4 · The Best Coherent Pulse Is Not Always the Best Physical Pulse
Lindblad dynamics, ensemble robustness, Rydberg exposure, and a reversal of the closed-system ranking
Three meanings of error
By the end of Part 3 we have several high-fidelity pulses, but “high fidelity” still means “under the Hamiltonian used by the optimizer.” This chapter separates three errors that are easy to collapse into one word:
- Coherent control error: the unitary misses the desired CZ because of duration, leakage, discretization, or a suboptimal waveform.
- Dissipative error: Rydberg decay and dephasing make the evolution nonunitary even when the waveform is exactly calibrated.
- Model uncertainty: the experiment implements a slightly different Rabi rate, detuning, interaction, or transfer function from the nominal model.
A pulse can be excellent against one and poor against another. Robust control does not mean adding one universal “robustness term”; it means deciding which uncertainty set or noise channel matters and optimizing an appropriate aggregate.
Open-system dynamics
The open-system comparison uses a Lindblad master equation
,
\[\dot\rho=-i[H(t),\rho] +\sum_j\left(L_j\rho L_j^\dagger -\frac12\{L_j^\dagger L_j,\rho\}\right).\]For each atom, the model includes Rydberg relaxation $\lvert r\rangle\rightarrow\lvert1\rangle$ at $0.01\,\mu\mathrm s^{-1}$ and Rydberg dephasing at $0.05\,\mu\mathrm s^{-1}$. These are the Pulser simulation defaults used by the simulator configuration, not fitted constants for a particular machine.
Writing $n_r=\lvert r\rangle\langle r\rvert$, the four collapse operators are
\[\begin{aligned} L_{1,\mathrm{rel}}&=\sqrt{\gamma_1}\,|1\rangle\langle r|\otimes I,& L_{2,\mathrm{rel}}&=\sqrt{\gamma_1}\,I\otimes|1\rangle\langle r|,\\ L_{1,\phi}&=\sqrt{\gamma_\phi}\,n_r\otimes I,& L_{2,\phi}&=\sqrt{\gamma_\phi}\,I\otimes n_r, \end{aligned}\]with $\gamma_1=0.01\,\mu\mathrm s^{-1}$ and $\gamma_\phi=0.05\,\mu\mathrm s^{-1}$. Relaxation changes population; the projector collapse operator damps coherences involving $\lvert r\rangle$ without directly removing Rydberg population. The Liouvillian is exponentiated on every saved time slice. Before using it, the audit checks trace preservation on the full nine-level space, checks the zero-collapse limit against coherent propagation, and verifies a representative coherence against an independent master-equation integrator.
Reading the collapse operators physically
The relaxation operator $\sqrt{\gamma_1}\lvert1\rangle\langle r\rvert$ has a simple direction: it removes amplitude from the Rydberg level and deposits population in the addressed logical level. In a quantum-trajectory picture, a jump reveals that the atom was in $\lvert r\rangle$ and destroys the phase relation with branches that did not jump. The master equation averages over observed and unobserved jump histories, producing both population transfer and loss of coherence.
The dephasing operator $\sqrt{\gamma_\phi}n_r$ is different. Because it is diagonal, it does not directly move population between levels. It damps an off-diagonal element such as $\lvert1\rangle\langle r\rvert$, which encodes phase coherence between logical and Rydberg components. A global gate that temporarily places two computational branches at different Rydberg populations is therefore vulnerable even if both branches return at the endpoint.
Tensoring each single-atom operator with the identity produces independent local noise. This model does not include correlated laser noise, collective decay, black-body transfer to other Rydberg states, or loss outside the three-level basis. Nor does a quoted rate by itself specify the physical fidelity: the waveform determines how much population and coherence are exposed to that rate over time. The following comparison holds rates fixed and lets the trajectories supply that missing exposure information.
