The Rydberg feature model¶

This page specifies a small simulated array of neutral atoms whose excitation probabilities serve as precomputed features for a classical head. It fixes every convention before any code: units, Hamiltonian, geometry, encoding, pulse, readout and noise. The register and Hamiltonian (#496), the solver (#497) and the feature map (#499) implement exactly what is written here, and each convention comes with a closed-form limit that their tests check (see Closed-form limits).

The defaults are chosen so that the interaction strength and the dephasing rate are the only swept quantities, and so that the model without interactions stays a closed-form reference:

	Default	Section
Atoms	`N = d`, one per input, `4 ≤ N ≤ 6` by default, open chain at spacing `a`	Geometry
Interaction	`C6 / r⁶` between every pair (full tail), `C6 = 2π × 862 690 MHz µm⁶`	Hamiltonian
Encoding	`Δ_i = Δ_max · σ(x_i)`, one-signed, `Δ_max = √3 Ω`	Encoding
Pulse	constant, `ΩT = π`, global Ω	Pulse
Readout	exact `⟨n_i⟩`, optionally `⟨n_i n_j⟩` and a stated number of shots	Readout
Noise	Markovian dephasing, `L_i = √γ n_i`, no decay	Noise
Swept	`V/Ω` through the spacing, and `γ/Ω`	Geometry, Noise

Everything here is a simulation of an idealised model. Which assumptions a current device supports is listed under Hardware realism.

Units¶

ħ = 1, so energies are angular frequencies. Angular frequencies are in rad/µs, lengths in µm, times in µs. A frequency of f MHz is 2π f rad/µs; C6 is in rad/µs · µm⁶.
Results are reported in dimensionless groups: the pulse area ΩT, the detuning Δ/Ω, the interaction V/Ω with V the nearest-neighbour interaction, and the dephasing rate γ/Ω.
Ω only sets the scale. Dividing the master equation below by Ω and measuring time as Ωt leaves Δ_i/Ω, V_ij/Ω, γ/Ω and ΩT as its only parameters, so the features depend on nothing else. The same features result at any Ω when the spacing is rescaled to keep V/Ω. A value of Ω is needed only to convert to µm and µs: the reference value is Ω = 4π rad/µs (2π × 2 MHz), the Rabi frequency of Bernien et al. (2017) and inside the range of the device compared against below.

Hamiltonian¶

Each atom is a two-level system of its ground state |g⟩ = |0⟩ and one Rydberg state |r⟩ = |1⟩. In the frame rotating at the laser frequency, with the rotating-wave approximation:

H = Ω/2 · Σ_i X_i  −  Σ_i Δ_i n_i  +  Σ_{i<j} V_ij n_i n_j

n_i = |r⟩⟨r|_i = (1 − Z_i)/2        V_ij = C6 / |r_i − r_j|⁶

For one atom, in the basis (|g⟩, |r⟩), this is the matrix [[0, Ω/2], [Ω/2, −Δ]]. The evolution starts with every atom in |g⟩.

