Address review comments

docs/source/programming.md

@@ -1,20 +1,35 @@
-# Programming a neutral-atom Quantum Computer
+# Programming a neutral-atom QPU
 
 ## Introduction
 
 ### 1. Atoms encode the state
 
 Neutral atoms store the quantum information in their energy levels (also known as [_eigenstates_](./conventions.md)). When only two eigenstates are used to encode information, each atom is a qubit. If these eigenstates are $\left|a\right>$ and $\left|b\right>$, then the state of an atom is described by $\left|\psi\right> = \alpha \left|a\right> + \beta \left|b\right>$, with $|\alpha|^2 + |\beta|^2 = 1$.
 
-When multiple atoms are used, the state of the system is described by a linear combination of the _eigenstates_ of the multi-atom system, whose set is obtained by making the cross product of the set of eigenstate of each atom. If each atom is described by $d$ eigenstates labelled ${\left|a_1\right>, \left|a_2\right>...\left|a_d\right>}$ (each atom is a _qudit_), and if there are $N$ atoms in the system, then the state of the whole system is provided by $\left|\Psi\right> = \sum_{i_1, ..., i_N \in [1, ..., d]} c_{i_1, ..., i_N} \left|a_{i_1}...a_{i_N}\right>$, with $\sum_{i_1, ..., i_N \in [1, ..., d]} |c_{i_1, ..., i_N}|^2 = 1$.
+When multiple atoms are used, the state of the system is described by a linear combination of the _eigenstates_ of the multi-atom system, whose set is obtained by making the cross product of the set of eigenstate of each atom. If each atom is described by $d$ eigenstates labelled ${\left|a_1\right>, \left|a_2\right>...\left|a_d\right>}$ (each atom is a _qudit_), and if there are $N$ atoms in the system, then the state of the whole system is provided by 
+
+$$
+\begin{align}
+\left|\Psi\right> &= \sum_{i_1, ..., i_N \in [1, ..., d]} c_{i_1, ..., i_N} \left|a_{i_1}...a_{i_N}\right> \\
+&= c_{1, 1, ..., 1}\left|a_{1}a_{1}...a_{1}\right> + ... + c_{1, 1, ..., d}\left|a_{1}a_{1}...a_{d}\right> + ... + c_{d, d, ..., d}\left|a_{d}a_{d}...a_{d}\right>
+\end{align}
+$$
+
+where $\sum_{i_1, ..., i_N \in [1, ..., d]} |c_{i_1, ..., i_N}|^2 = 1$. If $d=2$ and $N=1$, you have the state of a qubit as above. If $d=2$, then for any number of atoms $N$ this becomes
+
+$$
+\left|\Psi\right> = c_{1, 1, ..., 1}\left|a_{1}a_{1}...a_{1}\right> + c_{1, 1, ..., 2}\left|a_{1}a_{1}...a_{2}\right> + ... + c_{2, 2, ..., 2}\left|a_{2}a_{2}...a_{2}\right>
+$$
 
 ### 2. Hamiltonian evolves the state
 
-In quantum physics, the state of a quantum system evolves along time following the Schrödinger equation: $i\frac{d\left|\Psi\right>(t)}{dt} = \frac{H(t)}{\hbar} \left|\Psi\right>(t)$, where $H(t)$ is the Hamiltonian describing the evolution of the system. 
+In quantum physics, the state of a quantum system evolves along time following the Schrödinger equation: 
 
-**Pulser provides you the tools to program this Hamiltonian**. 
+$$i\frac{d\left|\Psi\right>(t)}{dt} = \frac{H(t)}{\hbar} \left|\Psi\right>(t)$$
+
+Here $H(t)$ is the Hamiltonian describing the evolution of the system. For a system of atoms in the initial state $\left|\Psi_0\right>$, the final state of the system after a time $\Delta t$ is:
 
-#### 2.1. The Hamiltonian
+$$ \left|\Psi_f\right> = \exp\left(-\frac{i}{\hbar}\int_0^{\Delta t} H(t) dt\right)\left|\Psi_0\right>$$
 
 The Hamiltonian describing the evolution of the system can be written as
 
@@ -26,98 +41,132 @@ where $H^D_i$ is the driving Hamiltonian for atom $i$ and
 $H^\text{int}_{ij}$ is the interaction Hamiltonian between atoms $i$
 and $j$.
 
