
• Discuss implications for problem embedding, non-planarity, and potential quantum/classical performance benefits.
• Highlight how Pegasus enables efficient minor-embedding of certain quadratization gadgets without extra qubits.

Proposed method

• Start from Z layers of Chimera graphs arranged in an X by Y by Z array of K4,4 cells.
• Label each qubit with six indices (x, y, z, i, j, k) to describe position and cell side within a K4,4.
• Define Chimera edges within and between cells as in the original Chimera graph.
• Add Pegasus-specific edges by first connecting corresponding k=0 to k=1 within each K4,4 cell.
• Connect K4,4 cells across Chimera layers using 64 inter-layer edges organized into eight K2,4 graphs, following explicit rules that map Pegasus edges across layers (Equations 5–11 and 12–19 in the text).
• Compress the graph conceptually to one vertex per cell with edges representing K2,4 groups to aid visualization (as shown in figures).
• Provide open-source code to generate Pegasus figures and variants, and supplementary material with multiple visualizations.

Experimental results

Research questions

• RQ1 How does Pegasus connectivity compare to Chimera in terms of qubit degree and inter-cell connections?
• RQ2 Can Pegasus connectivity preserve non-planarity while increasing embedding capacity for large-scale problems?
• RQ3 What embedding advantages does Pegasus offer for quadratization gadgets and minor-embedding without extra auxiliary qubits?
• RQ4 How can Pegasus be visualized and analyzed to reveal its structural properties?
• RQ5 What open-source tools exist to generate and depict Pegasus graphs?

Key findings

• Each Pegasus qubit gains additional connections, increasing typical degree to 15 for interior cells, compared with Chimera’s degree of 6.
• Pegasus connections across layers preserve non-planarity, enabling embedding of non-planar problems that resist polynomial-time classical solutions.
• Pegasus allows quadratization gadgets requiring one auxiliary qubit to be embedded without adding extra qubits because Pegasus contains K4, enabling direct connections among three logical qubits and the auxiliary qubit.
• An explicit, repeatable rule set (Equations 5–11 and 12–19) generates the Pegasus graph from Chimera layers, yielding a scalable, layered connectivity.
• The authors provide open-source code and supplemental visualizations to illustrate Pegasus properties and variants.
• Supplementary materials include multiple 2D/3D visualizations of Pegasus and Chimera references for verification.