Programming a neutral-atom QPU

Pulser establishes a framework for programming Quantum Processing Units (QPUs) based on neutral atoms (also named cold-atom Quantum Processing Units). In this page, you will learn:

• What mathematical objects are you programming with Pulser? In quantum computing, the evolution of a quantum state is defined by an Hamiltonian. What is a quantum state in Pulser? What are the Hamiltonians that can be defined in Pulser?

• How to program these mathematical objects with Pulser? How do you define the quantum state in Pulser? How do you define the Hamiltonian? We give you a step-by-step guide on how to create a quantum program using Pulser.

Introduction

1. Atoms encode the state

Neutral atoms store the quantum information in their energy levels (also known as eigenstates). 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 \(|\alpha|^2 + |\beta|^2 = 1\).

Definition of a quantum state with multiple atoms

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

\[\begin{split} \begin{align} \left|\Psi\right> &= \sum_{i_1, ..., i_N \in [1, ..., d]} c_{i_1, ..., i_N} \left|a_{i_1}...a_{i_N}\right> \\ &= c_{1, 1, ..., 1}\left|a_{1}a_{1}...a_{1}\right> + ... + c_{1, 1, ..., d}\left|a_{1}a_{1}...a_{d}\right> + ... + c_{d, d, ..., d}\left|a_{d}a_{d}...a_{d}\right> \end{align} \end{split}\]

where \(\sum_{i_1, ..., i_N \in [1, ..., d]} |c_{i_1, ..., i_N}|^2 = 1\). If \(d=2\), then this becomes

\[ \left|\Psi\right> = c_{1, 1, ..., 1}\left|a_{1}a_{1}...a_{1}\right> + c_{1, 1, ..., 2}\left|a_{1}a_{1}...a_{2}\right> + ... + c_{2, 2, ..., 2}\left|a_{2}a_{2}...a_{2}\right> \]

If \(d=2\) and \(N=1\), you have the state of a qubit as above \(\left|\Psi\right> = c_{1}\left|a_{1}\right> + c_{2}\left|a_{2}\right>\).

2. Hamiltonian evolves the state

In quantum physics, the state of a quantum system evolves along time following the Schrödinger equation:

\[i\frac{d\left|\Psi\right>(t)}{dt} = \frac{H(t)}{\hbar} \left|\Psi\right>(t)\]

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:

\[ \left|\Psi_f\right> = \mathcal{T}\exp\left(-\frac{i}{\hbar}\int_0^{\Delta t} H(t) dt\right)\left|\Psi_0\right>\]

Tip

The equation below uses the Indexed Operator notation.

The Hamiltonian describing the evolution of the system can be written as

\[ H(t) = \sum_i \left (H^D_i(t) + \sum_{j<i}H^\text{int}_{ij} \right), \]

where \(H^D_i\) is the driving Hamiltonian for atom \(i\) and \(H^\text{int}_{ij}\) is the interaction Hamiltonian between atoms \(i\) and \(j\).

Important

Pulser provides you the tools to program this Hamiltonian.

2.1. Driving Hamiltonian

The driving Hamiltonian describes the effect of a pulse on two energy levels of an individual atom, \(|a\rangle\) and \(|b\rangle\). A pulse is determined by its duration \(\Delta t\), its Rabi frequency \(\Omega(t)\), its detuning \(\delta(t)\) and its phase \(\phi\) (constant along the duration of the pulse). Between \(0\) and \(\Delta t\), the driving Hamiltonian is:

\[ H^D(t) / \hbar = \frac{\Omega(t)}{2} e^{-i\phi} |a\rangle\langle b| + \frac{\Omega(t)}{2} e^{i\phi} |b\rangle\langle a| - \delta(t) |b\rangle\langle b| \]

Rotations on the Bloch sphere

In the Bloch sphere representation, this Hamiltonian describes a rotation around the axis \(\overrightarrow{\Omega}(t) = (\Omega(t)\cos(\phi), -\Omega(t)\sin(\phi), -\delta(t))^T\), with angular velocity \(\Omega_{eff}(t) = |\overrightarrow{\Omega}(t)| = \sqrt{\Omega^2(t) + \delta^2(t)}\).

For a resonant pulse (\(\delta(t)=0\)) of duration \(\Delta t\), we rotate of an angle \(\int_0^{\Delta t} \Omega (t) dt\) around the fixed axis \((\cos(\phi), -\sin(\phi), 0)\) (on the equator of the Bloch sphere).

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 \(\Omega(t)\), detuning \(\delta(t)\) and phase \(\phi\).

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.

2.2. 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.