-#### 2.2. Driving Hamiltonian
+:::{important}
+**Pulser provides you the tools to program this Hamiltonian**. 
+:::
 
-The driving Hamiltonian describes the effect of a pulse of Rabi frequency $\Omega(t)$, detuning $\delta(t)$ and phase $\phi(t)$ on an individual atom, driving the transition
-between two of its energies levels, $|a\rangle$ and $|b\rangle$.
+#### 2.1. Driving Hamiltonian
 
-:::{figure} files/two_level_ab.png
+The driving Hamiltonian describes the effect of a pulse on two of energies levels of an individual atom, $|a\rangle$ and $|b\rangle$. A pulse is determined by its Rabi frequency $\Omega(t)$, its detuning $\delta(t)$ and its phase $\phi(t)$.
+
+$$
+H^D(t) / \hbar = \frac{\Omega(t)}{2} e^{-i\phi(t)} |a\rangle\langle b| + \frac{\Omega(t)}{2} e^{i\phi(t)} |b\rangle\langle a| - \delta(t) |b\rangle\langle b|
+$$
+
+In the Bloch sphere representation, this Hamiltonian describes a rotation around the axis **$\Omega$**$(t) = (\Omega(t)\cos(\phi), -\Omega(t)\sin(\phi), -\delta(t))^T$, with angular velocity $\Omega_{eff}(t) = |$**$\Omega$**$(t)| = \sqrt{\Omega^2(t) + \delta^2(t)}$.
+
+:::{figure} files/bloch_rotation_a_b.png
 :align: center
-:alt: The energy levels for the driving Hamiltonian.
+:alt: Representation of the drive Hamiltonian's dynamics as a rotation in the Bloch sphere.
 :width: 200
 
-The coherent excitation is driven between a lower energy level, $|a\rangle$, and a higher energy level,
+Representation of the drive Hamiltonian's dynamics as a rotation in the Bloch sphere. The coherent excitation is driven between a lower energy level, $|a\rangle$, and a higher energy level,
 $|b\rangle$, with Rabi frequency $\Omega(t)$ and detuning $\delta(t)$.
 :::
 
-$$
-H^D(t) / \hbar = \frac{\Omega(t)}{2} e^{-i\phi(t)} |a\rangle\langle b| + \frac{\Omega(t)}{2} e^{i\phi(t)} |b\rangle\langle a| - \delta(t) |b\rangle\langle b|
-$$
 
 :::{important}
 With Pulser, you program the driving Hamiltonian by setting $\Omega(t)$, $\delta(t)$ and $\phi(t)$, all the while Pulser ensures that you respect the constraints of your chosen device.
 :::
 
-#### 2.3. Interaction Hamiltonian
-
+#### 2.2. Interaction Hamiltonian
 
 The interaction Hamiltonian depends on the distance between the atoms $i$ and $j$, $R_{ij}$, and the energy levels in which the information is encoded in these atoms, that define the interaction between the atoms $\hat{U}_{ij}$
 
 $$
 H^\text{int}_{ij} = \hat{U}_{ij}(R_{ij})
 $$
 
-The interaction operator $\hat{U}_{ij}$ is composed of an entangling operator and an interaction strength:
-- If the Rydberg state $\left|r\right>$ is involved in the computation, then $\hat{U}_{ij}(R_{ij}) = \frac{C_6}{R_{ij}^6} \hat{n}_i \hat{n}_j$, where the interaction strength is $\frac{C_6}{R_{ij}^6}$, with $C_6$ a coefficient that depends on the principal quantum number of the Rydberg state. The entangling operator between atom $i$ and $j$ is $\hat{n}_i\hat{n}_j = |r\rangle\langle r|_i |r\rangle\langle r|_j$. Together with the driving Hamiltonian, this interaction encodes the _Ising Hamiltonian_ and is the **most common choice in neutral-atom devices.**
-- If the information is stored in the $\left|0\right>$ and $\left|1\right>$ states of the `XY` basis, $\hat{U}_{ij}(R_{ij}) =\frac{C_3}{R_{ij}^3} (|1\rangle\langle 0|_i |0\rangle\langle 1|_j + |0\rangle\langle 1|_i |1\rangle\langle 0|_j)$, where the interaction strength is $\frac{C_3}{R_{ij}^3}$, with $C_3$ a coefficient that depends on the energy levels used to encode $\left|0\right>$ and $\left|1\right>$. The entangling operator between atom $i$ and $j$ is $\hat{\sigma}_i^{+}\hat{\sigma}_j^{-} + \hat{\sigma}_i^{-}\hat{\sigma}_j^{+} = |1\rangle\langle 0|_i |0\rangle\langle 1|_j + |0\rangle\langle 1|_i |1\rangle\langle 0|_j$. This interaction hamiltonian is associated with the _XY Hamiltonian_ and is a less common mode of operation, usually accessible only in select neutral-atom devices.
+The interaction operator $\hat{U}_{ij}$ is composed of an entangling operator and an interaction strength.
+
+:::{note}
+The interaction Hamiltonian is time-independent. It is always on, no matter the values of the drive Hamiltonian (even if the values of the parameters $\Omega$, $\delta$, $\phi$ are equal to $0$ over a time $\Delta t$).
+:::
+
+##### Ising Hamiltonian
+
+If the Rydberg state $\left|r\right>$ is involved in the computation, then 
 