Ω is the Rabi frequency, not half of it. The coefficient of X is Ω/2, so a resonant atom has ⟨n⟩(t) = sin²(Ωt/2): the population oscillates at angular frequency Ω, and ΩT = π is a π pulse. Off resonance, with Ω_eff = √(Ω² + Δ²),
```
⟨n⟩(t) = Ω²/Ω_eff² · sin²(Ω_eff t / 2)
```
With the other common convention, H = Ω X, the same pulse would be a 2π pulse.
Δ is the laser frequency minus the atomic transition frequency. In the laboratory frame the atom has energy ω₀ n; rotating at the laser frequency ω_L subtracts ω_L n, which leaves −(ω_L − ω₀) n = −Δ n. A positive Δ therefore lowers the energy of |r⟩. The interaction enters with the opposite sign: an atom whose neighbours are excited sees the effective detuning Δ_i − Σ_j V_ij n_j, and the doubly excited state of two atoms has the diagonal entry −Δ_0 − Δ_1 + V_01.
Basis ordering. Atom i (counted from 0) is tensor factor i from the left and PennyLane wire i. The basis state |b_0 b_1 … b_{N−1}⟩, with b_i = 1 for |r⟩, has the index Σ_i b_i 2^(N−1−i): atom 0 is the most significant bit. For two atoms the order is |gg⟩, |gr⟩, |rg⟩, |rr⟩, and n_0 = diag(0, 0, 1, 1). This is the matrix qml.matrix(op, wire_order=range(N)) returns.
C6. The default is C6 = 2π × 862 690 MHz µm⁶ = 5 420 441 rad/µs · µm⁶, positive (repulsive), for two ⁸⁷Rb atoms in 70S₁/₂. It is the default of PennyLane's qml.pulse.rydberg_interaction, so a cross-check against PennyLane needs no coefficient passed. The Aquila device description (Wurtz et al. 2023) states 5 420 503 for the same state, 1.1 × 10⁻⁵ larger, which changes no number on this page. As an independent check, Bernien et al. (2017) measured V = 2π × 24.4 MHz between 70S atoms about 5.7 µm apart; the default gives 2π × 24.1 MHz at 5.74 µm. The interaction is taken as isotropic.
The same model in other software. With a drive phase φ the first term reads Ω/2 · Σ_i (cos φ X_i − sin φ Y_i) = Ω/2 · Σ_i (e^{iφ} |g⟩⟨r|_i + h.c.), the form of the trainable analog layer (#420), of qml.pulse.rydberg_drive and of Wurtz et al. (2023). This page fixes φ = 0 and constant Ω and Δ, the special case of one pulse segment; nothing here conflicts with a time-dependent Ω(t), Δ_i(t), φ(t). PennyLane takes its amplitudes in MHz and multiplies by 2π internally; with that conversion its matrix equals this one.

What the two-level model leaves out:

Intermediate-state scattering. The transition is driven with two photons through 6P₃/₂. In Bernien et al. (2017), scattering from that state occurs on a timescale of 40 µs and gives, together with the Rydberg lifetime, an effective lifetime of 50 µs.
Other Rydberg levels. One interaction channel C6/r⁶ stands for the full pair-state structure. It is a long-distance form; this page does not use it below 4 µm.
Atomic motion. Atoms at 12 µK have random Doppler shifts of about 2π × 50 kHz (Bernien et al. 2017), and their positions vary from shot to shot (0.2 µm in Wurtz et al. 2023), which changes V_ij. See Noise for why the Doppler shift is not the dephasing modelled here.
Decay of the Rydberg state, preparation and detection errors, pulse ramps and inhomogeneity of Ω and Δ across the array.

Geometry¶

One shape: an open chain. N atoms on a line at spacing a, atom i at r_i = (i · a, 0), in the order of the inputs: input x_i goes to atom i, so neighbouring inputs sit on neighbouring atoms. N equals the input dimension d. The default sizes are 4 ≤ N ≤ 6; nothing on this page depends on that range. Every formula holds for any N ≥ 1, and the closed-form limits use one to three atoms, so an implementation limits N only from above.

The full 1/r⁶ tail is kept. V_ij = V / |i − j|⁶ with V = C6/a⁶ the nearest-neighbour interaction: next-nearest neighbours interact with V/64, the next with V/729. At this size the tail costs nothing, and truncating to nearest neighbours would be one more approximation to justify.
Blockade radius. R_b = (C6/Ω)^(1/6) is the distance at which two atoms interact with V_ij = Ω, so V/Ω = (R_b/a)⁶ and a = R_b · (V/Ω)^(−1/6). At the reference Ω, R_b = 8.69 µm. A unit-disk graph at radius R_b (the gate-based entangler of #418) connects the pairs with V_ij > Ω. A pair at exactly R_b has V_ij = Ω and lies on the boundary of that graph, as the nearest neighbours do in the V/Ω = 1 column below; which side it falls on is for #418 to fix, and this model does not depend on it.
The regime is not fixed: V/Ω is swept through the spacing, from weakly interacting through V ≈ Ω into the blockade. At the reference Ω:

