# Solutions¶

samplers sample from low-energy states of a problem’s objective function—BQM samplers sample from low-energy states in models such as those defined by an Ising equation or a QUBO problem—and return an iterable of samples, in order of increasing energy.

When the D‑Wave quantum computer solves a problem, it uses quantum phenomena such as superposition and tunneling to explore all possible solutions simultaneously and find a set of the best ones. At the end of the computation (anneal), a single solution is sampled from a set of good solutions, with some probability, and returned. Because the sampled solution is probabilistic, different solutions may be returned in different runs. The standard way of submitting a problem to the system requests many samples, not just one. This not only returns multiple “best” answers but also reduces the probability of settling on a suboptimal answer.

Some classical samplers might return non-probabilistic solutions; for example,
the dimod `ExactSolver`

deterministically
returns the best solution or solutions to small problems by calculating the result for
every configuration of variable values. Such samplers are called solvers.

Some Ocean functions might return a single best solution; for example, some dwave-networkx graph algorithms return only the lowest-energy sample.

## SampleSets¶

Ocean uses the dimod `SampleSet`

class to hold samples and some additional information (e.g., timing information from some
samplers).

For the simple example three-variable “triangular” BQM,

might be solved directly on a D-Wave 2000Q system by sampling 1000 times as follows, where the
`EmbeddingComposite`

composite maps the symbolic BQM to
qubits on the quantum processor, which is called by the
`DWaveSampler`

sampler:

```
>>> bqm = dimod.BQM({}, {('s0', 's1'): -1, ('s0', 's2'): -1, ('s1', 's2'): 1}, 0, dimod.Vartype.SPIN)
>>> sampler = EmbeddingComposite(DWaveSampler(solver={'qpu': True}))
>>> sampleset = sampler.sample(bqm, num_reads=1000)
>>> print(sampleset)
s0 s1 s2 energy num_oc. chain_b.
0 -1 -1 +1 -1.0 141 0.0
1 +1 +1 +1 -1.0 132 0.0
2 -1 -1 -1 -1.0 159 0.0
3 -1 +1 -1 -1.0 143 0.333333
4 +1 +1 -1 -1.0 91 0.0
5 -1 +1 -1 -1.0 86 0.0
6 +1 +1 +1 -1.0 129 0.333333
7 +1 -1 +1 -1.0 119 0.0
['SPIN', 8 rows, 1000 samples, 3 variables]
```

The returned `SampleSet`

, in this case, shows eight solutions of
equal energy \(-1.0\). Solution \(s_0=-1, s_1=-1, s_2=+1\) occurred
in 141 of the 1000 samples. Two solutions, shown in line 3 and 6,
were based on a broken chain of qubits that represented one of the variables.

For this submission to a D-Wave 2000Q, the sampleset also contained the following additional information:

```
>>> print(sampleset.info.keys())
dict_keys(['timing', 'problem_id', 'embedding_context', 'warnings'])
```

For example, the timing information for the problem might look something like:

```
>>> print(sampleset.info["timing"])
{'qpu_sampling_time': 314960,
'qpu_anneal_time_per_sample': 20,
'qpu_readout_time_per_sample': 274,
'qpu_access_time': 324321,
'qpu_access_overhead_time': 5362,
'qpu_programming_time': 9361,
'qpu_delay_time_per_sample': 21,
'total_post_processing_time': 409,
'post_processing_overhead_time': 409,
'total_real_time': 324321,
'run_time_chip': 314960,
'anneal_time_per_run': 20,
'readout_time_per_run': 274}
```