-:::{important}
-With Pulser, you program the interaction Hamiltonian by setting the distance between atoms, $R_{ij}$, and the choice of eigenstates used in the computation and the Rydberg level(s) targeted by the device. Most commonly, the ground and Rydberg states ($\left|g\right>$ and $\left|r\right>$) are used, such that
 $$
-H^\text{int}_{ij} = \frac{C_6}{R_{ij}^6} \hat{n}_i\hat{n}_j
+\hat{U}_{ij}(R_{ij}) = \frac{C_6}{R_{ij}^6} \hat{n}_i \hat{n}_j
 $$
-When providing the distance between atoms, Pulser ensures that you respect the constraints of your chosen device.
-:::
 
-#### 2.4. Evolution of the system's state
+- The interaction strength is $\frac{C_6}{R_{ij}^6}$, with $C_6$ a coefficient that depends on the principal quantum number of the Rydberg state.
+- The entangling operator between atom $i$ and $j$ is $\hat{n}_i\hat{n}_j = |r\rangle\langle r|_i |r\rangle\langle r|_j$. 
+
+Together with the driving Hamiltonian, this interaction encodes the _Ising Hamiltonian_ and is the **most common choice in neutral-atom devices.**
+
+##### XY Hamiltonian
+
+If the information is stored in the Rydberg states $\left|0\right>$ and $\left|1\right>$ (the so-called `XY` basis), then
+
+$$
+\hat{U}_{ij}(R_{ij}) =\frac{C_3}{R_{ij}^3} (|1\rangle\langle 0|_i |0\rangle\langle 1|_j + |0\rangle\langle 1|_i |1\rangle\langle 0|_j)
+$$
+
+- The interaction strength is $\frac{C_3}{R_{ij}^3}$, with $C_3$ a coefficient that depends on the energy levels used to encode $\left|0\right>$ and $\left|1\right>$. 
+- The entangling operator between atom $i$ and $j$ is $\hat{\sigma}_i^{+}\hat{\sigma}_j^{-} + \hat{\sigma}_i^{-}\hat{\sigma}_j^{+} = |1\rangle\langle 0|_i |0\rangle\langle 1|_j + |0\rangle\langle 1|_i |1\rangle\langle 0|_j$. 
+
+This interaction hamiltonian is associated with the _XY Hamiltonian_ and is a less common mode of operation, usually accessible only in select neutral-atom devices.
 
-For a system of atoms in the initial state $\left|\Psi_0\right>$, the final state of the system after a time $\Delta t$ is:
-$$ \left|\Psi_f\right> = \exp\left(-\frac{i}{\hbar}\int_0^{\Delta t} H(t) dt\right)$$
+:::{important}
+With Pulser, you program the interaction Hamiltonian by setting the distance between atoms, $R_{ij}$. Additionally, the choice of eigenstates used in the computation and the Rydberg level(s) targeted by the device fully determine the interaction strength. Most commonly, the ground and Rydberg states ($\left|g\right>$ and $\left|r\right>$) are used, such that   
+$H^\text{int}_{ij} = \frac{C_6}{R_{ij}^6} \hat{n}_i\hat{n}_j$   
+When providing the distance between atoms, Pulser ensures that you respect the constraints of your chosen device.
+:::
 
 ## Writing a Pulser program
 
-As outlined above, Pulser lets you program an Hamiltonian so that you can manipulate the quantum state of a system of atoms, that encode the states. The series of necessary instructions is encapsulated in the so-called Pulser `Sequence`. Here is a step-by-step guide to create your own Pulser `Sequence`.
+As outlined above, Pulser lets you program an Hamiltonian ([the Hamiltonian $H$](./programming.md#2-hamiltonian-evolves-the-state)) so that you can manipulate the quantum state of a system of atoms. The series of necessary instructions is encapsulated in the so-called Pulser `Sequence`. Here is a step-by-step guide to create your own Pulser `Sequence`.
 
 ### 1. Pick a Device
 
 ```{mermaid}
 
 flowchart TB
   A[Picking a Device] --> B{Do I want to run on a QPU ?}
-  B -->|Yes| C["`Pick the Device from the [QPU backend](./tutorials/backends.nblink)`"]
+  B -->|Yes| C["`Pick the Device from the QPU backend`"]
   B -->|No| D["`Do I want to be constrained by all the QPU specs ?`"]
   D -->|Yes| E["`Use a Device (like _pulser.AnalogDevice_)`"]
-  D --> |No| F["`Use a [_VirtualDevice_](./tutorials/virtual_devices.nblink) (like _pulser.MockDevice_)`"]
+  D --> |No| F["`Use a _VirtualDevice_ (like _pulser.MockDevice_)`"]
 
 ```
 
-The `Device` you select will dictate some parameters and constrain others. For instance, the value of the $C_6$ and $C_3$ coefficients of the interaction Hamiltonian are defined by the device. For a complete view of the constraints introduced by the device, [check its description](./apidoc/core.rst).
+The `Device` you select will dictate some parameters and constrain others. For instance, the value of the $C_6$ and $C_3$ coefficients of the [interaction Hamiltonian](./programming.md/#22-interaction-hamiltonian) are defined by the device. For a complete view of the constraints introduced by the device, [check its description](./apidoc/core.rst).
 