V/Ω 0.1 0.3 1 3 10 30 100

a in µm 12.76 10.62 8.69 7.24 5.92 4.93 4.03
The non-interacting control is V_ij = 0 exactly, the Hamiltonian without its third term, not a large spacing. Everything else stays the same.

Encoding¶

One input per atom, in a one-signed local detuning:

Δ_i(x_i) = Δ_max · σ(x_i)        σ(x) = 1 / (1 + e^(−x))        Δ_max = √3 Ω

Data and controls. The detunings Δ_i carry the data and nothing else does. Ω, T, the positions and C6 are fixed controls, the same for every sample.
Bounds. Δ_i lies in the open interval (0, Δ_max) for every finite input. The bound is enforced by the logistic function, not by clipping: clipping would map every input beyond the range to the same detuning. A non-finite input is an error, not a value to squash. This holds in exact arithmetic. In double precision σ(x) rounds to 1 for x ≳ 37 and underflows to 0 for x below about −700, so such inputs land on a bound and are no longer told apart. In floating point the interval is therefore closed, and the strict monotonicity stated in the next section holds for inputs short of those values.
Why the logistic function. Any strictly increasing map onto (0, Δ_max) keeps the properties below; the logistic function is fixed here because it uses the range for both input scales the library produces. Standardised inputs within ±2 cover 12 % to 88 % of the range, and PCANormalizer outputs in (−π, π) cover 4 % to 96 %. No scale parameter is added: scaling the inputs is preprocessing.
Why local detuning. It keeps the control informative: without interactions each feature is the single-atom formula of that atom's own input (next section), and no other input enters it.
Why one-signed. Without interactions the excitation probability depends on Δ only through Δ² (the formula above contains Δ only in Ω_eff). A symmetric range would discard the sign of the input in the control. Interactions shift the effective detuning by −Σ_j V_ij n_j and would recover the sign, which would credit them with information the encoding itself threw away.

Pulse¶

Constant Ω and Δ_i for a time T, with ΩT = π. Without interactions and noise, feature i is then, with s_i = Ω_eff,i / Ω = √(1 + (Δ_i/Ω)²),

⟨n_i⟩ = f(s_i),        f(s) = sin²(π s / 2) / s²,        s_i ∈ (1, 2)

f falls strictly from 1 at Δ = 0 to 0 at Δ_max. Its derivative is sin(πs/2) · [π s cos(πs/2) − 2 sin(πs/2)] / s³; on 1 < s < 2 the sine is positive and the cosine negative, so it is negative. The control therefore loses no information about its input: the composition with σ is strictly decreasing in x_i, from 1 as x_i → −∞ to 0 as x_i → +∞, and 0.437 at x_i = 0.
Why Δ_max = √3 Ω. It is the detuning with s = 2, the first zero of f. Beyond it f rises again (to 0.116 at Δ = 2.68 Ω), and two inputs would give the same feature.
Why ΩT = π. It is the shortest pulse with full contrast: the feature reaches 1 at Δ = 0.
The ends are flat. f ≈ 1 − (Δ/Ω)² near Δ = 0 and f ≈ (3π²/64) · (√3 − Δ/Ω)² near Δ_max: injective, but with vanishing slope at both ends.
Time-dependent schedules are out of scope.