Note

The interaction Hamiltonian is constant over time. 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

\[ \hat{U}_{ij}(R_{ij}) = \frac{C_6}{R_{ij}^6} \hat{n}_i \hat{n}_j \]

Interaction strength and entangling operator

• The interaction strength is \(\frac{C_6}{R_{ij}^6}\), with \(C_6\) the Ising interaction 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

\[ \hat{U}_{ij}(R_{ij}) =\frac{C_3}{R_{ij}^3} (|1\rangle\langle 0|_i |0\rangle\langle 1|_j + |0\rangle\langle 1|_i |1\rangle\langle 0|_j) \]

Interaction strength and entangling operator

• 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.

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 (the Hamiltonian \(H\)) 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

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. Notably, the Device defines the list of Channels that can be used in the computation, which have a direct impact on the Hamiltonian that can be implemented.

See also

For a complete view of the constraints introduced by the device, check its description.

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.

See also

For an in-depth view on Register creation, check out this page.

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 the Ising Hamiltonian

\[H^\text{int}_{ij} =\frac{C_6}{R_{ij}^6}|r\rangle\langle r|_i |r\rangle\langle r|_j\]

• If the selected Channel is the Microwave channel, the system is initialized in \(\left|00...0\right>\) and the interaction Hamiltonian is the XY Hamiltonian

\[H^\text{int}_{ij} =\frac{C_3}{R_{ij}^3}|1\rangle\langle 0|_i |0\rangle\langle 1|_j + |0\rangle\langle 1|_i |1\rangle\langle 0|_j\]

Important

At this stage, the interaction Hamiltonian is fully determined.

A Channel is also characterized by its addressing, which defines the number of atoms that are going to be targeted by a pulse. If the addressing of a Channel is Global, all the atoms will experience the same pulse targetting the same transition. In the Hamiltonian \(H\), all the driving Hamiltonians \(H^D_i\) are expressed as

\[ H^D_i(t) / \hbar = \frac{\Omega(t)}{2} e^{-j\phi(t)} |a\rangle_i \langle b|_i + \frac{\Omega(t)}{2} e^{j\phi(t)} |b\rangle_i\langle a|_i - \delta(t) |b\rangle_i\langle b|_i \]

If the addressing of a Channel is Local, then only certain atoms (the “targets”) will experience the pulse and have their evolution follow \(H^D_i\). The driving Hamiltonian of the other atoms is \(H^D_i = \hat{0}_i\). The Hamiltonian \(H\) can also be rewritten:

\[ H(t) = \sum_{i \in targets} H^D_i(t) + \sum_i \sum_{j<i}H^\text{int}_{ij}, \]

Important

The addressing of a Channel defines the number of atoms whose transition will be targeted by a pulse.

The most common addressing for a Channel is the Global one: all the atoms evolve under the same driving Hamiltonian.

See also

For a thorough description of what a Channel is and some tips on how to select a Channel, check this page.

4. Add the Pulses

By adding pulses to a channel, we incrementally construct the 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.

See also

For an in-depth view on Pulse creation, check out this page.

Conclusion

We have successfully defined the Hamiltonian \(H\) describing the evolution of the system over time, by:

• Picking a Device, which defines the value of the \(C_6\) or \(C_3\) coefficients.

• Creating a Register of atoms, which defines the number of atoms used and the distance between them, \(R_{ij}\).

• Selecting the Channels of the Device to use, which define the energy levels of the atoms to use - this step completely defines the interaction Hamiltonian. The addressing property of each Channel also dictates the atoms that will be targeted by the Pulse.

• Adding Pulse and delays to the Channels defines the driving Hamiltonian of each atom along time.

You can now simulate your first Hamiltonian by programming your first Sequence! In this tutorial, you will simulate the evolution of the state of an atom initialized in \(\left|g\right>\) under a Hamiltonian \(H(t)=\frac{\Omega(t)}{2} |g\rangle \langle r|+\frac{\Omega(t)}{2} |r\rangle\langle g|\), with \(\Omega\) chosen such that the final state of the atom is the excited state \(\left|r\right>\).

Many concepts have been introduced here and you might want further explanations.

• The Device object contains all the constraints and physical quantities that are defined in a QPU. This section in the fundamentals details these and provides examples of Devices. The VirtualDevices were also mentioned in this document (here), which is a more advanced feature described here.

• There are multiple ways of defining a Register, as is further detailed in this section.

• The energy levels associated with each Channel and the interaction Hamiltonian they implement are summed up in the conventions page. The channels contain lots of constraints and physical informations, they are detailed in the same section as the Device.

• The quantities in a Pulse are defined using Waveforms, you can read more about these on this page.