 ### 2. Create the Register
 
 The `Register` defines the position of the atoms. This determines:
 
 - the number of atoms to use in the quantum computation, i.e, the size of the system (let's note it $N$).
-- the distance between the atoms, the $R_{ij} (1\le i, j\le N)$ parameters in the interaction Hamiltonian.
+- the distance between the atoms, the $R_{ij} (1\le i, j\le N)$ parameters in the [interaction Hamiltonian](./programming.md/#22-interaction-hamiltonian).
 
 ### 3. Pick the Channels
 
 A `Channel` targets the transition between two energy levels. Therefore, picking channels defines the energy levels that will be used in the computation. The channels must be picked from the `Device.channels`, so your device selection should take into account the channels it supports.
 
-Picking the channel will initialize the state of the system, and fully determine the interaction Hamiltonian:
-- If the selected Channel is the `Rydberg` or the `Raman` channel, the system is initialized in $\left|gg...g\right>$ and the interaction Hamiltonian is $H^\text{int}_{ij} =\frac{C_6}{R_{ij}^6}|r\rangle\langle r|_i |r\rangle\langle r|_j$.
-- If the selected Channel is the `Microwave` channel, the system is initialized in $\left|00...0\right>$ and the interaction Hamiltonian is $H^\text{int}_{ij} =\frac{C_3}{R_{ij}^3}|1\rangle\langle 0|_i |0\rangle\langle 1|_j + |0\rangle\langle 1|_i |1\rangle\langle 0|_j$.
+Picking the channel will initialize the state of the system, and fully determine the [interaction Hamiltonian](./programming.md#22-interaction-hamiltonian):
+- If the selected Channel is the `Rydberg` or the `Raman` channel, the system is initialized in $\left|gg...g\right>$ and the interaction Hamiltonian is the [Ising Hamiltonian](./programming.md#ising-hamiltonian)
 
-Note: It is possible to pick a `Rydberg` and a `Raman` channel, the information will then be encoded in 3 energy levels $\left|r\right>$, $\left|g\right>$ and $\left|h\right>$: the atoms are no-longer qubits but qutrits. However, it is not possible to pick the `Microwave` channel and another channel.
+$$H^\text{int}_{ij} =\frac{C_6}{R_{ij}^6}|r\rangle\langle r|_i |r\rangle\langle r|_j$$
 
-At this stage, the interaction strength of the Hamiltonian is fully determined.
+- If the selected Channel is the `Microwave` channel, the system is initialized in $\left|00...0\right>$ and the interaction Hamiltonian is the [XY Hamiltonian](./programming.md#xy-hamiltonian) 
+$$H^\text{int}_{ij} =\frac{C_3}{R_{ij}^3}|1\rangle\langle 0|_i |0\rangle\langle 1|_j + |0\rangle\langle 1|_i |1\rangle\langle 0|_j$$
+
+:::{note}
+It is possible to pick a `Rydberg` and a `Raman` channel, the information will then be encoded in 3 energy levels $\left|r\right>$, $\left|g\right>$ and $\left|h\right>$: the atoms are no-longer qubits but qutrits. However, it is not possible to pick the `Microwave` channel and another channel.
+:::
+
+:::{important}
+At this stage, the [interaction Hamiltonian](./programming.md#22-interaction-hamiltonian) is fully determined.
+:::
 
 ### 4. Add the Pulses
 
-By adding pulses to a channel, we incrementally construct the driving Hamiltonian:
+By adding pulses to a channel, we incrementally construct the [driving Hamiltonian](./programming.md#21-driving-hamiltonian):
 - Each `Pulse` defines the Rabi frequency $\Omega(t)$, the detuning $\delta(t)$ and the phase $\phi$ of the driving Hamiltonian over a duration $\Delta t$. Similarly, a delay sets all these parameters to zero for a given amount of time.
 - The channel dictates the states $\left|a\right>$ and $\left|b\right>$ of the driving Hamiltonian.
 
 By applying a series of pulses and delays, one defines the entire driving Hamiltonian of each atom over time.
 
-**Conclusion**: We have successfully defined the hamiltonian $H$ describing the evolution of the system over time.
+**Conclusion**: We have successfully defined the [Hamiltonian](./programming.md#2-hamiltonian-evolves-the-state) $H$ describing the evolution of the system over time.