Optimal Control Pulse Engineering Grape

Short answer: Optimal control is the practice of shaping the microwave or laser pulses that drive qubit gates so the gate is as accurate as possible. GRAPE (gradient ascent pulse engineering) adjusts a pulse sample by sample using gradients of a cost function. Experimental papers from 2018 to 2021 report using such methods on superconducting qubits to reach high fidelity, to suppress leakage out of the qubit states, and to cope with imperfect models of the hardware. The figures are from single laboratory devices.

Why pulse shape matters

An npj Quantum Information paper from 2020 says high-speed, high-fidelity qubit operations need precise control over the shape of the underlying pulses. For weakly anharmonic systems such as superconducting transmon qubits, short gates lead to leakage to states outside the computational subspace. The authors note that fast gates are required to lower the limits on gate errors set by decoherence, and that reducing leakage is paramount for error correction, since correcting such errors takes significantly more resources. They add that open-loop optimal control may reduce leakage, but model inaccuracies can severely limit the usability of such pulses.

How GRAPE works

A 2021 arXiv paper (2104.11936) describes the idea. A gate operator is equivalent to a time evolution operator that depends fully on the control pulse. By discretising time, the pulse becomes a sequence of amplitudes, and the derivative of the gate operator with respect to each amplitude can be computed numerically. This gives the GRAPE algorithm. The authors say that compared with the Nelder-Mead algorithm, GRAPE searches much faster because of the guidance of gradient vectors, and allows flexibility in the dimensionality of the parameter space. They also note that the original design needs an accurate theoretical description of the quantum dynamics, which is not always available in real experiments because of systematic errors, and that a hybrid approach using experimental measurement can partly get around this.

Data-driven and closed-loop experiments

The 2021 paper reports optimizing a two-qubit controlled-Z gate with two data-driven GRAPE protocols, one aimed at the gate operator and one at the output state. The gradients use a mix of the input pulse and experimental measurements. Starting from an imperfect flat-top pulse, the authors report that the gate quickly improves and that fidelities for the two optimized pulses are around 99%.

The 2020 paper implements a closed-loop optimization that adapts all control parameters at once, based on measurements of a cost function built from Clifford gates, directly optimizing the amplitude and phase of each sample of the digitized pulse. They report a 4.16 ns single-qubit pulse with 99.76% fidelity and 0.044% leakage, which they describe as a sevenfold reduction in leakage and a threefold reduction in standard error compared with the best DRAG pulse they calibrated at such short durations on the same system.

Pulses that can be calibrated

A 2018 Physical Review Letters paper points out that superconducting qubits need pulses with simple parameterizations so they can be calibrated in the experiment to compensate for uncertain parameters. It presents a gradient-based technique called gradient optimization of analytic controls (GOAT) and uses it to optimize fast, coherence-limited pulses for flux-tunable transmons and for fixed-frequency transmons with tunable couplers. The results are numerical demonstrations.

What this does not tell you

These are results from specific devices and simulations. A fidelity measured on one chip with one gate does not carry over to a full processor, and the comparisons with DRAG pulses apply only to the system tested.

FAQ

What does GRAPE stand for?

Gradient ascent pulse engineering.

Why not use a model only?

Real hardware has errors the model misses, so some papers use measurements in the loop.

What is leakage?

Population escaping outside the computational states, which short gates on transmon qubits can cause.

What results have been reported?

One experiment reports 99.76% single-qubit fidelity with a 4.16 ns pulse, another around 99% for a controlled-Z gate.

Is this specific to superconducting qubits?

The papers here concern them, but the approach is described generally.