Fastest pulses that implement dynamically corrected single-qubit phase gates

Dynamically correcting for unwanted interactions between a quantum system and its environment is vital to achieving the high-fidelity quantum control necessary for a broad range of quantum information technologies. In recent work, we uncovered the complete solution space of all possible driving fields that suppress transverse quasistatic noise errors while performing single-qubit phase gates. This solution space lives within a simple geometrical framework that makes it possible to obtain globally optimal pulses subject to a set of experimental constraints by solving certain geometrical optimization problems. In this work, we solve such a geometrical optimization problem to find the fastest possible pulses that implement single-qubit phase gates while canceling transverse quasistatic noise to second order. Because the time-optimized pulses are not smooth, we provide a method based on our geometrical approach to obtain experimentally feasible smooth pulses that approximate the time-optimal ones with minimal loss in gate speed. We show that in the presence of realistic constraints on pulse rise times, our smooth pulses significantly outperform sequences based on ideal pulse shapes, highlighting the benefits of building experimental waveform constraints directly into dynamically corrected gate designs.

Figure 1

An example of a closed curve, with angle subtended by the two legs $\varphi$. Any such curve, parameterized by Eq. (A4), is associated with a single-qubit phase gate robust to first-order error.

Figure 2

(a) Curve that yields the fastest qubit $z$ rotation by angle $4\pi /3$ (the angle subtended at the origin is $\varphi =\pi /3$) while canceling the first-order noise error and respecting the pulse constraint, $\left|\mathrm{\Omega }\left(t\right)\right|\le {\mathrm{\Omega }}_{max}$. The curve is made up of three circular arcs that connect tangentially to each other. (b) Composite square pulse obtained from the curve in (a). The minimal gate time in this case is $T_{min}=6.59/{\mathrm{\Omega }}_{max}$.

Figure 3

The minimal time to perform quantum rotations while canceling first-order noise for different values of $\varphi$.

Figure 4

(a) Curve that yields the fastest qubit $\pi$ rotation (subtended angle $\varphi =0$) while canceling the second-order noise error and respecting the pulse constraint $\left|\mathrm{\Omega }\left(t\right)\right|\le {\mathrm{\Omega }}_{max}$. (b) Corresponding driving pulse. (c) Parameter $k\left(\varphi \right)$ as a function of the subtended angle $\varphi$.

Figure 5

The minimal time to perform quantum rotations while canceling second-order noise for different values of $\varphi$.

Figure 6

Toolbox for smoothing the curve. (a) $f_1(\mathcal{P},x)=tanh\left(\mathcal{P}x\right)$. (b) $f_2(\mathcal{P},x)=1-f_1(\mathcal{P},x)$. Together with $f_1$, we use this to guarantee that the second derivative of the final smoothed function (and hence the driving pulse) starts from zero and ends at zero. (c) $f_3(\mathcal{Q},x)={\left[1+exp(-\mathcal{Q}x)\right]}^{-1}$. (d) $f_4(\mathcal{Q},x)=1-f_3(\mathcal{Q},x)$. Together with function $f_3$, we use this to smoothly connect different pieces of the curve such that the end result is infinitely differentiable everywhere.

Figure 7

Comparison of two types of pulse smoothing. (a) The original time-optimal composite square pulse with $\varphi =0$ (blue), the pulse obtained by curve smoothing (red), and the pulse obtained from direct smoothing (green). (b) Fidelity vs noise strength for the two smoothed driving pulses shown in (a). Here, CS denotes curve smoothing and DS direct smoothing. The numbers 450, 525, 600, and 675 are the maximum slopes of the driving pulses.

Figure 8

Three piecewise curves that give time-optimal pulses. (a) An arc tangential to two straight lines. The length of such a curve is $\pi +\varphi +2cot\left(\frac{\varphi }{2}\right)$. (b) Two arcs tangential to one vertical line. The length is $3\pi -\varphi +2cot\left(\frac{\varphi }{2}\right)$. (c) Three arcs tangential to each other. The length is $4\psi +\pi -\varphi$, where $\psi =arccos[cos(\varphi /2)/2]$. (d) Comparison of the lengths of the three different curves. The curve shown in (c) is always the shortest.

Figure 9

An example of a second-order error-canceling curve made up of externally tangential arcs.