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# Example: Introduction to the Simulator

True-Q™ offers a highly versatile `Simulator`

that allows
users to simulate arbitrary circuits with a wide variety of noise models and options
for noise model customization. In particular, the simulator can be used to:

Simulate the final quantum state of a circuit. This will be a pure state vector if the noise model is ideal or has only unitary noise, and will be a density matrix otherwise.

Simulate the total effective operator of a circuit. This will be a unitary matrix if the noise model is ideal or has only unitary noise, and will be a superoperator otherwise.

Sample a given number of bitstrings (or ditstrings if higher energy levels are defined) from the distribution defined by the final simulated quantum state of a circuit. These results can be returned, or can be automatically populated into the respective

`results`

attributes of the given circuits.

In this example, we go through the basic features of the True-Q™ simulator. We begin by instantiating a circuit to play with:

```
[2]:
```

```
import numpy as np
import trueq as tq
circuit = tq.Circuit([{0: tq.Gate.x, 1: tq.Gate.y}, {(0, 1): tq.Gate.cz}])
circuit.measure_all()
circuit.draw()
```

```
[2]:
```

## Simulator Basics

In its simplest configuration, a (noiseless) simulator can be instantiated from
True-Q™’s `Simulator`

class as follows:

```
[3]:
```

```
sim = tq.Simulator()
```

We can use it to simulate the final state of the circuit, with all initial states prepared as \(|0\rangle\) by default. Since the simulator is noiseless, the output state is a pure state; a noisy simulator will generally return a density matrix instead.

```
[4]:
```

```
sim.state(circuit).mat()
```

```
[4]:
```

```
array([0.+0.j, 0.+0.j, 0.+0.j, 0.-1.j])
```

If we want to force the output to be a density matrix, we can use:

```
[5]:
```

```
sim.state(circuit).upgrade().mat()
```

```
[5]:
```

```
array([[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
[0.-0.j, 0.-0.j, 0.-0.j, 1.+0.j]])
```

We can also use it to compute the overall action of the circuit. Since the simulator is currently noiseless, the output is a unitary matrix; a noisy simulator will return a superoperator (in the rowstacked basis).

```
[6]:
```

```
tq.plot_mat(sim.operator(circuit).mat())
```

Finally, we can use a simulator to populate the `results`

attribute of the circuit. Currently, this attribute is just an empty dictionary:

```
[7]:
```

```
circuit.results
```

```
[7]:
```

```
Results({}, dim=None)
```

But after calling the `run()`

method, it has
`results`

that are randomly sampled bitstrings from the final
state of a state simulation:

```
[8]:
```

```
sim.run(circuit, n_shots=100)
circuit.results
```

```
[8]:
```

```
Results({'11': 100})
```

In this case, all 100 shots end in the `11`

state because the final state is a
computational eigenstate.

When re-running the simulation, it will overwrite the existing
`results`

unless
otherwise specified. Another option is to call the simulator’s
`sample()`

method which returns a `Results`

object directly without affecting the circuit’s `results`

attribute:

```
[9]:
```

```
results = sim.sample(circuit, n_shots=100)
results
```

```
[9]:
```

```
Results({'11': 100})
```

We can also specify infinite shots to get the expectation values of each bitstring:

```
[10]:
```

```
sim.run(circuit, n_shots=np.inf, overwrite=True)
circuit.results
```

```
[10]:
```

```
Results({'11': 1.0})
```

## Adding Noise Sources

We construct a noisy simulator by appending noise sources to a noiseless simulator. The example below demonstrates this using two of True-Q™'s built-in noise sources:

```
[11]:
```

```
# Add an overrotation noise, which causes single qubit gates to be simulated as U^1.02
sim.add_overrotation(single_sys=0.02)
# Add a depolarizing noise source at a rate of 0.8% per acted-on qubit per cycle
sim.add_depolarizing(p=0.008)
# Note that noisy simulators can be constructed as one-liners
other_sim = tq.Simulator().add_overrotation(single_sys=0.02).add_depolarizing(p=0.008)
```

Note

Note that simulation is cycle-based.
Each noise source is called to add noise to the quantum state (or to the
superoperator if `operator()`

is called) for each cycle
in a circuit. The order in which noise sources are applied is dictated by the
order in which they were added to the simulator.

Now, since the simulator applies a depolarizing noise source, we get a density operator rather than a pure state:

```
[12]:
```

```
tq.plot_mat(sim.state(circuit).mat())
```

After calling the `run()`

method (overwriting the
`results`

from above), we end up with noisy outcomes:

```
[13]:
```

```
sim.run(circuit, n_shots=100, overwrite=True)
circuit.results
```

```
[13]:
```

```
Results({'01': 2, '11': 98})
```

For more information on both built-in and custom noise sources, check out Example: Advanced Simulator Usage next.

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