The metric also changes. Let $\mathcal E$ be the full channel and $P$ the computational projector. Projecting the output back to the four-dimensional computational space gives a generally trace-decreasing map when leakage is present. For target $CZ_\theta$, define $F_{\mathrm{pro}}$ from the sixteen computational matrix units and define survival
\[s=\frac{1}{4}\operatorname{tr}[P\mathcal E(P)].\]The unconditional Haar-average fidelity is then
\[F_{\mathrm{avg}}^{\mathrm{uncond}} =\frac{4F_{\mathrm{pro}}+s}{5},\]not $(4F_{\mathrm{pro}}+1)/5$ unless the projected process is trace preserving. The original numerical analysis used the latter legacy convention. This revision preserves that value for provenance and independently propagates every saved pulse to record $F_{\mathrm{pro}}$, $s$, and the corrected unconditional score. The target phase $\theta$ is optimized in the same CZ coset for every method.
There is also a conditional question—fidelity given that the population survived—but it answers a different experimental protocol and is not used for the ranking below. A single reported “average fidelity” is therefore incomplete unless it states the target, projection, survival convention, and whether the result is conditional.
The matrix-unit construction makes the bookkeeping concrete. For computational basis operators $E_{ij}=\lvert i\rangle\langle j\rvert$ and target $U_\theta=CZ_\theta$,
\[\begin{aligned} F_{\mathrm{pro}}(\theta)=\frac{1}{16} \sum_{i,j=0}^{3} \operatorname{tr}\!\bigl[ &(U_\theta E_{ij}U_\theta^\dagger)^\dagger\\ &\times P\mathcal E(E_{ij})P \bigr]. \end{aligned}\]The scorer maximizes this expression over $\theta$. Survival uses the four diagonal $E_{ii}$ inputs and measures their average retained trace. For a noiseless but leakage-bearing unitary, the projected map is still trace decreasing, so the survival correction is not solely a Lindblad issue.
Numerically, the full Liouville superoperator acts on an 81-component vectorized density matrix. The audit fixes column-major vectorization and checks $\mathrm{vec}(A\rho B)=(B^\mathsf T\otimes A)\mathrm{vec}(\rho)$. A transpose in the wrong place can preserve trace and still corrupt phase-sensitive process fidelity. This is why trace preservation, the coherent limit, and a representative off-diagonal operator are tested separately.
Why survival appears in average fidelity
For a trace-preserving channel on a $d$-dimensional system, the familiar relation $F_{\mathrm{avg}}=(dF_{\mathrm{pro}}+1)/(d+1)$ contains a one because every input leaves unit trace in the scored space. Projection changes that premise. If ten percent of an input ends in a Rydberg state, the projected computational output has trace 0.9. Giving the missing branch an implicit perfect score by retaining the one would overestimate the unconditional gate quality.
The correction replaces that one with average survival $s$. For $d=4$, imagine a projected process that performs the target perfectly whenever it survives but retains only $s=0.9$ on average. Its process overlap scales with the surviving part, and the unconditional expression counts loss as failure. A conditional fidelity could divide by survival and answer “how correct was the output among retained events?” That can be appropriate for a heralded protocol, but an ordinary deterministic gate cannot silently postselect leakage away.
Process-map validation also needs more than propagating four populations. The diagonal matrix units test how basis populations move; the off-diagonal $E_{ij}$ carry relative phase and coherence. A map can preserve every computational population and still dephase the superpositions needed for a gate. Propagating all sixteen matrix units reconstructs the projected linear map without assuming it remains unitary.
Several numerical checks catch complementary mistakes. Hermiticity preservation tests whether Hermitian inputs stay Hermitian. Complete positivity can be inspected through the Choi matrix, allowing small negative eigenvalues only at integration tolerance. Full-space trace preservation checks the Lindblad implementation, while projected trace loss is a physical leakage signal rather than a failure. Finally, the zero-rate limit must agree with unitary propagation under the same samples and time convention.
