# Simulation of Sequences

```
[1]:
```

```
import numpy as np
import qutip
import matplotlib.pyplot as plt
import pulser
from pulser_simulation import QutipEmulator
```

To illustrate the simulation of sequences, let us study a simple one-dimensional system with periodic boundary conditions (a ring of atoms):

```
[2]:
```

```
# Setup
L = 14
Omega_max = 2.3 * 2 * np.pi
U = Omega_max / 2.3
delta_0 = -3 * U
delta_f = 1 * U
t_rise = 2000
t_fall = 2000
t_sweep = (delta_f - delta_0) / (2 * np.pi * 10) * 5000
# Define a ring of atoms distanced by a blockade radius distance:
R_interatomic = pulser.MockDevice.rydberg_blockade_radius(U)
coords = (
R_interatomic
/ (2 * np.tan(np.pi / L))
* np.array(
[
(np.cos(theta * 2 * np.pi / L), np.sin(theta * 2 * np.pi / L))
for theta in range(L)
]
)
)
reg = pulser.Register.from_coordinates(coords, prefix="atom")
reg.draw(blockade_radius=R_interatomic, draw_half_radius=True, draw_graph=True)
```

We use the drawing capabilites of the `Register`

class to highlight the area **half** the blockade radius away from each atom, which makes it so that overlapping circles correspond to interacting atoms. This is further fleshed out by the graph edges drawn using the `draw_graph`

option.

In this register, we shall act with the following pulser sequence, which is designed to reach a state with *antiferromagnetic order*:

```
[3]:
```

```
rise = pulser.Pulse.ConstantDetuning(
pulser.RampWaveform(t_rise, 0.0, Omega_max), delta_0, 0.0
)
sweep = pulser.Pulse.ConstantAmplitude(
Omega_max, pulser.RampWaveform(t_sweep, delta_0, delta_f), 0.0
)
fall = pulser.Pulse.ConstantDetuning(
pulser.RampWaveform(t_fall, Omega_max, 0.0), delta_f, 0.0
)
seq = pulser.Sequence(reg, pulser.MockDevice)
seq.declare_channel("ising", "rydberg_global")
seq.add(rise, "ising")
seq.add(sweep, "ising")
seq.add(fall, "ising")
seq.draw()
```

## 1. Running a Simulation

First we define our `QutipEmulator`

object, which creates an internal respresentation of the quantum system, including the Hamiltonian which will drive the evolution:

```
[4]:
```

```
sim = QutipEmulator.from_sequence(seq, sampling_rate=0.1)
```

Notice we have included the parameter `sampling_rate`

which allows us to determine how many samples from the pulse sequence we wish to simulate. In the case of the simple shapes in our sequence, only a very small fraction is needed. This largely accelerates the simulation time in the solver.

To run the simulation we simply apply the method `run()`

. At the time of writing of this notebook, the method uses a series of routines from **QuTiP** for solving the Schröedinger equation of the system. It returns a `SimulationResults`

object, which will allow the study or post-processing of the states for each time step in our simulation. Additionally, we can include a progress bar to have an estimate of how the simulation is advancing:

```
[5]:
```

```
results = sim.run(progress_bar=True)
```

```
10.0%. Run time: 1.46s. Est. time left: 00:00:00:13
20.0%. Run time: 2.98s. Est. time left: 00:00:00:11
30.0%. Run time: 4.74s. Est. time left: 00:00:00:11
40.0%. Run time: 6.65s. Est. time left: 00:00:00:09
50.0%. Run time: 8.81s. Est. time left: 00:00:00:08
60.0%. Run time: 11.24s. Est. time left: 00:00:00:07
70.0%. Run time: 14.04s. Est. time left: 00:00:00:06
80.0%. Run time: 16.65s. Est. time left: 00:00:00:04
90.0%. Run time: 19.08s. Est. time left: 00:00:00:02
Total run time: 21.23s
```

## 2. Using the `SimulationResults`

object

The `SimulationResults`

object that we created contains the quantum state at each time step. We can call them using the `states`

attribute:

```
[6]:
```

```
results.states[23] # Given as a `qutip.Qobj` object
```

```
[6]:
```

We can sample the final state directly, using the `sample_final_state()`

method from the `SimulationResults`

object. We try it with \(1000\) samples and discard the less frequent bitstrings:

```
[7]:
```

```
counts = results.sample_final_state(N_samples=1000)
large_counts = {k: v for k, v in counts.items() if v > 5}
plt.figure(figsize=(15, 4))
plt.xticks(rotation=90, fontsize=14)
plt.title("Most frequent observations")
plt.bar(large_counts.keys(), large_counts.values())
```

```
[7]:
```

```
<BarContainer object of 41 artists>
```

Notice how the most frequent bitstrings correspond to the antiferromagnetic order states.

We can also compute the expectation values of operators for the states in the evolution, using the `expect()`

method, which takes a list of operators (in this case, the local magnetization acting on the \(j\)-th spin):

```
[8]:
```

```
def magnetization(j, total_sites):
prod = [qutip.qeye(2) for _ in range(total_sites)]
prod[j] = qutip.sigmaz()
return qutip.tensor(prod)
magn_list = [magnetization(j, L) for j in range(L)]
```

```
[9]:
```

```
expect_magnetization = results.expect(magn_list)
for data in expect_magnetization:
plt.plot(sim.evaluation_times, data)
```

Notice how the local magnetization on *each* atom goes in the same way from \(-1\) (which corresponds to the ground state) to \(0\). This is expected since as we saw above, the state after the evolution has antiferromagnetic-order, so at each site, there is a compensation of magnetization. The parity (even) and the boundary conditions (periodic) allow for two lowest-energy states, whose superposition is similar to that of the perfectly antiferromagnetic state:
\(\Big(|grgr\cdots \rangle + |rgrg\cdots \rangle\Big)/\sqrt{2}\)