Fast adiabatic qubit gates using only ${\sigma }_z$ control

A controlled-phase gate was demonstrated in superconducting Xmon transmon qubits with fidelity reaching 99.4%, relying on the adiabatic interaction between the $|11\rangle$ and $|02\rangle$ states. Here we explain the theoretical concepts behind this protocol, which achieves fast gate times with only ${\sigma }_z$ control of the Hamiltonian, based on a theory of nonlinear mapping of state errors to a power spectral density and use of optimal window functions. With a solution given in the Fourier basis, optimization is shown to be straightforward for practical cases of an arbitrary state change and finite bandwidth of control signals. We find that errors below ${10}^{-4}$ are readily achievable for realistic control wave forms.

Figure 1

Geometric picture of nonadiabatic errors. (a) Control field of constant $H_x$ and variable $H_z\left(t\right)$ gives control angle $\theta$ from the $z$ axis. (b) Bloch sphere representation gives a ground eigenstate parallel to the control vector (solid arrow). For a small step in $\theta$, the quantum state (gray arrow) points slightly away, with change $\Delta \theta$ (small gray arrow). This state rotates around the eigenstate at frequency $\omega$ (gray dotted lines).

Figure 2

Wave form and power spectral density $S\left({\omega }_0\right)$, for a small change in control $\theta$ such that the qubit frequency is approximately constant, $\omega ={\omega }_0$. Here, the spectral density is dimensionless and proportional to the qubit error $P_e\simeq (1/4)S\left({\omega }_0\right)$, valid in the limit of large $t_p$. (a) Plot of wave form $d\theta /dt$ versus $t/t_p$ for a rectangular pulse (i, black), Hanning window (1, blue), Slepian (ii, gray), and optimized Fourier coefficients with two (2, red), four (4, green), and ten (10, cyan) terms. Note the similarity of the Slepian curves to the optimized Fourier functions, except near $t/t_p=0,1$. (b) Plot of power spectral density obtained from a Fourier transform of the wave forms in (a). All wave forms are normalized to have unit area. The vertical dashed line shows the spectral cutoff for minimization of integrated noise.

Figure 3

Wave form and power spectral density $[S\left({\omega }_0\right)\simeq 4P_e]$ as for Fig. 2, but with a trajectory from ${\theta }_i$ to ${\theta }_f$ and back. (a) Plot of wave form $d\theta /dt$ versus $t/t_p$ for optimized Fourier coefficients with one (1, blue), two (2, red), and three (3, green) terms, as well as the Slepian (i, gray). (b) Plot of power spectral density from Fourier transforms of the wave forms in (a). Wave forms normalized using $2\sum {\lambda }_n^{\prime }=1$.

Figure 4

Wave form and error for $H_z$ changing from $-10H_x$ to $10H_x$. (a) Blue lines are plot of control $\theta \left(t\right)$ versus $t/t_p$ for two (solid) and three (dashed) Fourier coefficients. The right scale shows the wave form $H_z\left(t\right)$ as green lines. (b) Plot of nonadiabatic error versus pulse time $t_p$ for linearized and exact solutions, with both two and three optimized coefficients. Low errors can be achieved for times greater than $t_p=2\pi /{\omega }_x$, corresponding to one oscillation at the avoided level crossing. We define ${\omega }_x=2H_x/\hslash$.

Figure 5

Wave form and error for $H_z$ changing from $10H_x$ to $1.17H_x$, and then back to $10H_x$. (a) Black line (1) is for optimal wave form, whereas blue line (2) includes the effect of wave form rounding by control electronics, here accounted for by a convolution. (b) Plot of nonadiabatic error versus pulse time $t_p$ for the two cases, using exact solutions. Optimal coefficients are $\{{\lambda }_2^{\prime },{\lambda }_3^{\prime }\}=\{-0.19,0\}$ for black (1) and $\{-0.16,0.04\}$ for blue (2) lines.

Figure 6

Average probability of 2-state population versus pulse time $t_p$ for a qubit $\pi$ pulse. The time is normalized with the qubit nonlinearity $\left|\Delta \right|/2\pi$. Parameters are chosen to minimize qubit errors near $t_p\left|\Delta \right|/2\pi =2.5$. The DRAG terms $D=$ 0, $-0.48$, and $-1.20$ are shown in black (1), red (3), and green (4), respectively. Optimizations of the wave form $x\left(t\right)$ with Fourier coefficient ${\lambda }_2$ are shown in blue (2) and purple (5).