Readout¶

Features are excitation probabilities, F_i = ⟨n_i⟩ = Tr[ρ(T) n_i], one per atom, in the order of the atoms.
Pair correlations are optional: ⟨n_i n_j⟩ for i < j, appended in the order (0,1), (0,2), …, (N−2,N−1), which gives N + N(N−1)/2 features.
Both come from the diagonal of ρ alone, as Σ_b b_i ρ_bb and Σ_b b_i b_j ρ_bb over the basis states b. This is a measurement of every atom in the (|g⟩, |r⟩) basis.
Exact expectation values by default. As an option, the features are estimated from S bitstrings sampled from the diagonal of ρ, as the mean of b_i (and of b_i b_j). The estimate is unbiased with standard error √(p(1 − p)/S) ≤ 1/(2√S) for a feature of value p. S has no default and is stated with every result that uses it, so the effect of finite statistics is shown separately from that of the model.

Noise¶

One mechanism: Markovian dephasing during the evolution, given by a rate γ and the evolution time T. The state follows the Lindblad equation

dρ/dt = −i [H, ρ] + Σ_i ( L_i ρ L_i† − ½ { L_i† L_i , ρ } ),        L_i = √γ n_i

with one collapse operator per atom, independent of the others.

What γ means. Between basis states a and b, n_i ρ n_i contributes a_i b_i ρ_ab and the anticommutator −½ (a_i + b_i) ρ_ab, together −½ (a_i − b_i)² ρ_ab. Summed over the atoms,
```
(dρ_ab/dt)_dephasing = −(γ/2) · d_H(a, b) · ρ_ab
```
with d_H the Hamming distance. The coherence of a single atom decays at rate γ/2 (T₂ = 2/γ without a drive), and populations are untouched. Over a time τ the dissipator alone multiplies ρ elementwise by exp(−γ τ d_H(a, b) / 2), exactly.
Other ways to write it. √γ n_i and √(γ/4) Z_i give the same equation. A collapse operator √κ Z_i corresponds to γ = 4κ.

One atom. With p = ρ_rr and c = ρ_gr:

dp/dt = Ω · Im c
dc/dt = −(γ/2 + iΔ) c − i (Ω/2)(2p − 1)

At Δ = 0 this is a damped oscillator, q'' + (γ/2) q' + Ω² q = 0 for q = p − ½, with q(0) = −½ and q'(0) = 0:

γ < 4Ω:   ⟨n⟩(t) = ½ − ½ e^(−γt/4) [ cos λt + γ/(4λ) · sin λt ],     λ = √(Ω² − γ²/16)
γ = 4Ω:   ⟨n⟩(t) = ½ − ½ e^(−Ωt) (1 + Ωt)
γ > 4Ω:   ⟨n⟩(t) = ½ − ½ e^(−γt/4) [ cosh κt + γ/(4κ) · sinh κt ],   κ = √(γ²/16 − Ω²)

Rabi oscillations decay with the envelope e^(−γt/4), and γ = 4Ω is critical damping. After the π pulse, ⟨n⟩ ≈ 1 − πγ/(8Ω) for γ ≪ Ω.

Why local operators. The elementwise action above is exact, and without interactions both H and the dissipator are sums of single-atom terms, so the state stays a product of single-atom states. The control stays analytic under noise: each of its features solves the two equations above for its own Δ_i. Its range shrinks with γ (from 0.72 to 0.21 at γ = Ω), and it stayed strictly decreasing in Δ_i at every rate checked numerically (γ/Ω from 0.03 to 10); that is an observation, not a proof.
Strong dephasing. Every L_i is Hermitian, so the generator is unital and the maximally mixed state 1/2^N is stationary for any H. For Ω > 0 and γ > 0 it is the only stationary state. The Hamiltonian part does not change the purity and the dissipator gives d/dt Tr ρ² = −γ Σ_i ‖[n_i, ρ]‖² (Hilbert-Schmidt norm), so a stationary state commutes with every n_i. An operator that commutes with every n_i is diagonal, and the dissipator vanishes on it, so a stationary one also commutes with H; a diagonal operator that commutes with H is a multiple of the identity, because the drive connects all basis states. So ρ(t) → 1/2^N, every ⟨n_i⟩ → ½ and every ⟨n_i n_j⟩ → ¼ as t → ∞, whatever the input and the interaction.

