Improving quantum gate performance through neighboring optimal control

Successful implementation of a fault-tolerant quantum computation on a system of qubits places severe demands on the hardware used to control the many-qubit state. It is known that an accuracy threshold $P_a$ exists for any quantum gate that is to be used in such a computation. Specifically, the error probability $P_e$ for such a gate must fall below the accuracy threshold: $P_e<P_a$. Estimates of $P_a$ vary widely, although $P_a\sim {10}^{-4}$ has emerged as a challenging target for hardware designers. In this paper, we present a theoretical framework based on neighboring optimal control that takes as input a good quantum gate and returns a new gate with better performance. We illustrate this approach by applying it to all gates in a universal set of quantum gates produced using nonadiabatic rapid passage that has appeared in the literature. Performance improvements are substantial, both for ideal and nonideal controls. Under suitable conditions detailed in this paper, all gate error probabilities fall well below the target threshold of ${10}^{-4}$.

Figure 1

The control modification $x$ component $\Delta F_x\left(\tau \right)$ used to implement a neighboring optimal control improved approximation to the Hadamard gate. Here, $\tau$ is dimensionless time.

Figure 2

The Fourier transform $\Delta {\mathcal{F}}_x\left(\omega \right)$ of $\Delta F_x\left(\tau \right)$ for the Hadamard gate as target. Here, $\omega$ is dimensionless frequency.

Figure 3

The noise-averaged value of $\mathrm{Tr}P$ with NOC versus timing jitter ${\sigma }_t={\sigma }_{\varphi }/\left(2\pi f_{\mathrm{clock}}\right)$ for the Hadamard gate. For each ${\sigma }_t$, 10 realizations of phase noise were generated, and for each realization, gate performance was determined by numerical simulation of the Schrödinger dynamics generated by the control field ${\mathbf{F}}^{\prime }\left(\tau \right)$ that includes the noisy TRP nominal control ${\mathbf{F}}_0^{\prime }\left(\tau \right)$ and the NOC modification $\Delta \mathbf{F}\left(\tau \right)$ (see text). The average and standard deviation were determined for the resulting 10 $\text{Tr}P$ values. For each value of ${\sigma }_t$, the average of $\text{Tr}P$ is plotted, and the standard deviation is used to specify the error bar. To obtain ${\sigma }_t$, we have assumed that $f_{\mathrm{clock}}=1\mathrm{GHz}$ (see text).

Figure 4

The $x$ component of the control-field modification $\Delta F_x\left(\tau \right)$ for the not gate.

Figure 5

The Fourier transform of the $x$ component of the control-field modification $\Delta {\mathcal{F}}_x\left(\omega \right)$ for the not gate.

Figure 6

The $x$ component of the control-field modification $\Delta F_x\left(\tau \right)$ for the modified phase gate.

Figure 7

The Fourier transform of the $x$ component of the control-field modification $\Delta {\mathcal{F}}_x\left(\omega \right)$ for the modified phase gate.

Figure 8

The $x$ component of the control-field modification $\Delta F_x\left(\tau \right)$ for the modified $\pi /8$ gate.

Figure 9

The Fourier transform of the $x$ component of the control-field modification $\Delta {\mathcal{F}}_x\left(\omega \right)$ for the modified $\pi /8$ gate.

Figure 10

The $x$ component of the control-field modification $\Delta F_x\left(\tau \right)$ for modified controlled-phase gate.

Figure 11

The Fourier transform of the $x$ component of the control-field modification $\Delta {\mathcal{F}}_x\left(\omega \right)$ for the modified controlled-phase gate.