5 · From Optimized Arrays to Hardware Candidates
Pulser sequences, clocks, channel modulation, geometry, anonymized SPAM calibration, and an unexecuted run plan
A deliberate change of task
Parts 1–4 optimize a gate in an effective three-level basis ${\lvert0\rangle,\lvert1\rangle,\lvert r\rangle}$ for each atom. The available Pulser AnalogDevice abstraction exposes a globally driven two-level ${\lvert g\rangle,\lvert r\rangle}$ manifold. It does not, by itself, encode the same computational qubit, local phase corrections, and three-level CZ semantics.
Pretending otherwise would make the cleanest-looking result in the series scientifically wrong. The hardware bridge therefore uses a smaller hardware-native problem: prepare
\[|\Psi^+\rangle=\frac{|gr\rangle+|rg\rangle}{\sqrt2}\]from $\lvert gg\rangle$, using global $\Omega(t)\ge0$, detuning $\Delta(t)$, and the same $C_6/R^6$ interaction. Bell preparation exercises blockade, entanglement, waveform translation, noise, and measurement without claiming that the earlier CZ was executed.
This change of task is a model-interface decision, not a statement that Bell preparation is equivalent to a two-qubit gate. A state-transfer experiment tests one input and one target ray. A gate must act correctly on every superposition in the computational subspace. In particular, the Bell target is insensitive to several phases that a CZ objective must retain. High Bell fidelity can validate blockade dynamics and the delivery pipeline while leaving the native-CZ question open.
The distinction also prevents a basis-label shortcut. Relabeling $\lvert g\rangle$ as $\lvert1\rangle$ does not create the dark $\lvert0\rangle$ level or the local phase corrections assumed by the qutrit model. A hardware-facing demonstration must either use a device abstraction that represents those levels and couplings or specify an encoded protocol that realizes the same logical map. Until then, the honest claim is a hardware-feasible blockade-mediated entangled-state candidate.
Arrays are not waveforms
The two Bell-state candidates begin as arrays in rad/µs:
| Spacing | Optimizer duration | Slices | Coherent Bell infidelity |
|---|---|---|---|
| 5.0 µm | 300 ns | 30 | below $10^{-8}$ |
| 6.5 µm | 1500 ns | 75 | below $10^{-8}$ |
The exporter converts each into a Pulser Sequence
. Two transformations are already lossy:
- Duration must lie on the channel’s 4 ns clock. The exporter rounds down rather than silently exceeding the device duration after fall time is added.
- Slice-center values become an interpolated waveform. The export path uses shape-preserving interpolation rather than treating the optimizer’s array as an analog signal.
If the channel clock is $\tau_c=4$ ns, a requested duration becomes an integer $N_c\tau_c$. The rounding policy is part of the experiment: rounding upward can violate a maximum duration once the fall time is included, whereas rounding downward changes the integrated pulse area. The audit records both the optimizer grid and serialized clock grid so this change is inspectable.
A clock-rounding example
Suppose an optimizer returns a 350 ns pulse while the channel accepts durations only in 4 ns increments. The neighboring clock-compatible durations are 348 ns and 352 ns. If an 85 ns fall-time allowance must also fit beneath a hard sequence limit, choosing 352 ns may make the complete request invalid even though it is closer to the optimizer value. Choosing 348 ns avoids that overrun but removes two nanoseconds of pulse area and shifts every normalized sample time.
That two-nanosecond change is small compared with the gate, yet it is not mathematically zero. For a constant angular Rabi rate $\Omega$, shortening by $\delta t$ changes rotation area by $\Omega\delta t$. For a shaped waveform it also changes interpolation locations and the phase accumulated under detuning. The exported sequence must therefore be scored after quantization; recording only the nominal optimizer fidelity would skip the first hardware transformation.
The time-grid convention matters just as much. Thirty optimizer values over 300 ns can mean thirty 10 ns holds, thirty slice-center samples, or thirty knots including endpoints. Those interpretations have different supports. The exporter treats the stored values according to their recorded convention, constructs the requested continuous curve, and then samples it on the device grid. A provenance record should preserve all three axes: optimizer time, programmed clock time, and delivered-channel time.