How fast is set by g, the smallest nonzero decay rate of the Liouvillian (minus the largest real part among its nonzero eigenvalues): at long times the distance from 1/2^N falls as e^(−gt), times a power of t where that eigenvalue is defective (t e^(−gt) at critical damping, γ = 4Ω and Δ = 0), so the limit is reached when gt ≫ 1. γt ≫ 1 does not ensure that. For one atom at Δ = 0 the closed form above gives g = γ/4 up to γ = 4Ω and g = γ/4 − κ above it, exactly. g is largest at γ = 4Ω, where it equals Ω, and falls beyond it, approaching 2Ω²/γ only for γ ≫ 4Ω (the exact γ/4 − κ is 1.64 times that at γ = 4.1 Ω, 1.04 times at 10 Ω): at γ = 100 Ω and Ωt = 10, γt = 1000 and ⟨n⟩ = 0.0905.

The rest of this item is numerical, from the spectrum and the exponential of the Liouvillian of one to four atoms, and is not derived here. Without interactions g is the smallest of the single-atom values; over Δ ∈ [0, Δ_max] and γ/Ω from 0.01 to 1000 a single-atom g was never below 1/3.5 of its Δ = 0 value (lowest ratio at γ = 4Ω, Δ = Δ_max, where g = 0.289 Ω), and a small detuning can raise it. Strong interactions lower g without bound: for V ≫ Ω, γ it is of order γΩ²/V² (at Δ = 0, 0.667 γΩ²/V² for two atoms and 0.432 γΩ²/V² for three, within 2.5 % for V/Ω ≥ 10 and γ/Ω from 0.1 to 3), so two atoms at γ = Ω and Ωt = 40 still have ⟨n_i⟩ = 0.372 at V/Ω = 10 and 0.334 at 100. A point that does reach the limit: γ = 4Ω, Ωt = 100, V/Ω of 0 or 1 puts every element of ρ within 10⁻¹² of 1/2^N for one to four atoms, at every set of detunings in [0, Δ_max] that was sampled.

Two idealisations are part of this choice:

Doppler noise is a different model and is left out. An atom's velocity is constant over one pulse, so its Doppler shift is a detuning offset that is fixed within a shot and random between shots. Its effect is an average of unitary evolutions over that offset, not a Lindblad term.
Decay of the Rydberg state is neglected, which requires T ≪ τ_r. For 70S of ⁸⁷Rb, τ_r ≈ 150 µs at 300 K including blackbody radiation (Bernien et al. 2017, from Beterov et al. 2009). At the reference Ω, T = π/Ω = 0.25 µs, so T/τ_r = 1.7 × 10⁻³: an excited atom decays during the pulse with a probability below 0.2 %. T ≤ τ_r/100 holds for Ω ≥ 2.1 rad/µs.

Closed-form limits¶

Each convention above is fixed by a limit that the tests of #496 and #497 check; the last row is checked by the feature map (#499).

