Geometric formalism for constructing arbitrary single-qubit dynamically corrected gates

Implementing high-fidelity quantum control and reducing the effect of the coupling between a quantum system and its environment is a major challenge in developing quantum information technologies. Here, we show that there exists a geometrical structure hidden within the time-dependent Schrödinger equation that provides a simple way to view the entire solution space of pulses that suppress noise errors in a system's evolution. In this framework, any single-qubit gate that is robust against quasistatic noise to first order corresponds to a closed three-dimensional space curve, where the driving fields that implement the robust gate can be read off from the curvature and torsion of the space curve. Gates that are robust to second order are in one-to-one correspondence with closed curves whose projections onto three mutually orthogonal planes each enclose a vanishing net area. We use this formalism to derive examples of dynamically corrected gates generated from smooth pulses. We also show how it can be employed to analyze the noise-cancellation properties of pulses generated from numerical algorithms such as grape. A similar geometrical framework exists for quantum systems of arbitrary Hilbert space dimension.

Figure 1

Dynamically corrected Clifford gate $R(-\widehat{x}+\widehat{y}+\widehat{z},2\pi /3)$. (a) Closed space curve. Here, the curve is constructed as $r\left(l\right)=(1-l)r_1\left(l\right)+l{r}_2\left(l\right)$, where $r_1\left(l\right)=\sqrt{2}sin\left(\pi l\right)[0,{sin}^2\left(\frac{\pi l}{2}\right),{cos}^2\left(\frac{\pi l}{2}\right)]$ and $r_2\left(l\right)=\sqrt{2}sin\left(\pi l\right)\left[{sin}^2\left(\frac{\pi l}{2}\right),{cos}^2\left(\frac{\pi l}{2}\right),0\right]·R_z\left(q\right)$, and where $l$ ranges from 0 to 1. Here, $R_z\left(q\right)$ is the rotation matrix around the $z$ axis for angle $q$. Changing $q$ deforms the curve but does not alter the relative orientation of the initial and final tangent vectors, and thus does not alter the final values of $\chi$ and $\varphi$. Tuning $q$ does, however, alter the final value of $\theta$. We have determined numerically that $q=1.6054$ achieves the desired value in this example. (b) The pulse shape. Here, ${\mathrm{\Omega }}_x=\mathrm{\Omega }cos\mathrm{\Phi }$, ${\mathrm{\Omega }}_y=\mathrm{\Omega }sin\mathrm{\Phi }$. (c) Comparison of the log-log infidelity between the shaped pulse and naive square pulse.

Figure 2

Single-qubit identity gate for Hamiltonian $\mathcal{H}\left(t\right)=\mathrm{\Omega }\left(t\right){\sigma }_x+(\mathrm{\Delta }+\delta \beta ){\sigma }_z$ that is robust to first order.

Figure 3

Single-qubit identity gate robust against errors up to second order. (a) The curve (blue) and its projections onto the $xy$, $yz$, and $xz$ planes (gray). All three projected curves have zero enclosed area. (b) The pulses obtained from the curvature and torsion of the curve in (a). Here, ${\mathrm{\Omega }}_x=\mathrm{\Omega }cos\mathrm{\Phi }$, ${\mathrm{\Omega }}_y=\mathrm{\Omega }sin\mathrm{\Phi }$.

Figure 4

Using space curves to analyze pulses obtained from grape. (a)–(d) The space curves and their projections onto the $xy$, $yz$, and $xz$ planes corresponding to four different microwave pulses (e)–(h) designed to implement four different single-qubit gates [identity, $\pi /2$ rotation about $x$ ($X/2$), $\pi /2$ rotation about $z$ ($Z/2$), and Hadamard gate ($H$)] while canceling noise in a silicon quantum dot spin qubit [37]. Arrows on the curves represent the phase $\theta \left(t\right)$ in the evolution operator. For example, the $I$ gate in (a) has $\theta \left(T\right)=0$, while the $Z/2$ gate in (c) has $\theta \left(T\right)=\pi /2$. In (e)–(h), the dashed orange and green curves are ${\mathrm{\Omega }}_x\left(t\right)/2$ and ${\mathrm{\Omega }}_y\left(t\right)/2$, while the solid blue curve is the total magnitude of the pulse envelope, as in previous figures.