The exposure budget
Dissipation acts mainly while population occupies a Rydberg level. Define the integrated Rydberg exposure
\[\mathcal E_r=\int_0^T \operatorname{tr}\!\left[\rho(t)(n_r^{(1)}+n_r^{(2)})\right]dt.\]Across the saved pulse family, the observed noisy infidelity is well summarized by
\[1-F_{\mathrm{avg}}^{\mathrm{uncond}} \approx (1-F_{\mathrm{coh}})+\kappa\mathcal E_r.\]This is an empirical law for the selected model and pulses, not a new universal decoherence formula. Its usefulness is conceptual: coherent accuracy and Rydberg exposure are two budgets. Improving one while spending much more of the other can make the physical answer worse.
The fitted $\kappa$ in the revised figure is a descriptive slope after subtracting each pulse’s independently propagated coherent trace error. It is not either collapse rate. Different trajectories distribute exposure between one- and two-Rydberg states and between populations and coherences, so equal $\mathcal E_r$ need not imply equal loss. The near-linearity is useful for intuition and screening; the full master equation is the score used for ranking.
Exposure as an accounting exercise
If one atom carried Rydberg population 0.5 for 200 ns and otherwise stayed in the computational space, its contribution to exposure would be $0.5\times0.2\,\mu\mathrm s=0.1\,\mu\mathrm s$. If both atoms simultaneously carried population 0.5 for the same interval, the definition would count $0.2\,\mu\mathrm s$. A trajectory may reach the same endpoint with a short, highly excited excursion or a longer, weakly excited excursion; exposure integrates those choices into one first-order budget.
For small rates, the probability of a relaxation event is approximately the rate times the relevant integrated population. This perturbative intuition motivates a linear trend. Dephasing depends on coherences as well as populations, double excitation changes the weighting, and finite rates modify the trajectory itself, so the exact answer is not simply $(\gamma_1+\gamma_\phi)\mathcal E_r$. The fitted $\kappa$ absorbs those differences only for this pulse family and noise model.
Exposure is therefore a useful design diagnostic but not a replacement objective by itself. Driving exposure to zero would forbid the Rydberg mechanism that creates the entangling phase. The meaningful question is whether the pulse accumulates the required phase efficiently: how much useful nonlocal action is obtained per unit of dissipative exposure, subject to coherent fidelity and control constraints.
The GRAPE pulse is the clearest example. Its saved coherent error is below numerical reporting precision, but it accumulates $0.464\,\mu\mathrm s$ of Rydberg exposure and has corrected infidelity $1.868\times10^{-2}$. The legacy score was $1.625\times10^{-2}$. CRAB carries only $0.245\,\mu\mathrm s$ exposure; its corrected infidelity is $1.083\times10^{-2}$ rather than the legacy $9.38\times10^{-3}$.
The constrained $F\ge0.9999$ collocation pulse lies between them: coherent error $8.97\times10^{-5}$, exposure $0.328\,\mu\mathrm s$, and corrected noisy infidelity $1.411\times10^{-2}$. Hardware smoothness and decoherence efficiency are related, but they are not the same objective.
Open-system and ensemble optimization
There are two direct responses.
Open-system GRAPE differentiates through the density-matrix propagator and optimizes the noisy objective itself. The open-system GRAPE result at $1\,\mu\mathrm s$ has coherent infidelity $1.43\times10^{-5}$, exposure $0.254\,\mu\mathrm s$, and corrected noisy infidelity $1.100\times10^{-2}$. It learns to avoid expensive Rydberg occupation while retaining coherent accuracy. Its noisy score is close to CRAB’s shorter pulse; the small difference is less important than their shared improvement over coherent-only GRAPE.
Ensemble GRAPE keeps unitary dynamics but optimizes the mean over an uncertainty grid. Here the grid is a $\pm2\%$ amplitude scale error:
\[J_{\mathrm{ens}}(u)=\frac{1}{5}\sum_{\epsilon\in \{-0.02,-0.01,0,0.01,0.02\}} \left[1-F\bigl((1+\epsilon)u\bigr)\right].\]The recorded mean coherent infidelity is $1.03\times10^{-4}$. This number measures amplitude robustness, not Lindblad noise. To compare pulse families consistently, the saved robust pulse is nevertheless propagated afterward through the same Lindblad map; that evaluation is not an open-system reoptimization. An ensemble can also include detuning offsets, interaction uncertainty, or a small set of transfer functions, but the cost grows with every member.
