Two-qubit controlled-Z gates robust against charge noise in silicon while compensating for crosstalk using neural network

The fidelity of two-qubit gates using silicon spin qubits is limited by charge noise. When attempting to dynamically compensate for charge noise using single-qubit echo pulses, crosstalk can cause complications. We present a method of using a deep neural network to optimize the components of an analytically designed composite pulse sequence, resulting in a two-qubit gate robust against charge noise errors while also taking crosstalk into account. We analyze two experimentally motivated scenarios. For a scenario with strong electron dipole spin resonance driving and negligible crosstalk, the composite pulse sequence yields up to an order of magnitude improvement over a simple cosine pulse. In a scenario with moderate electron spin resonance driving and appreciable crosstalk such that simple analytical control fields are not effective, optimization using the neural network approach allows one to maintain order-of-magnitude improvement despite the crosstalk.

Figure 1

Schematic representation of a neural network where the input, output, and hidden layers are shown as circles and the weights are shown as arrows.

Figure 2

Analytical control fields vs time for the pulse sequence of Eq. (4) when crosstalk is negligible. Top: Exchange $J$, and EDSR tone amplitudes ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ where $h$ is Planck's constant. Bottom: Accompanying phases of the EDSR tones.

Figure 3

Neural network designed control fields vs time for single-shot pulse when crosstalk is negligible. Top: Exchange $J$, and EDSR tone amplitudes ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$. Bottom: Accompanying phase modulation of the EDSR tones.

Figure 4

Top: Infidelity vs barrier fluctuations, $\delta V$. Bottom: Infidelity vs average Zeeman fluctuation.

Figure 5

Top: Filter function, $\mathcal{F}\left(\omega \right)$, for the charge noise, of the single-shot pulse and the half cosine pulse vs noise frequency $\omega$ in the top plot. Bottom: Average fidelity of the pulse sequence, the single-shot pulse, and the half cosine pulse vs cutoff frequency, with daily calibration and $\delta V_{\text{rms}}=3$ mV.

Figure 6

Shaped exchange pulses $J_{\pi /4}\left(t\right)$ and $J_{\zeta }\left(t\right)$ for an adiabatic $e^{-i(\pi /4)ZZ}$ and rotation represented by the dashed blue line and the solid red line. The strength of the exchange $J_{\zeta }\left(t\right)$ on the left side of the plot is scaled by a factor of $\approx 1.63$ compared to the strength of the exchange $J_{\pi /4}\left(t\right)$ shown on the right.

Figure 7

Shaped pulse for an $XX$ gate robust against Zeeman fluctuations. Top: Two-tone ESR amplitude modulation. Bottom: Accompanying phase modulation.

Figure 8

Shaped pulse for a $e^{-i\left\{\right[\pi /2\left)\right]XI+[(\theta -\pi )/2]IX\}}$ rotation robust against Zeeman fluctuations. Top: Two-tone ESR amplitude modulation. Bottom: Accompanying phase modulation.

Figure 9

Shaped pulse for a $e^{-i\left\{\right(\pi /2)XI-[(\pi +\eta )/2\left]IX\right\}}$ rotation robust against Zeeman fluctuations. Top: Two-tone ESR amplitude modulation. Bottom: Accompanying phase modulation.

Figure 10

Shaped pulse for a $e^{-i(\eta /2)IX}$ rotation robust against Zeeman fluctuations. Top: Two-tone ESR amplitude modulation. Bottom: Accompanying phase modulation.

Figure 11

Infidelity of the neural network optimized pulse sequence vs quasistatic fluctuations in barrier gate voltage (top) and in average Zeeman energy (bottom). For comparison the naive square pulse infidelity is also plotted.

Figure 12

Simultaneous shaped pulse of exchange and two-tone ESR. Top: Amplitudes vs time. Bottom: Accompanying phase modulation of the ESR tones.