Convention	Closed-form check
Factor of 2 in Ω	One atom, `Δ = γ = 0`: `⟨n⟩(t) = sin²(Ωt/2)`; the matrix is `[[0, Ω/2], [Ω/2, −Δ]]`
Sign of Δ, sign of `C6`	Two atoms at distance `r`: the diagonal entry of the doubly excited state is `−Δ_0 − Δ_1 + C6/r⁶`
Detuned Rabi formula	One atom, `γ = 0`: `⟨n⟩(t) = Ω²/Ω_eff² · sin²(Ω_eff t/2)`
Basis ordering	A detuning on atom `i` alone changes exactly the diagonal entries with `b_i = 1`
Full tail	Three atoms in a chain: the next-nearest coupling is 1/64 of the nearest
Interactions off	`H` is the Kronecker sum of single-atom Hamiltonians, and ρ their tensor product, with and without dephasing
Blockade	Two atoms, `V ≫ Ω`, `Δ = γ = 0`: `⟨n_0 + n_1⟩ → sin²(√2 Ωt/2)` as `Ω/V → 0`, and the doubly excited population stays below `(Ω/V)²` (its maximum over `Ωt ≤ 20` was 0.57, 0.52 and 0.51 times that at `V/Ω` = 10, 30 and 100)
What γ means	No drive: a single-atom coherence decays as `e^(−γt/2)` and `ρ_ab` as `e^(−γ t d_H(a,b)/2)`
Dephasing against the drive	One atom, `Δ = 0`: the damped oscillation above, in all three cases
Strong dephasing	`Ω > 0`, `γ > 0`, any input and any `V`: `ρ → 1/2^N` and every `⟨n_i⟩ → ½` once `gt ≫ 1`, with `g` the smallest nonzero decay rate of the Liouvillian (one atom, `Δ = 0`: `γ/4` up to `γ = 4Ω`, `γ/4 − κ` above). `γt ≫ 1` alone is not sufficient. Reached at `γ = 4Ω`, `Ωt = 100`, `V/Ω` of 0 or 1
Pulse and bound	Interactions off, `γ = 0`: feature `i` equals `f(√(1 + 3σ(x_i)²))` and does not change with any other input

Separation measures¶

A classification score does not say why a fixed feature map helps or fails. The measures below are properties of the map on a set of inputs, computed without any trained head: on the states ρ(x) (RydbergFeatureMap.states), or on the feature matrix F the head actually sees (RydbergFeatureMap.transform, one row per sample), with binary labels y. The feature measures are in hqnn_forge.diagnostics (separation_measures computes all of them); their definitions, derivations and edge cases are in that module's docstring.

Measure	Computed from	Range	Statement it supports
State kernel `K(x, x′) = Tr[ρ(x) ρ(x′)]`, and its centred alignment with the labels	`kernel_from_density_matrices`, `kernel_target_alignment`	`K ∈ [0, 1]`, purity on the diagonal; alignment in `[0, 1]`	An upper bound on what any measurement of these states could distinguish (below)
Pairwise feature distances `‖F(x) − F(x′)‖`, within and between classes, and the ratio of the two means	`pairwise_distances`	Distances in `[0, √N]`; ratio in `(0, ∞]`, about 1 without class information	How far apart this readout puts two inputs, in units of excitation probability: the scale to hold against the shot error `≤ 1/(2√S)` of Readout
Fisher discriminant ratio along the best linear direction, `J = δᵀ S_w⁻¹ δ` with the pooled within-class covariance `S_w` (the squared Mahalanobis distance between the class means)	`fisher_discriminant_ratio`	`[0, ∞]`, 0 for equal class means	How far apart a linear function of the readout puts the class means, in units of the spread within the classes
Effective rank of the feature covariance	`effective_rank`	`[1, min(N, M − 1)]` for `M` samples	In how many directions the readout varies: features collapsing onto fewer directions lower it
Centred alignment of the linear feature kernel `F Fᵀ` with the labels	`linear_feature_kernel`, `kernel_target_alignment`	`[0, 1]`, 1 for features that are affine in the label	How much of the readout's variation is the label, to set beside the alignment of the state kernel

