# D-Wave QPU Architecture: Topologies¶

The layout of the D-Wave QPU is critical to translating a QUBO or Ising objective function into a format that a D-Wave system can solve. We know that binary objective functions can be represented as graphs; this chapter explains the mapping between a problem graph and the QPU topology.

Note

Although Ocean software automates this mapping, you should understand it if you are directly programming the QPU because it has implications for the problem-graph size and solution quality. If you are sending your problem to a Leap quantum-classical hybrid solver, the solver handles all interactions with the QPU.

Leap hybrid solvers are described here: Leap’s Hybrid Solvers

The D-Wave quantum processing unit (QPU) is a lattice of interconnected qubits. While some qubits connect to others via couplers, the D-Wave QPU is not fully connected. Instead, the qubits of D-Wave 2000Q and earlier generations of QPUs interconnect in a topology known as Chimera while Advantage QPUs incorporate the Pegasus topology.

## Chimera Graph¶

In the D-Wave 2000Q and earlier systems, qubits are “oriented” on the QPU vertically or horizontally as shown in Figure 12.

Fig. 12 Qubits represented as horizontal and vertical loops. This graphic shows three rows of 12 vertical qubits and three columns of 12 horizontal qubits for a total of 72 qubits, 36 vertical and 36 horizontal.

For QPUs with the Chimera topology it is conceptually useful to categorize couplers as follows:

• Internal couplers.

Internal couplers connect pairs of orthogonal (with opposite orientation) qubits as shown in Figure 13. The Chimera topology has a recurring structure of four horizontal qubits coupled to four vertical qubits in a $K_{4,4}$ bipartite graph, called a unit cell.

Fig. 13 Green circles at the intersections of qubits signify internal couplers; for example, the upper leftmost vertical qubit, highlighted in green, internally couples to four horizontal qubits, shown bolded. The translucent green squares provide a helpful way to envision a recurring structure of this topology: a division of couplings into unit cells of $K_{4,4}$ bipartite graphs.

A unit cell is typically rendered as either a cross or a column as shown in Figure 14.

Fig. 14 Chimera unit cell. In each of these renderings there are two sets of four qubits. Each qubit connects to all qubits in the other set but to none in its own, forming a $K_{4,4}$ graph; for example, the green qubit labeled 0 connects to bolded qubits 4 to 7.

• External couplers.

External couplers connect colinear pairs of qubits—pairs of parallel qubits in the same row or column—as shown in Figure 15.

Fig. 15 External couplers, shown as connected blue circles, couple vertical qubits to adjacent vertical qubits and horizontal qubits to adjacent horizontal qubits; for example, the green horizontal qubit in the center couples to the two blue horizontal qubits in adjacent unit cells. (It is also coupled to the bolded qubits in its own unit cell by internal couplers.)

The $K_{4,4}$ unit cells formed by internal couplers are connected by external couplers as a lattice: this is the Chimera topology. Figure 16 shows two unit cells that form part of a larger Chimera graph.

Fig. 16 A cropped view of two unit cells of a Chimera graph. Qubits are arranged in 4 unit cells (translucent green squares) interconnected by external couplers (blue lines).

Chimera qubits are characterized as having:

• nominal length 4—each qubit is connected to 4 orthogonal qubits through internal couplers
• degree 6—each qubit is coupled to 6 different qubits

The notation CN refers to a Chimera graph consisting of an $N{\rm x}N$ grid of unit cells. The D-Wave 2000Q QPU supports a C16 Chimera graph: its 2048 qubits are logically mapped into a $16 {\rm x} 16$ matrix of unit cells of 8 qubits. The $2 {\rm x} 2$ Chimera graph of Figure 14 is denoted C2.

## Pegasus Graph¶

In the Pegasus topology, qubits are “oriented” vertically or horizontally, as in Chimera, but similarly aligned qubits are also shifted, as illustrated in Figure 17.

Fig. 17 A cropped view of the Pegasus topology with qubits represented as horizontal and vertical loops. This graphic shows approximately three rows of 12 vertical qubits and three columns of 12 horizontal qubits for a total of 72 qubits, 36 vertical and 36 horizontal.

For QPUs with the Pegasus topology it is conceptually useful to categorize couplers as internal, external, and odd. Figure 96 and Figure 19 show two views of the coupling of qubits in this topology.

Fig. 18 Coupled qubits (represented as horizontal and vertical loops): the horizontal qubit in the center, shown in red and numbered 1, with its odd coupler and paired qubit also in red, is internally coupled to vertical qubits, in pairs 3 through 8, each pair and its odd coupler shown in a different color, and externally coupled to horizontal qubits 2 and 9, each shown in a different color.

Fig. 19 Coupled qubits “roadway” graphic (qubits represented as dots and couplers as lines): the qubit in the upper center, shown in red and numbered 1, is oddly coupled to the (red) qubit shown directly below it, internally coupled to vertical qubits, in pairs 3 through 8, each pair and its odd coupler shown in a different color, and externally coupled to horizontal qubits 2 and 9, each shown in a different color.

### Pegasus Couplers¶

• Internal couplers.

Internal couplers connect pairs of orthogonal (with opposite orientation) qubits as shown in Figure 20. Each qubit is connected via internal coupling to 12 other qubits.

Fig. 20 Junctions of horizontal and vertical loops signify internal couplers; for example, the green vertical qubit is coupled to 12 horizontal qubits, shown bolded. The translucent green square represents a Chimera unit cell structure (a $K_{4,4}$ bipartite graph of internal couplings).

• External couplers.

External couplers connect vertical qubits to adjacent vertical qubits and horizontal qubits to adjacent horizontal qubits as shown in Figure 21.

Fig. 21 External couplers connect similarly aligned adjacent qubits; for example, the green vertical qubit is coupled to the two adjacent vertical qubits, highlighted in blue.

• Odd couplers.

Odd couplers connect similarly aligned pairs of qubits as shown in Figure 22.

Fig. 22 Odd couplers connect similarly aligned pairs of qubits; for example, the green vertical qubit is coupled to the red vertical qubit by an odd coupler.

Pegasus features qubits of degree 15 and native $K_4$ and $K_{6,6}$ subgraphs. Pegasus qubits are considered to have a nominal length of 12 (each qubit is connected to 12 orthogonal qubits through internal couplers) and degree of 15 (each qubit is coupled to 6 different qubits).

As we use the notation $C_n$ to refer to a Chimera graph with size parameter N, we refer to instances of Pegasus topologies by $P_n$; for example, $P_3$ is a graph with 144 nodes.

## Chains and Minor Embedding¶

The nodes and edges on the graph that represents an objective function translate to the qubits and couplers in the system graph. Each logical qubit, in the graph of the objective function, may be represented by one or more physical qubits. The process of mapping the logical qubits to physical qubits is known as minor embedding.

Note

While tools for minor embedding are available in the Ocean SDK, you can also do this manually as explained in the Minor-Embedding a Problem onto the QPU chapter.