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University of Chicago

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Efficient control pulses for continuous quantum gate families through coordinated re-optimization back to home

# Efficient control pulses for continuous quantum gate families through coordinated re-optimization

University of Chicago
Frederic T. Chong
University of Chicago
University of Chicago
Frederic T. Chong
University of Chicago
University of Chicago
Frederic T. Chong
University of Chicago

## Abstract

We present a general method to quickly generate high-fidelity control pulses for any continuously-parameterized set of quantum gates after calibrating a small number of reference pulses. We pick several reference operations in the gate family of interest and directly optimize pulses that implement these operations, then iteratively re-optimize these reference pulses to guide pulse shapes to be similar for operations that are closely related. A straightforward interpolation method can then obtain pulses for arbitrary operations in the continuous set with high average fidelity. We demonstrate this procedure on the three-parameter Cartan decomposition of all two-qubit gates to obtain control pulses for any two-qubit gate, up to single-qubit operations, with consistently high fidelity. The method generalizes to any number of parameters and can be used with any pulse optimization algorithm.

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## Selected Figures

Figure 1. Example of the re-optimization scheme. Axes on the top plots correspond to the three Cartan coordinates of two- qubit gates. Reference pulses are generated by optimal control software for 14 reference points in parameter space. Pulses for any point in the chamber can be obtained by linear interpolation between reference pulses. Left: interpolated pulse infi- delities at 285 test points after initial optimization of reference points. Black lines connect reference points and each colored point represents an interpolated pulse. The yellow-colored region indicates poor interpolation quality. Mean infidelity is $5.1 \pm 15 \times 10^{−2}$ . Right: results after repeatedly re-optimizing each reference point to be near the average of its neighbors. Mean infidelity is improved to $5.5 \pm 5.7 \times 10^{−4}$ .

Figure 2. Comparison of pulse shapes for two adjacent reference points before and after several rounds of re-optimization. Point A corresponds to Cartan coordinates $(\frac 1 4, 0, 0)$ and point B corresponds to coordinates $(\frac 1 4, \frac 1 4, \frac 1 4)$ in the same results as displayed in Figure 1. Controls from the two points are shown for the $\sigma_x \sigma_x$ control (top) and $\sigma_z^{(1)}$ control (bottom) of the Hamiltonian (Equation (8)). Inset: locations of points A and B in the Weyl chamber. Left: the pulse shapes are initially significantly different between points A and B. Right: the pulses become far more similar after re-optimization, making interpolation easier, but still retain certain differences in their shapes that account for the differences in the resulting operations.

Figure 3. Different reference point granularities translate to varying amounts of classical computation time needed to optimize all reference pulses. Consecutive points with the same granularity $g$ correspond to subsequent re-optimization rounds, which can further improve average (and maximum) infidelity at the cost of more optimizer iterations. Each optimization iteration corresponds to one system evolution. The points indicated by the plus sign indicate the worst infidelity of any test point.

Figure S1. Infidelities at 2197 test points within the $[0,1]^3$ Cartan coordinate box for 0 to 3 rounds of neighbor-average re-optimization. Average and worst-case infidelities are shown in Table III.

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