The state kernel bounds what any measurement could distinguish. Its entries give the Hilbert–Schmidt distance of two states, ‖ρ − ρ′‖² = Tr[(ρ − ρ′)²] = K(x, x) + K(x′, x′) − 2 K(x, x′) (the diagonal is the purity, not 1). For any observable A and any number c, because Tr[ρ − ρ′] = 0 and by the Cauchy–Schwarz inequality, with ‖·‖ the Hilbert–Schmidt norm throughout,
```
| Tr[A ρ] − Tr[A ρ′] | = | Tr[(A − c) (ρ − ρ′)] | ≤ ‖A − c‖ · ‖ρ − ρ′‖
```
For the readout, n_i − ½ = −Z_i / 2, and the Z_i / √(2^N) are orthonormal in the Hilbert–Schmidt inner product, so the squared feature differences sum to at most the squared norm of ρ − ρ′ (Bessel's inequality) times 2^N / 4:
```
‖F(x) − F(x′)‖ ≤ ½ √(2^N) · ‖ρ(x) − ρ(x′)‖
```
with equality when ρ − ρ′ is a combination of the Z_i, as for one atom and two diagonal states. This holds for the N excitation probabilities, not for the appended pair correlations. Two inputs with the same state give the same value of every measurement, and the bound is far from tight in general: the state has 4^N − 1 real parameters, the readout N. - The feature measures describe what this readout distinguishes, and nothing about the state beyond it: the readout can leave a large state distance unused, which the bound allows and the distances show. The alignments of the two kernels are not ordered either: a readout that drops variation unrelated to the label can align better than the state kernel. - None of them predicts the test score of a particular head. They are computed on one sample with its labels, without held-out data, a model or a threshold. The Fisher ratio and the feature alignment see linear structure only, which a nonlinear head is not limited to, and both grow on a small sample without any class difference behind them (J = ∞ with fewer samples than features plus two). For M samples of d features in classes of n0 and n1 that do not differ, J is about d · M/(n0 n1) on average, roughly d over the size of the rarer class: values of J from sets of different size or class balance compare only where they are large against that level, and with unequal class covariances J weights the larger class more. The Fisher ratio gives an error rate, Φ(−√J / 2), only for two Gaussian classes of equal covariance and prior, which excitation probabilities are not. - Under strong dephasing every distance falls to 0, since every ⟨n_i⟩ → ½ (see Noise). The ratios, the rank and the alignments do not depend on the scale of the features, so a collapse alone does not change them; they change because inputs lose their trace in the state at different rates. On the way to the limit they moved towards their no-information values in the cases tried (γ = 4Ω, three atoms, V/Ω of 0 and 1: Fisher ratio from 19 and 17 after the π pulse to 1.6 and 1.1 at Ωt = 30); that is an observation. - No monotonicity is claimed. The Hamiltonian depends on the input, so two inputs do not pass through the same channel and the data-processing inequality does not apply to their states directly. Whether dephasing contracts the state or feature distances with γ in this model is an open question, not an assumption of the code.

Hardware realism¶

The comparison is with QuEra's Aquila, an analog device of ⁸⁷Rb atoms using the same Rydberg state, as described by Wurtz et al. (2023) and the Amazon Braket documentation (AWS 2024). A device run of this model is out of scope; this section only says which assumptions one would meet.

