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Adaptive hybrid optimal quantum control for imprecisely characterized systems
D. J. Egger, F. K. Wilhelm
TL;DR
Imprecise system parameters and control errors complicate optimal quantum control. Ad-HOC pairs model-based gradient search with experimental, model-free pulse updates; gradient initialization reduces search effort, while fidelity-estimation noise limits the terminal fidelity.
Problem
Imprecise system parameters and control errors can degrade optimized pulse fidelity, while complex setups also involve a control transfer chain.
Source: pp. 1, 3Method
Ad-HOC uses model-based gradient search to approach a favorable pulse, then updates similar pulses using experimental performance and the Nelder-Mead algorithm.
Source: p. 2Results
About 50 gradient-search steps per order-of-magnitude error reduction are needed, while noisy feedback can halt optimization when gains fall below its noise threshold.
Source: pp. 3–4Takeaways & Limitations
Ad-HOC can be used with any fidelity measure that captures the desired time evolution, and higher fidelity can be gained by increasing estimation precision.
Source: p. 4Ad-HOC’s robustness is limited by fidelity-estimation reliability; optimization may halt when averaged fidelity gains fall below the noise threshold.
Source: p. 4
Abstract
from arXiv · showhide
Optimal quantum control theory carries a huge promise for quantum technology. Its experimental application, however, is often hindered by imprecise knowledge of the its input variables, the quantum system's parameters. We show how to overcome this by Adaptive Hybrid Optimal Control (Ad-HOC). This protocol combines open- and closed-loop optimal by first performing a gradient search towards a near-optimal control pulse and then an experimental fidelity measure with a gradient-free method. For typical settings in solid-state quantum information processing, Ad-Hoc enhances gate fidelities by an order of magnitude hence making optimal control theory applicable and useful.
Supplementary material
The supplementary material expands on random two-level-system optimization, rapid calibration with few parameters, and calibration beyond randomized benchmarking.
Supplementary material
- It provides further details on optimizing control pulses for random two-level systems.Source: Supplementary material · p. 5
- It shows that a pulse with few parameters can be calibrated extremely quickly.Source: Supplementary material · p. 5
- It adds qubit-bus-qubit details and demonstrates CZ calibration using a gate-tailored fidelity measure rather than randomized benchmarking.Source: Supplementary material · p. 5
Modeling of the Control Transfer Chain
The control transfer chain changes the relationship between AWG output voltages and the fields received by the quantum system. Modeling this chain requires precise characterization of its calibration and impulse response.
Modeling of the Control Transfer Chain
- AWG voltages are not identical to the control fields: calibration curve C and transfer-chain impulse response h mediate and distort the signals.Source: Modeling of the Control Transfer Chain · p. 5
- The transfer-chain model can account for these effects when C and h are precisely known.Source: Modeling of the Control Transfer Chain · p. 5
- In practice, C, h, and the assumed linear signal transfer are difficult to verify precisely, and the authors know of no approach for uncertain transfer functions.Source: Modeling of the Control Transfer Chain · p. 5
Control of Random Two Level Systems
The supplied examples apply closed-loop calibration to random two-level systems, a weakly nonlinear three-level system, and a qubit-bus-qubit architecture. They show how controllability, model uncertainty, and the choice of fidelity measure shape calibration outcomes.
Control of Random Two Level Systems
- Poorly converging random two-level realizations have small drift-control commutator norms, and smaller η corresponds to harder control.Two very small-commutator instances would have required a much longer gate time.Source: Control of Random Two Level Systems · p. 6
- At very small η, some random two-level systems are difficult or impossible to control within the chosen gate time, and two low-η instances were excluded after failing to converge.For controllable systems, the number of fidelity evaluations is low, including from a zero-control starting point.Source: Control of Random Two Level Systems · p. 6, Figure 6 · p. 6
Few Parameter Pulse Closed-Loop Optimization Example: Finding DRAG
- Closed-loop optimization found an X gate in 76 fidelity evaluations from an initial pulse too weak to drive transitions, while minimizing leakage.The initial and final pulses and their population evolutions are shown in Fig. 7.Source: Few Parameter Pulse Closed-Loop… · p. 7, Figure 7 · p. 7, Figure 8 · p. 8
Control Distance Characterization
- At current-experiment parameter precisions, the qubit-bus model and system optimal controls remain relatively close, with θ(ξ = 1) ≃0.2.The comparison is between their optimal controls and the more distant GRAPE initial guess.Source: Control Distance Characterization · pp. 8–9, Figure 9 · p. 9
Application of Ad-HOC to the Qubit-Bus-Qubit Architecture
- Ad-HOC increased CZ-gate fidelity across a wide range of control DC-offset errors, while the central high-fidelity region remained unchanged at the 99.9% calibration target.The figure compares the fidelity before and after Ad-HOC calibration as the DC offset error varies.Source: Additional Example · pp. 9–10, Figure 10 · p. 10
Optimization with Gate-Taylored Quality Parameters
- The gate-tailored Φcz measure can calibrate a CZ pulse, and the intrinsic gate fidelity follows it closely during model-free optimization.The initial pulse was GRAPE-optimized to 80% fidelity, and the target Φcz was 99.9%.Source: Optimization with Gate-Taylored… · p. 10, Figure 11 · p. 10
Noise Threshold
Closed-loop optimization stops when fidelity changes are no longer distinguishable from experimental noise. The threshold reflects the experiment’s ability to resolve changes, while the depolarization probability is estimated from measured trials.
Noise Threshold
- The closed loop stops when fidelity changes can no longer be distinguished, on average, from noise in the experiment.The stopping condition is tied to the experiment’s ability to resolve fidelity differences.Source: Noise Threshold · p. 11
- When the average improvement for the worst pulse falls below ΔΦth, the experiment cannot distinguish fidelities sufficiently to improve them.The threshold is applied to an operation on the simplex.Source: Noise Threshold · p. 11
- The depolarization probability is estimated as n/m, with uncertainty p̄ ± σp; the factor 12 in the uncertainty comes from a uniform distribution.Here n counts depolarizations among m trials.Source: Noise Threshold · p. 11