# D-Wave QPU Architecture: Topologies#

The layout of the D-Wave quantum processing unit (QPU) is critical to formulating an objective function in a format that a D-Wave annealing quantum computer can solve, as described in the Solving Problems with Quantum Samplers section. Although Ocean software automates the mapping from the linear and quadratic coefficients of a quadratic model to qubit bias and coupling values set on the QPU, you should understand it if you are using QPU solvers directly because it has implications for the problem-graph size and solution quality.

Note

If you are sending your problem to a Leap quantum-classical hybrid solver, the solver handles all interactions with the QPU.

The D-Wave 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 annealing quantum computers interconnect in one of the following topologies:

## The Working Graph#

Some small number of qubits and couplers in a QPU may not meet the specifications
to function as desired. These are therefore removed from the programmable fabric
that users can access. The subset of the QPU’s graph available to users is the
*working graph*. The yield[1] of the working graph is the percentage of
working qubits that are present.

## Topology Concepts and Chimera#

This section introduces concepts in topology by describing elements of the
*Chimera* topology used on D-Wave’s previous generation of quantum computer,
the D-Wave 2000Q. The Chimera topology’s relative simplicity is helpful for
understanding the more complex topologies of newer QPUs: in addition to shared
concepts, the newer topologies are often described in terms derived from Chimera.

### Qubit Orientation#

Qubits on D-Wave QPUs are “oriented” vertically or horizontally as shown, for the Chimera topology, in Figure 11.

Qubits on newer QPUs, such as those with Zephyr topology, are also oriented vertically and horizontally.

### Coupler Types#

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 12.

**External couplers**.External couplers connect colinear pairs of qubits—pairs of parallel qubits in the same row or column—as shown in Figure 13.

**Odd couplers**.Odd couplers connect similarly aligned pairs of qubits as shown in Figure 22 of the Pegasus Graph section. The Chimera topology does not support such couplers but newer topologies do.

### Unit Cells#

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*. Figure 14 shows
three unit cells.

A unit cell is typically rendered as either a cross or a column as shown in Figure 15.

### Chimera Structure#

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.

### Notations#

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.

For example, the D-Wave 2000Q QPU supported a C16 Chimera graph: its more than 2000 qubits were logically mapped into a \(16 {\rm x} 16\) matrix of unit cells of 8 qubits. The \(2 {\rm x} 2\) Chimera graph of Figure 12 is denoted C2.

## Pegasus Graph#

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

For QPUs with the Pegasus topology it is conceptually useful to categorize couplers as internal, external, and odd. Figure 18 and Figure 19 show two views of the coupling of qubits in this topology.

### 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.

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

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

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 15 different qubits).

As the notation \(C_n\) refers to a Chimera graph with size parameter N, \(P_n\) refers to instances of Pegasus topologies; for example, \(P_3\) is a graph with 144 nodes. A Pegasus unit cell contains twenty-four qubits, with each qubit coupled to one similarly aligned qubit in the cell and two similarly aligned qubits in adjacent cells, as shown in Figure 23. An Advantage QPU is a lattice of \(16x16\) such unit cells, denoted as a \(P_{16}\) Pegasus graph.

More formally, the Pegasus unit cell consists of 48 halves of qubits that are divided between adjacent such unit cells, as shown in Figure 24.

## Zephyr Graph#

D-Wave is currently developing its next-generation QPU with the Zephyr topology: qubits are “oriented” vertically or horizontally, as in Chimera and Pegasus, and are shifted and connected with three coupler types as in Pegasus, but this new graph achieves higher nominal length (16) and degree (20). A qubit in the Zephyr topology has sixteen internal couplers connecting it to orthogonal qubits and two external couplers and two odd couplers connecting it to similarly aligned qubits.

Zephyr topology enables native \(K_4\) and \(K_{8,8}\) subgraphs.

Figure 25 shows the 20 couplers of a qubit in a Zephyr graph.

As the notations \(C_n\) and \(P_n\) refer to Chimera and Pegasus graphs with size parameter N, \(Z_n\) refers to instances of Zephyr topologies; specifically, \(Z_n\) is a \((2n+1) \times (2n+1)\) grid of unit cells. For example, \(Z_3\) is a graph with 336 nodes.

A Zephyr unit cell, as shown in Figure 26, contains two groups of eight half qubits, with each qubit in the cell coupled either to four oppositely aligned qubits and one similarly aligned qubit (four \(K_{4,4}\) complete graphs with their internal and external couplings) or to eight oppositely aligned qubits and one similarly aligned qubit (a \(K_{8,8}\) complete graph with its internal and odd couplings).

Figure 27 shows a Zephyr \(Z_1\) \(3X3\) grid of unit cells.

## Ocean’s Graph Tools#

Ocean provides for all supported topologies the following graph tools:

graph generation creates graphs for the supported topologies of various sizes.

drawing visualizes the graphs you create.

indexing helps translate coordinates of the supported graphs.

## Further Information: Technical Reports#

You can learn more about these topologies and their implications in the following technical reports: