Programming a neutral atom QPU

Qoolqit model

In neutral-atom systems, atoms interact through a combination of distance-dependent interactions and laser-driven controls. The interaction strength between two atoms decreases rapidly with their separation \(r\) (as \(1/r^6\)), while the laser beams determine how strongly each atom is driven.

As a result, the behavior of the system is not set by absolute values alone, but by the interplay between geometry (distances) and control strength (laser power). Different combinations of these quantities can lead to equivalent physical behavior, as long as their relative scales are preserved.

QoolQit introduces a dimensionless reference frame where all quantities are expressed in function of an interaction reference.

The quantum system is thus described by the sum of two energetic contribution, the interaction and the driving one, as described by the following Hamiltonian:

\[ \tilde{H}(t) = \underbrace{\sum_{i<j} \tilde{J}_{ij}\,\hat{n}_i \hat{n}_j}_{\text{interactions}} + \underbrace{\sum_i \frac{\tilde{\Omega}(t)}{2} \left( \cos\phi(t)\,\hat{\sigma}^x_i - \sin\phi(t)\,\hat{\sigma}^y_i \right)}_{\text{global drive}} - \underbrace{\sum_i \left( \tilde{\delta}(t) + \epsilon_i\,\tilde{\Delta}(t) \right) \hat{n}_i}_{\text{detuning}}. \]

Here, \(\hat{n}_i = \frac{1}{2}(1 + \hat{\sigma}^z_i)\) is the Rydberg occupation operator of atom \(i\), and the \(\hat{\sigma}^{x,y,z}_i\) are the Pauli operators:

\[ \sigma^x=\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}, \qquad \sigma^y=\begin{pmatrix} 0 & -i \\ i & 0\end{pmatrix}, \qquad \sigma^z=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}. \]

The interaction \(\tilde{J}_{ij}\) follows the \(1/r^6\) Rydberg scaling, normalized so that the maximum equals \(1\): \(\tilde{J}_{ij} = \tilde{r}_{ij}^{-6}\) and \(\max(\tilde{J}_{ij}) = 1\).
\(\tilde{\Omega}(t)\) and \(\tilde{\delta}(t)\) are laser parameters (amplitude and detuning) and are measured relative to the maximum interaction strength, which is equal to \(1\).
Times \(\tilde{t}\) are measured relative to the interaction timescale.

This means that programs are hardware-independent until compilation: drive strengths are naturally expressed as multiples of the interaction strength, and the same program can be compiled to different devices without modification.

The actual physical scale — such as the precise distances or laser amplitudes — is determined only later, during the compilation step, when targeting a specific device.

More details about the connection to physical units are provided in the section Adimensionalization.

The following table summarizes the parameters appearing in the Hamiltonian and their allowed ranges.

Symbol	Description	Range
\(\tilde{J}_{ij}\)	Dimensionless coupling between sites \(i\) and \(j\)	\([0,\,1]\)
\(\tilde{\Omega}(t)\)	Global drive amplitude, affecting all sites equally	\(\geq 0\)
\(\tilde{\delta}(t)\)	Global detuning, affecting all sites equally	any real value
\(\tilde{\Delta}(t)\)	Local detuning amplitude	\(\leq 0\)
\(\phi(t)\)	Global phase	\([0,\,2\pi]\)
\(\epsilon_i\)	Local detuning weight for site \(i\)	\([0,\,1]\)
\(\tilde{t}\)	Dimensionless time	\(> 0\)

Drive regimes

Because \(\tilde{\Omega}\) is expressed relative to the maximum interaction strength, strong and weak drive regimes are defined independently of the specific geometry:

Regime	Condition	Intuition
Strong drive	\(\tilde{\Omega} \gg 1\)	Controls dominate; interactions are a perturbation
Balanced	\(\tilde{\Omega} \sim 1\)	Controls and interactions compete
Weak drive	\(\tilde{\Omega} \ll 1\)	Interactions dominate; blockade and correlation effects are strong

Time regimes

Time is expressed in QoolQit in units of the maximum interaction energy.