Interpolation answers a separate question. An optimizer value may represent a slice center, a left endpoint, or a knot of a continuous interpolant. Treating those conventions as interchangeable creates a half-step timing error. Shape-preserving cubic interpolation avoids some overshoot, but it does not manufacture bandwidth feasibility; it only defines the requested continuous curve sampled by Pulser.
The sequence then checks amplitude, detuning, clock alignment, minimum and maximum duration, atom spacing, radial extent, and delivered slew. It is serialized to Pulser’s abstract representation and reconstructed; the nominal samples are required to survive that round trip exactly.
That serialization round trip checks units, duration, waveform support, register coordinates, and phase segmentation. It does not check the physics of the emitted optical field. The abstract representation is a device-facing program, not a certificate that the requested array is reachable after the analog channel.
The analytic Levine–Pichler CZ supplies a useful representation test. Its one phase jump can be represented by two consecutive constant-phase pulses. A continuously phase-modulated CZ pulse would require many such segments. Exact syntax for a phase boundary does not guarantee accurate device physics between boundaries.
A complex control can be written $\Omega_x+i\Omega_y=Ae^{i\phi}$. Pulser’s global channel expresses nonnegative amplitude plus phase. A piecewise-constant phase jump can therefore be segmented exactly at a clock boundary. A rapidly winding phase would need many segments or a phase-modulation primitive, and finite phase-update bandwidth would become another transfer function. This is why the quadrature plots in Part 2 cannot be translated by plotting $\lvert\Omega\rvert$ alone.
Phase conversion has a delicate corner at vanishing amplitude. The angle $\phi=\operatorname{atan2}(\Omega_y,\Omega_x)$ is undefined when both quadratures are zero, even though the physical drive is off. An exporter must carry the previous phase, choose a documented default, or omit a zero-amplitude segment; arbitrary phase jumps at zero can otherwise create needless instructions. Phase unwrapping is only a display and segmentation aid because phases differing by $2\pi$ describe the same complex field.
Segmentation also introduces a resolution choice. Approximating a continuously varying phase with more constant-phase pieces reduces representation error but increases program complexity and may exceed update-rate constraints. Fewer pieces are easier to serialize but can alter the trajectory. The correct number is not determined by how smooth a plot looks; it is established by convergence of the delivered-waveform rollout under finer segmentation and by the actual phase-control specification.
What the channel delivers
The programmed sequence is not the emitted optical field. The channel has an 8 MHz modulation bandwidth and an 85 ns rise time. Pulser’s modulation model adds rise and fall tails and smooths fast structure. The analysis samples both the nominal and modulated sequences, then independently propagates the delivered arrays.
The delivered grid is longer than the programmed grid because the filter rings up and down. In the figure, the blue curve ends at the programmed duration while the orange curve continues through the shaded tail on its own time axis. Comparing both arrays against optimizer slice index would hide this extra evolution.
The shaded tail is dynamically active. Once the programmed samples stop, the filtered field does not instantly become zero. The state continues to evolve under the residual drive, detuning, and interaction until the modeled output settles. Truncating propagation at the nominal optimizer duration would therefore score a field that the channel never delivers. Conversely, appending zeros to the optimizer array without filtering would not reproduce the same ring-down.
Bandwidth changes phase as well as amplitude. Filtering $\Omega_x$ and $\Omega_y$ separately can alter the instantaneous angle of the complex envelope. A model expressed as nonnegative amplitude and phase may implement a different internal transfer path. The export audit should follow the representation used by the device abstraction and compare the reconstructed complex field, not assume that smoothing the amplitude alone captures the channel.
The modest fidelity reduction for these Bell candidates is an observed property of this specific translation. It should not be read backward as proof that the irregular samples were physically smooth, or forward as a guarantee for other pulses. A resonance between filter delay and a phase jump could produce a much larger error. Delivered-waveform propagation is therefore a required scoring stage for every candidate, not a one-time validation of the export code.
At $R=5\,\mu\mathrm m$, the sequence representation gives $F=0.999980$ before modulation and $F=0.999915$ after it. At $R=6.5\,\mu\mathrm m$, the corresponding values are $0.999967$ and $0.999058$. The programmed Bell arrays are visibly irregular. They nevertheless retain high simulated Bell fidelity after this particular Pulser modulation model. That is an empirical result for these two saved arrays, this state-transfer target, and this filter; it is not “smoothness by construction,” and it cannot be generalized to the jagged CZ pulses from Part 2 without translating and independently propagating those pulses too.
