Neutral-Atom Pulse Control
Five experiments in turning a Rydberg Hamiltonian into a physical two-qubit pulse
The question
A neutral-atom gate begins as a compact equation and ends as voltages, optical fields, clock ticks, and counts. Between those endpoints sit several distinct optimization problems. A pulse that is perfect for a closed Schrödinger equation may be fragile to a one-percent calibration shift. A pulse that is smooth in an optimizer may be reshaped by the control channel. A reduced state representation may optimize quickly while silently forgetting a phase that makes the gate a CZ.
This series follows those failure modes rather than hiding them. The first four chapters study a locally phase-equivalent CZ in a two-qutrit model: first as a coherent gate, then as a constrained trajectory, and finally as an open-system process. The fifth chapter deliberately changes the task. It translates a hardware-feasible Bell-state pulse to Pulser and develops a proposed validation run. That Bell pulse is not presented as a native hardware execution of the earlier CZ.
The intended reader knows bras, kets, matrices, and elementary quantum gates. Control theory is built from the ground up: state and control, costate and Pontryagin’s maximum principle, forward/backward gradients, randomized bases, and direct collocation.
Reading map
1 · Model
From the physical $g-e-r$ ladder to the effective $\lvert0\rangle,\lvert1\rangle,\lvert r\rangle$ Hamiltonian, blockade, CZ, and PMP.
Read Part 1 →2 · Pulse search
GRAPE, Krotov, and CRAB solve related—but not identical—finite-dimensional searches.
Read Part 2 →3 · Trajectories
Direct collocation and Piccolo make the trajectory and its constraints decision variables.
Read Part 3 →5 · Hardware
Clock grids, modulation, geometry, SPAM, and an explicitly unexecuted run plan.
Read Part 5 →What is being optimized
The control vector is
\[u(t)=\bigl(\Omega_x(t),\Omega_y(t),\Delta(t)\bigr),\]with a hard Rabi-amplitude limit and, in the collocation experiments, endpoint, slew, and curvature limits. The coherent CZ objective uses the computational block $M$ of the propagated unitary and optimizes over a local single-qubit phase,
\[F_{\mathrm{tr}}=\max_\theta\frac{\left|\operatorname{tr} \left(CZ_\theta^\dagger M\right)\right|^2}{16},\qquad CZ_\theta=\operatorname{diag}(1,e^{i\theta},e^{i\theta},-e^{2i\theta}).\]This is a squared trace-overlap convention. Part 4 reports process fidelity, survival, and the leakage-aware unconditional average fidelity
\[F_{\mathrm{avg}}^{\mathrm{uncond}} =\frac{4F_{\mathrm{pro}}+s}{5},\qquad s=\frac14\operatorname{tr}[P\mathcal E(P)].\]The earlier trace-preserving approximation $(4F_{\mathrm{pro}}+1)/5$ is retained only for provenance. Part 5 reports Bell-state population/fidelity. These labels are intentionally different: the numbers should not be placed in one leaderboard as though they were the same metric.
Experiment map
| Chapter | Scientific question | Main artifact |
|---|---|---|
| 1 | What Hamiltonian and phase convention define the CZ? | analytic baselines, blockade scan, canonical constants |
| 2 | How do three pulse-search parameterizations behave? | GRAPE, Krotov, CRAB, and reference pulses |
| 3 | What changes when states and controls are co-optimized? | constrained CZ trajectories and Piccolo findings |
| 4 | How do dissipation and uncertainty reorder the solutions? | Lindblad scores, exposure, open- and robust-control pulses |
| 5 | What survives translation to a device-facing sequence? | delivered waveforms, geometry audit, anonymized run plan |
Conventions and provenance
Angular controls are stored in rad/µs and displayed as ordinary frequencies, $\Omega/2\pi$ and $\Delta/2\pi$, in MHz. Durations are shown in ns when discussing gates or hardware clocks and in µs for differential equations. Every quantitative figure is regenerated from the saved JSON artifacts; the optimizers are not rerun.
The series is a report of a particular experimental path, not a claim that one method is universally superior. The useful comparison is structural: which variables and constraints a method can express, which derivatives it needs, and which physical errors remain outside its model.