Assumption	On a current device	Status
Hamiltonian, units, `C6`	Aquila's native Hamiltonian has this form, with the phase convention above and `C6` of `70S`	Supported
Chain, spacings 4 to 15 µm	Sites at least 4 µm apart in a region 75 µm wide: a row of 6 atoms fits up to `a = 15 µm`, so `0.04 ≤ V/Ω ≤ 105` at the reference Ω	Supported
Fixed positions	Positions vary by 0.2 µm from shot to shot, which near `a = R_b` spreads `V` by about 20 % (`6 √2 · 0.2 µm / a`); the same expression gives about 42 % at `a = 4.03 µm`. Wurtz et al. advise placing atoms well inside or outside `R_b`	Idealisation; the spread of `V` grows as the spacing shrinks
Pulse bounds	`Ω ≤ 15.8 rad/µs`, `−125 ≤ Δ ≤ 125 rad/µs`, at most 4 µs: the reference `Ω = 12.57`, `Δ_max = 21.8 rad/µs` and `T = 0.25 µs` are inside	Supported
Constant pulse	Ω may change by at most 250 rad/µs², so each edge of the pulse takes 0.05 µs of the 0.25 µs	Idealisation
Local addressing: one detuning per atom	Ω and the phase are global. A local detuning `−Δ_local(t) · h_k n_k` with a static pattern `0 ≤ h_k ≤ 1` exists as an experimental capability, on request (AWS 2024); with the global detuning it reaches a one-signed range. Its magnitude limits and the resolution of `h_k` were not verified for this page	Idealisation
Dephasing rate	Aquila reports `T₂* = 5.8 µs`, an echo time of 11.4 µs and a Rabi decay time of 7.5 µs. Read as this model's `2/γ`, `2/γ` and `4/γ`, they give `γ` between 0.18 and 0.53 rad/µs, so `γ/Ω` between 0.014 and 0.042 at the reference Ω. The ratio grows as Ω falls: the slowest π pulse within 4 µs, `Ω = π/4 rad/µs`, gives 0.22 to 0.68, where a row of 6 atoms reaches only `V/Ω ≥ 0.6` (`R_b = 13.8 µm`); the whole sweep down to `V/Ω = 0.1` fits for `Ω ≥ 4.8 rad/µs`, where `γ/Ω ≤ 0.11`	Indicative only: those decays include shot-to-shot noise and loss, which are not Markovian dephasing. `γ/Ω` well above 0.7 is a stress test, not a device regime
No decay	`T/τ_r = 1.7 × 10⁻³` for the Rydberg lifetime, `5 × 10⁻³` against the 50 µs effective lifetime of Bernien et al.	Good approximation
No detection error	Aquila misreads a ground-state atom as excited with probability 0.01 and an excited atom as ground with 0.08; Bernien et al. report detection fidelities of 98 to 99 % and 93 %. With independent errors `ε` and `ε′` a feature becomes `ε + (1 − ε − ε′) ⟨n_i⟩`	Idealisation
Exact expectation values	1 to 1000 shots per task at fewer than 10 shots per second, and each input sample is its own task; about 100 shots are typical	Idealisation; the shots option models the statistics

The encoding is an idealisation. A detuning that is set freely and independently for each atom is more than current devices offer as a standard capability. A device-compatible variant would have to give up:

the exact Δ_i: each sample's detunings become one pattern h_k of finite resolution, scaled by a waveform shared by all atoms;
the square pulse and exact features: ramped edges, and at most 1000 shots per sample with the detection errors above;
on a device with global detuning only, one input per atom itself. The inputs would have to enter global pulse parameters or the positions, and the control without interactions would no longer have features that each depend on one input.

References¶

AWS (2024), "Submit an analog program using QuEra Aquila", Amazon Braket Developer Guide, https://docs.aws.amazon.com/braket/latest/developerguide/braket-quera-submitting-analog-program-aquila.html; and Komar, Bylinskii, Becker & Lin (2024), "Local detuning now available on QuEra's Aquila device with Braket Direct", AWS Quantum Technologies Blog, 17 April 2024.
Bernien, Schwartz, Keesling, Levine, Omran, Pichler, Choi, Zibrov, Endres, Greiner, Vuletić & Lukin (2017), "Probing many-body dynamics on a 51-atom quantum simulator", Nature 551, 579–584, doi:10.1038/nature24622, arXiv:1707.04344.
Beterov, Ryabtsev, Tretyakov & Entin (2009), "Quasiclassical calculations of blackbody-radiation-induced depopulation rates and effective lifetimes of Rydberg nS, nP, and nD alkali-metal atoms with n ≤ 80", Physical Review A 79, 052504, doi:10.1103/PhysRevA.79.052504.
Wurtz, Bylinskii, Braverman, Amato-Grill, Cantu, Huber, Lukin, Liu, Weinberg, Long, Wang, Gemelke & Keesling (2023), "Aquila: QuEra's 256-qubit neutral-atom quantum computer", arXiv:2306.11727.