This closes a loop back to Part 3. Slew limits did not merely make the collocation plots look nicer. They were a way of buying robustness to the physical channel before the Pulser representation existed.
Geometry is a constraint
The abstract sequence can place two atoms at any validated continuous coordinates. A real machine can expose a calibrated trap layout with only certain pair separations. In the selected device configuration, 5.0 µm is available but 6.5 µm is not; the next useful spacing is 8.66 µm.
That means the 6.5 µm pulse is a simulator candidate, not a machine-ready candidate. It should be reoptimized at a realizable spacing rather than rounded to the nearest trap. Because $V=C_6/R^6$, a geometric change is a Hamiltonian change, not a minor placement error.
The three points use the same recorded $C_6$, not a fitted curve. The 6.5 µm point is valuable for simulation but absent from the selected calibrated layout. The 8.66 µm point is available, yet its much weaker blockade defines a different control problem. A sound workflow enumerates feasible geometry first and optimizes on each candidate Hamiltonian.
Geometry also connects pulse-level control to analog neutral-atom programming. In an analog independent-set protocol, atom coordinates encode graph adjacency through the blockade radius; the unit-disk mapping tutorial develops that workflow in detail. Here only one pair is considered, but the design rule is the same: coordinates enter the Hamiltonian before they enter a drawing. A layout validator cannot repair a pulse optimized for the wrong $V$.
At 8.66 µm, the interaction is reduced by the sixth power of the spacing ratio. The single-excitation drive can then compete with, rather than be dominated by, the shift of $\lvert rr\rangle$. Population paths and conditional phase change together. Retuning duration alone may be insufficient; amplitude, detuning, and waveform shape should all be optimized for the available pair. If several calibrated pairs exist, geometry can be an outer discrete choice and pulse design an inner continuous optimization.
Real layouts add uncertainty around nominal coordinates. Thermal motion and trap calibration produce a distribution of $R$, which becomes a strongly nonlinear distribution of $V$. A hardware candidate can include several interaction strengths in its robust ensemble, but the range should come from position calibration. This is another reason to postpone the label “hardware pulse” until register choice and its uncertainty are part of the model.
Aggregate hardware calibration
The calibration model uses aggregate counts from two prior QPU batches totaling 1,100 shots. The public draft intentionally removes job IDs, account/team metadata, layout identifiers, and the submission recipe. A simple readout-and-preparation model fits:
| Parameter | Fitted probability |
|---|---|
| false positive | 1.388% |
| false negative | 8.179% |
| state-preparation error | 0.326% |
The asymmetry is important. A single symmetric “measurement error” would hide the large false-negative component. Within this simple model, the observed floor is dominated by state preparation and measurement (SPAM); the residual assigned to coherent dynamics is less than roughly two percentage points. This is an inference from a small calibration set, not a complete device-noise reconstruction.
For one atom, a minimal asymmetric readout model is
\[\begin{pmatrix}P(m=0)\\P(m=1)\end{pmatrix} = \begin{pmatrix} 1-p_{\mathrm{fp}} & p_{\mathrm{fn}}\\ p_{\mathrm{fp}} & 1-p_{\mathrm{fn}} \end{pmatrix} \begin{pmatrix}P(0)\\P(1)\end{pmatrix}.\]The false-positive and false-negative probabilities need not match. State-preparation error is applied before this confusion matrix, so it should not be folded into one “readout fidelity.” With aggregate counts, the fitted decomposition is useful for planning but not uniquely diagnostic of every physical mechanism.
For two atoms, the simplest extension takes a tensor product of the one-atom confusion matrices. That assumes independent readout errors. Crosstalk, correlated loss, or a state-dependent detection process would require additional parameters that aggregate counts may not identify. State-preparation error is likewise not uniquely separated from early dynamical error without dedicated calibration sequences. The fitted values should therefore be used to forecast observed populations, not narrated as a complete microscopic noise model.
