# Working With Different Topologies¶

The examples shows how to construct software samplers with the same structure as the QPU, and how to work with embeddings with different topologies.

The code examples below will use the following imports:

```
>>> import neal
>>> import dimod
>>> import dwave_networkx as dnx
>>> import networkx as nx
>>> import dwave.embedding
...
>>> from dwave.system import DWaveSampler, EmbeddingComposite
```

## Creating a Chimera Sampler¶

As detailed in Classical Solvers, you might want to use a classical solver while developing your code or writing tests. However, it is sometimes useful to work with a software solver that behaves more like a quantum computer.

One of the key features of the quantum computer is its working graph, which defines the connectivity allowed by the binary quadratic model.

To create a software solver with the same connectivity as a D-Wave 2000Q quantum computer
we first need a representation of the Chimera graph which can be obtained
from the dwave_networkx project using the
`chimera_graph()`

function.

```
>>> C16 = dnx.chimera_graph(16)
```

Next, we need a software sampler. We will use the
`neal.SimulatedAnnealingSampler`

found in dwave_neal,
though the `tabu.TabuSampler`

from dwave-tabu
would work equally well.

```
>>> classical_sampler = neal.SimulatedAnnealingSampler()
```

Now, with a classical sampler and the desired graph, we can use
dimod’s `dimod.StructuredComposite`

to create
a Chimera-structured sampler.

```
>>> sampler = dimod.StructureComposite(classical_sampler, C16.nodes, C16.edges)
```

This sampler accepts Chimera-structured problems. In this case we create an Ising problem.

```
>>> h = {v: 0.0 for v in C16.nodes}
>>> J = {(u, v): 1 for u, v in C16.edges}
>>> sampleset = sampler.sample_ising(h, J)
```

We can even use the sampler with the `dwave.system.EmbeddingComposite`

```
>>> embedding_sampler = EmbeddingComposite(sampler)
```

Finally, we can confirm that our sampler matches the `dwave.system.DWaveSampler`

’s
structure. We make sure that our QPU has the same topology we have
been simulating. Also note that the working graph of the QPU is usually
a subgraph of the full hardware graph.

```
>>> qpu_sampler = DWaveSampler(solver={'qpu': True, 'num_active_qubits__within': [2000, 2048]})
>>> QPUGraph = nx.Graph(qpu_sampler.edgelist)
>>> all(v in C16.nodes for v in QPUGraph.nodes)
True
>>> all(edge in C16.edges for edge in QPUGraph.edges) # doctest: +SKIP
True
```

## Creating a Pegasus Sampler¶

Another topology of interest is the Pegasus topology.

As above, we can use the generator function `dwave_networkx.pegasus_graph()`

found in
dwave_networkx and the
`neal.SimulatedAnnealingSampler`

found in dwave_neal
to construct a sampler.

```
>>> P6 = dnx.pegasus_graph(6)
>>> classical_sampler = neal.SimulatedAnnealingSampler()
>>> sampler = dimod.StructureComposite(classical_sampler, P6.nodes, P6.edges)
```

## Working With Embeddings¶

The example above using the `EmbeddingComposite`

hints that we might be interested in trying embedding with different
topologies.

One thing we might be interested in is the chain length when embedding our problem. Say that we have a fully connected problem with 40 variables and we want to know the chain length needed to embed it on a 2048 node Chimera graph.

We can use dwave-system’s
`find_clique_embedding()`

function to find the
embedding and determine the maximum chain length.

```
>>> num_variables = 40
>>> embedding = dwave.embedding.chimera.find_clique_embedding(num_variables, 16)
>>> max(len(chain) for chain in embedding.values())
11
```

Similarly we can explore clique embeddings for a 40-variables fully connected
problem with a 680 node Pegasus graph using
dwave-system’s
`find_clique_embedding()`

function

```
>>> num_variables = 40
>>> embedding = dwave.embedding.pegasus.find_clique_embedding(num_variables, 6)
>>> max(len(chain) for chain in embedding.values())
6
```