# Working With Different Topologies¶

The examples below show 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 uses the following imports:

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

## Creating a Pegasus 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 an Advantage quantum computer you first need a representation of the Pegasus graph which can be obtained from the dwave_networkx project using the `pegasus_graph()` function.

```>>> P16 = dnx.pegasus_graph(16)
```

Next, you need a software sampler and can use the `SimulatedAnnealingSampler` (`TabuSampler` works equally well).

```>>> classical_sampler = SimulatedAnnealingSampler()
```

Now, with a classical sampler and the desired graph, you can use dimod’s `StructureComposite` to create a Pegasus-structured sampler.

```>>> sampler = dimod.StructureComposite(classical_sampler, P16.nodes, P16.edges)
```

This sampler accepts Pegasus-structured problems. For example, create an Ising problem.

```>>> h = {v: 0.0 for v in P16.nodes}
>>> J = {(u, v): 1 for u, v in P16.edges}
>>> sampleset = sampler.sample_ising(h, J)
```

You can even use the sampler with the `EmbeddingComposite`.

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

Finally, you can confirm that the sampler matches the `DWaveSampler`‘s structure. Make sure that the QPU has the same topology you 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=dict(topology__type='pegasus'))
>>> qpu_graph = qpu_sampler.to_networkx_graph()
>>> qpu_graph.nodes <= P16.nodes
True
>>> qpu_graph.edges <= P16.edges
True
```

## Creating a Zephyr Sampler¶

Another topology of interest is the Zephyr topology.

As above, you can use the generator function `dwave_networkx.zephyr_graph()` found in dwave_networkx and the `SimulatedAnnealingSampler` to construct a sampler.

```>>> Z3 = dnx.zephyr_graph(3)
>>> classical_sampler = SimulatedAnnealingSampler()
>>> sampler = dimod.StructureComposite(classical_sampler, Z3.nodes, Z3.edges)
```

## Working With Embeddings¶

The example above using the `EmbeddingComposite` hints that you might be interested in trying embedding with different topologies.

One thing you might be interested in is the chain length when embedding your problem. For example, if you have a fully connected problem with 40 variables and you want to know the chain length needed to embed it on a 5000+ node Pegasus graph.

You 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.pegasus.find_clique_embedding(num_variables, 16)
>>> max(len(chain) for chain in embedding.values())
5
```

Similarly you can explore clique embeddings for a 40-variables fully connected problem with a 300+ node Zephyr graph using dwave-system’s `find_clique_embedding()` function

```>>> num_variables = 40
>>> embedding = dwave.embedding.zephyr.find_clique_embedding(num_variables, 3)
>>> max(len(chain) for chain in embedding.values())
4
```