Inverting a confusion matrix can estimate pre-readout populations, but mitigation trades bias for variance. A large false-negative asymmetry can amplify shot noise, and an unconstrained inverse may even produce negative estimated probabilities in a finite sample. A validation report should show raw counts first, document the calibration matrix and its uncertainty, and present mitigated values as a secondary analysis with propagated confidence intervals.
The roughly ten-percent gap between an ideal Bell target and a calibrated observed population is therefore not evidence that another decimal of coherent optimizer fidelity will help. Better discrimination, measurement mitigation, and a validation protocol that separates SPAM from dynamics are higher-leverage.
The proposed run
A robust-GRAPE Bell candidate at $R=5\,\mu\mathrm m$ is selected for the proposed run:
- 350 ns and 35 optimization slices;
- amplitude and detuning exported to a clock-compatible Pulser sequence;
- 1,000 proposed shots;
- calibrated prediction $P_{\mathrm{Bell}}\approx0.898$; and
- a preregistered comparison against aggregate prior behavior.
A useful validation would record, before looking at the outcome, the target statistic, confidence interval or acceptance band, treatment of invalid/missing atoms, and the decision rule for whether the new candidate improves on the control. Without that preregistration, it is too easy to move between parity, Bell population, raw success, and mitigated success after the counts arrive.
For 1,000 independent shots and a predicted success probability near 0.898, the binomial standard error is
\[\sqrt{p(1-p)/1000}\approx0.0096.\]A practical protocol should include a contemporaneous reference sequence, predeclare whether raw or SPAM-corrected counts determine acceptance, and attach a confidence interval to their difference. It should record atom-loss handling, register validation, pulse hash, software/device version, and the exact abstract sequence. These safeguards separate a failed waveform from drift, layout mismatch, or a post-processing choice.
One concrete protocol interleaves candidate and reference batches rather than running all reference shots first. Interleaving reduces confounding from slow drift. Each batch logs the validated register and sequence hash, and analysis uses a fixed rule for missing atoms. The primary endpoint can be the raw probability of the two single-excitation outcomes that define the Bell-population proxy; a phase-sensitive parity or tomography measurement would be required to certify coherence rather than population alone.
The acceptance rule should compare the candidate with the contemporaneous reference, not only with a historical point estimate. For example, predeclare a minimum improvement and require the lower confidence bound on the difference to exceed it. If SPAM correction is reported, calibrations collected in the same session should be propagated through the uncertainty analysis. With 1,000 shots, statistical precision is around one percentage point, so claims about improvements much smaller than that need either more shots or a paired design with lower variance.
Four evidence levels remain distinct:
| Level | What has been established |
|---|---|
| simulation | a mathematical pulse reaches the target in a recorded model |
| hardware-compatible | Pulser validates and serializes the selected device abstraction |
| prior aggregate calibration | anonymized historical counts constrain a simple SPAM model |
| proposed experiment | a 1,000-shot candidate and decision protocol exist; no job was submitted |
Moving down this table requires new evidence; it is not a change of wording. Simulation needs reproducible arrays, units, and a dynamical model. Hardware compatibility adds a validated device description and serialized sequence. A measured claim adds execution records, counts, contemporaneous calibration, and an analysis protocol. Passing one level does not imply the next, and failure at a later level does not erase what was learned at an earlier one.
This evidence ladder is useful beyond neutral atoms. It prevents a simulator object from being described as an emitted waveform and prevents a calibrated prediction from being described as an observation. Keeping hashes and transformations makes disagreements traceable: one can ask whether the optimizer array, exported sequence, delivered model, or measurement interpretation first changed the conclusion.
What remains for native CZ
The hardware bridge does not end the original problem. A native CZ demonstration still needs:
- a device model that represents the computational $\lvert0\rangle,\lvert1\rangle$ states and their coupling through $\lvert r\rangle$, or a defensible encoded protocol in the available interface;
- explicit handling of local $Z$ phases and the CZ coset;
- optimization at a realizable trap spacing;
- channel modulation inside the optimization or robust ensemble;
- a noise model calibrated from suitable experiments; and
- a measurement protocol capable of distinguishing a gate from one state-preparation trajectory.
That boundary is the main lesson of the series. Optimal-control software is necessary because a realistic pulse is a constrained trajectory through several models. It is not sufficient because every model transition—atomic reduction, target gauge, discretization, noise, channel, geometry, and measurement—must be validated in its own language.