The curve is more informative than a mean. Nominal GRAPE has a very deep and narrow minimum at zero calibration error. The ensemble pulse buys a flatter basin by accepting a higher nominal error. Whether that trade is favorable depends on the actual amplitude distribution and on errors not represented by a single scale factor.
The horizontal axis represents a coherent multiplicative bias applied to the complete drive, not random sample-by-sample noise. At $+2\%$, both quadratures are scaled together, so the optical phase is preserved while rotation rates and pulse areas change. A separate quadrature imbalance would distort phase; an additive offset would affect weak and strong segments differently. Robustness to one axis should not be assumed along those untested directions.
Response curves also reveal asymmetry. A pulse need not react equally to positive and negative scale errors because detuning, finite blockade, and time ordering make the dynamics nonlinear. Training on a symmetric grid can still produce an asymmetric curve. If calibration data show a biased distribution, centering or weighting the ensemble on that distribution is more relevant than enforcing visual symmetry.
Outside the training interval, the robust curve is an extrapolation, not a guarantee. An optimizer can flatten the specified region by moving sensitivity to its edges. A responsible audit therefore plots a somewhat wider range than the training grid and checks for sharp deterioration. It also evaluates nominal coherent fidelity and exposure, since a flatter calibration response can be purchased with a longer or more dissipative path.
These two strategies answer different questions:
| Strategy | Optimizes against | Does not automatically cover |
|---|---|---|
| open-system GRAPE | specified relaxation/dephasing model | calibration and model bias |
| ensemble GRAPE | selected parameter distribution | Markovian decay outside ensemble |
| constrained collocation | explicit path and hardware inequalities | unmodeled stochastic noise |
A realistic design may combine all three: an ensemble of open-system trajectories inside a constrained transcription. That is precisely the kind of formulation for which a trajectory-optimization framework becomes more than convenience.
The computational price follows from the number of trajectories. Closed-system GRAPE may propagate several basis columns. A channel objective propagates a basis of operators in a larger Liouville space. An ensemble repeats those propagations for every calibration member. Combining five amplitude errors with an open-system gate score can therefore multiply an already larger derivative calculation. Parallel propagation, symmetry reduction, and sparse trajectory abstractions become practical necessities rather than cosmetic software features.
The aggregate objective also encodes a risk preference. Minimizing the ensemble mean can sacrifice one edge member if the others improve. Minimizing the worst member creates a nonsmooth minimax problem but protects the stated range more evenly. Weighting members by a calibrated probability distribution emphasizes likely errors, while a uniform grid claims only bounded uncertainty. A “robust pulse” is incomplete terminology unless the ensemble and aggregate are reported.
Open-system gradients retain GRAPE’s forward/backward architecture, but the generator is the Liouvillian. For an interval $S_k=\exp(\mathcal L_k\Delta t)$, the derivative is again a Fréchet derivative $DS_k[D\mathcal L_k]$. The dimension grows from a 9-component ket to an 81-component vectorized operator, and a gate objective samples multiple input operators. Reduced representations and exact derivatives therefore matter even more in open-system control.
Parameter uncertainty is conceptually different. A fixed unknown amplitude scale $\epsilon$ produces a unitary member $U_\epsilon(T)$; an ensemble objective averages or minimizes fidelity over several members. Lindblad dephasing is not equivalent to drawing one static detuning error per experiment. One describes a Markovian channel, the other epistemic or slowly varying model mismatch. They can coexist, but combining them is justified only when calibration data support both models.