In an interacting many-body system, this gives \(\tilde{t}\) a natural physical interpretation: it measures evolution time relative to the timescale on which interactions generate correlations. Roughly speaking, a time \(\tilde{t} \sim 1\) is enough for nearest-neighbor sites to begin developing correlations. More generally, \(\tilde{t} \sim n\) can be interpreted as the timescale on which correlations may have propagated over a distance of order \(n\) lattice spacings.

This makes dimensionless time a convenient, geometry-independent way to describe how long the system evolves relative to its intrinsic interaction dynamics.

Regime	Condition	Intuition
Short time	\(\tilde{t} \ll 1\)	Evolution is too brief for interactions to significantly build up correlations
Intermediate time	\(\tilde{t} \sim 1\)	Interactions begin to visibly affect the dynamics; nearest-neighbor correlations can emerge
Long time	\(\tilde{t} \gg 1\)	Correlations and many-body interaction effects have had time to spread across the system

Compilation

As described above, a QoolQit program is written in dimensionless units. This means that the user specifies the problem in terms of dimensionless quantities, independently of any particular device.

However, the values that can actually be implemented are constrained by the hardware. Real devices only allow certain ranges of interaction strengths, drive amplitudes, detunings, and evolution times. Therefore, an important task of QoolQit is to take the dimensionless program specified by the user and map it to a set of parameters that can be realized on the chosen hardware. We refer to this step as compilation.

Compilation

Compilation is the step where QoolQit takes a dimensionless program and rescales it into hardware-realizable parameters, while preserving physical invariants.

A convenient way to understand this is to first work entirely in dimensionless units. As mentioned above, the key idea is that the program is defined by ratios, not by absolute scales. For example, fixing the ratio \(\frac{\max_{\tilde{t}}\tilde{\Omega}}{\tilde{J}}\) defines a line in the \((\tilde{J},\tilde{\Omega})\) plane. Moving along this line changes the overall scale of the program, but preserves its dimensionless structure (here \(\max_{\tilde{t}}\) stands for the maximum over time).

This means that compilation does not change the dimensionless physics of the program. Instead, it rescales the program so that it lies inside the region that can be implemented on a given device.

Example

Consider a program defined by \((\tilde{J},\max_{\tilde{t}}\tilde{\Omega}) = (1,0.4),\) so that \(\frac{\max_{\tilde{t}}\tilde{\Omega}}{\tilde{J}} = 0.4\).

Now assume that the valid compilation region is constrained by \(\tilde{J} \leq 1, \;\tilde{\Omega} \leq 0.2,\) as shown in the figure below.

Compilation in dimensionless units

In this figure, the green rectangle represents the valid compilation region which takes into account the limitations of a specific device. The diagonal line corresponds to all programs with fixed ratio \(\tilde{\Omega}/\tilde{J}=0.4\). The point \((1,0.4)\) is outside the valid region, because the drive amplitude is too large. To compile the program, QoolQit rescales it while preserving the ratio \(\max_{\tilde{t}}\tilde{\Omega}/\tilde{J} = 0.4\).

The largest valid point on this line is \((\tilde{J},\tilde{\Omega}) = (0.5,0.2).\) So the program is rescaled by a factor \(0.5\), but its dimensionless content is unchanged: the ratio between drive and interaction is the same, and therefore the underlying dimensionless problem is the same.

What changes under compilation?

What remains fixed is the dimensionless program. What changes is the reference scale used to map it to a physical device. A full derivation of the mapping and concrete numerical examples are given in Adimensionalization.

If the initial point \((1,0.4)\) corresponds to choosing the maximum interaction strength as the reference scale, then compiling to \((0.5,0.2)\) means that the program is realized with a smaller reference interaction, equal to \(0.5\) times the original one. It is important to keep in mind that compilation also rescales time.

In other words, compilation keeps the dimensionless problem unchanged, but changes the conversion between dimensionless quantities and physical ones.