Concrete time scales make the distinction clearer. If the Rabi calibration drifts slowly and remains nearly constant during each 1 µs gate, an ensemble of fixed scale factors is reasonable. If the optical phase fluctuates rapidly during a gate with a short correlation time, a Markovian dephasing channel may be a useful approximation. If it fluctuates slowly but randomly between shots, a quasi-static detuning ensemble is closer. Two models can produce similar decay curves in one calibration experiment yet predict different responses to shaped control, so robustness should follow measured noise spectra where possible.
The ranking reversal
Under the common noise evaluator, the leading recorded families are:
| Pulse | Survival $s$ | Legacy $1-F$ | Corrected $1-F_{\mathrm{avg}}^{\mathrm{uncond}}$ |
|---|---|---|---|
| CRAB | 0.992739 | 0.00938 | 0.01083 |
| open-system GRAPE | 0.992493 | 0.00949 | 0.01100 |
| collocation 0.9999 | 0.990528 | 0.01222 | 0.01411 |
| robust GRAPE | 0.989364 | n/a | 0.01464 |
| coherent GRAPE | 0.987827 | 0.01625 | 0.01868 |
| Krotov | 0.987733 | 0.01779 | 0.02024 |
| Levine–Pichler | 0.991180 | 0.02384 | 0.02560 |
This table is the strongest argument against optimizing one scalar in isolation. Coherent GRAPE “wins” Part 2 and loses to four alternatives here. The optimizer did not make a mistake; it was asked the wrong physical question.
Read the table in three passes. First compare survival. Coherent GRAPE and Krotov lose more projected trace than the shorter CRAB pulse, consistent with their larger exposure. Second compare the legacy and corrected columns. Their difference is not a change in the propagation; it is the cost of no longer pretending that leaked probability survived perfectly. Third compare methods whose corrected scores are close. CRAB and open-system GRAPE differ by less than two ten-thousandths in this model, far below the level at which generic rates justify a universal winner.
The ranking is conditional on the selected evaluator. Changing decay rates, adding laser noise, or filtering the controls can reorder it again. A useful decision process therefore keeps a Pareto set rather than one champion: coherent accuracy, duration, exposure, roughness, uncertainty response, and hardware feasibility. Candidates can then be rescored as the physical model improves without rerunning every search immediately.
This perspective also changes how to value an optimizer’s extra decimal. Once dissipative error is near one percent, reducing coherent error from $10^{-5}$ to $10^{-10}$ has little effect unless the change also reduces exposure or duration. Effort is better spent on a control objective aligned with the dominant error budget, or on experiments that identify which budget is actually dominant.
A practical selection loop begins with a coherent Pareto set, rescoring every candidate under the same channel and uncertainty models. Clearly dominated pulses can be dropped, but close candidates should survive until calibration improves. If two scores differ by less than the uncertainty in the rates or interaction strength, choosing the easier pulse to synthesize may be more defensible than declaring a numerical winner. The evaluator, its parameters, and its uncertainty are therefore part of the result.
This is where control theory meets experimental design. Sensitivity calculations identify which calibration parameter most changes the ranking; a targeted measurement of that parameter can be more valuable than another optimization run. Optimal control is not only the production of waveforms. It is a framework for deciding which model refinement or measurement would most improve the next waveform decision.
Laboratory note
This still is not a hardware prediction. The rates are a generic model, the waveform is not yet sampled on a device clock, and state preparation and measurement are absent. Part 5 crosses that boundary—and changes from the two-qutrit CZ task to a Bell-state hardware surrogate rather than pretending the available analog interface implements the same gate.
That handoff preserves the hierarchy of evidence. The open-system scorer asks what the saved mathematical controls do under one dissipative model. It does not establish that a control channel can emit them, that a calibrated register realizes the assumed $V$, or that measured counts resolve the modeled process fidelity. Each of those questions needs its own representation and validation step, which is why the final chapter begins from arrays rather than calling them waveforms. That vocabulary keeps numerical predictions useful without promoting them beyond the evidence actually supplied by the model. It also prevents false precision.