# Bandwidth-Limited and Noisy Pulse Sequences for Single Qubit Operations in Semiconductor Spin Qubits

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. ${R}_{x}\left(\theta \right)$ and ${R}_{z}\left(\theta \right)$ with Bandwidth-Limited Pulses

#### 2.1.1. Single Spin Qubit

#### 2.1.2. Singlet–Triplet Qubit

#### 2.1.3. Hybrid Qubit

#### 2.2. Fidelity Comparison

## 3. Discussion

## 4. Methods

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

QD | Quantum Dot |

SS | Single Spin |

ST | Singlet–Triplet |

HY | Hybrid |

## Appendix A. Analytical Gate Sequences

**Table A1.**Analytical gate sequences that realize ${R}_{x}\left(\theta \right)$ (second column) and ${R}_{z}\left(\theta \right)$ (third column). Each row refers to a qubit type.

Qubit | ${\mathit{R}}_{\mathit{x}}\left(\mathit{\theta}\right)$ | ${\mathit{R}}_{\mathit{z}}\left(\mathit{\theta}\right)$ |
---|---|---|

SS | ${t}_{x}\left(\theta \right)=\frac{\theta}{{\mathsf{\Omega}}_{x}}$ | ${t}_{y}=(-\pi /2)/{\mathsf{\Omega}}_{y}\phantom{\rule{142.26378pt}{0ex}}$ ${t}_{x}\left(\theta \right)=\theta /{\mathsf{\Omega}}_{x}\phantom{\rule{142.26378pt}{0ex}}$ ${t}_{y}=(\pi /2)/{\mathsf{\Omega}}_{y}$ |

ST | ${t}_{z}\left(\theta \right)=\left(\frac{\theta}{4\pi}+n\right)\frac{h}{\Delta {E}_{z}}\phantom{\rule{113.81102pt}{0ex}}$ with n integer | ${t}_{z}=\frac{n}{2}\frac{h}{\Delta {E}_{z}}$ ${t}_{J}\left(\theta \right)=\left(-\frac{\theta}{2\pi}+n\right)\frac{h}{J}$ |

HY | ${t}_{{J}_{1}}\left(\theta \right)=\left(\frac{n}{C}-\frac{1}{\sqrt{3}}\frac{\theta}{2\pi}\frac{1}{{J}^{max}}\right)h\phantom{\rule{113.81102pt}{0ex}}$ ${t}_{{J}_{2}}\left(\theta \right)=\left(\frac{n}{C}+\frac{1}{\sqrt{3}}\frac{\theta}{2\pi}\frac{1}{{J}^{max}}\right)h\phantom{\rule{113.81102pt}{0ex}}$ with $n=\u2308\frac{C}{{J}^{max}}\frac{1}{\sqrt{3}}\frac{\theta}{2\pi}\u2309$, $C={E}_{z}+\frac{3}{4}{J}^{max}$ and ${J}^{max}=max\left({J}_{1}\right)=max\left({J}_{2}\right)$ | ${t}_{{J}_{1}}\left(\theta \right)=\frac{1}{C}\left[\frac{\theta}{\pi}A+sign\left(\frac{2\pi}{3}-\theta \right)B\right]\frac{h}{{J}^{max}}\phantom{\rule{56.9055pt}{0ex}}$ ${t}_{{J}_{2}}\left(\theta \right)={t}_{{J}_{1}}\left(\theta \right)\phantom{\rule{142.26378pt}{0ex}}$ ${t}_{J}\left(\theta \right)=\left(2-\frac{\theta}{\pi}\right)\frac{h}{{J}^{max}}\phantom{\rule{113.81102pt}{0ex}}$ with $A=\frac{{E}_{z}}{2}+\frac{1}{8}{J}^{max}$ and $B=-{E}_{z}+\frac{1}{4}{J}^{max}$ |

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**Figure 1.**Bloch sphere. The blue arrow represents the initial condition. The red arrows represent the final conditions after the ${R}_{x}(\pi /2)$ and ${R}_{z}(\pi /2)$ gate operations.

**Figure 2.**SS qubit. (

**Left**) ${R}_{x}$ Infidelity as a function of $\theta $ and $\tau $ when bandwidth-limited input signals are considered. (

**Right**) The same for ${R}_{z}$. Qubit parameters: $\mathsf{\Omega}/2\pi $ = 1 MHz, $\Delta {\omega}_{z}$ = 0.

**Figure 3.**SS qubit. (

**Left**) ${R}_{x}$ infidelity as a function of $\theta $ when undisturbed input signals (solid line, blue) and disturbed input signals with ${\sigma}_{\mathsf{\Omega}/2\pi}$ = 0.05 MHz [7] and ${\sigma}_{\Delta {\omega}_{z}/2\pi}$ = 20 Hz [19] (dashed line, red) are considered. The value of $\tau $ is fixed to 100 ps. (

**Right**) The same for ${R}_{z}$.

**Figure 4.**ST qubit. (

**Left**) ${R}_{x}$ Infidelity as a function of $\theta $ and $\tau $ when bandwidth-limited input signals are considered. (

**Right**) The same for ${R}_{z}$. Qubit parameters: J = 700 neV, $\Delta {E}_{z}$ = 32 neV [20].

**Figure 5.**ST qubit. (

**Left**) ${R}_{x}$ Infidelity as a function of $\theta $ when undisturbed input signals (solid line, blue) and disturbed input signals with ${\sigma}_{J}$ = 1 neV, ${\sigma}_{\Delta {E}_{z}}$ = 4 neV [20] (dashed line, red) are considered. The value of $\tau $ is fixed to 100 ps. (

**Right**) The same for ${R}_{z}$.

**Figure 6.**HY qubit. (

**Left**) ${R}_{x}$ Infidelity as a function of $\theta $ and $\tau $ when bandwidth-limited input signals are considered. (

**Right**) The same for ${R}_{z}$. Qubit parameters: ${J}_{1}$ = ${J}_{2}$ = 1 $\mathsf{\mu}$eV, J = 0.5 $\mathsf{\mu}$eV [21].

**Figure 7.**HY qubit. (

**Left**) ${R}_{x}$ Infidelity as a function of $\theta $ when undisturbed input signals (solid line, blue) and disturbed input signals with ${\sigma}_{J}$ = 80 neV [22] (dashed line, red) are considered. The value of $\tau $ is fixed to 100 ps. (

**Right**) The same for ${R}_{z}$.

**Figure 8.**${R}_{x}(\pi /2)$ infidelity as a function of $\tau $ for SS (solid line, blue), ST (dashed line, red), HY (dot-dashed line, green) qubit and No-Operation (dotted line, black): (

**Left**) with undisturbed input signals; and (

**Right**) with disturbed input signals.

**Figure 9.**${R}_{z}(\pi /2)$ infidelity as a function of $\tau $ for SS (solid line, blue), ST (dashed line, red), HY (dot-dashed line, green) qubit and No-Operation (dotted line, black): (

**Left**) with undisturbed input signals; and (

**Right**) with disturbed input signals.

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**MDPI and ACS Style**

Ferraro, E.; De Michielis, M.
Bandwidth-Limited and Noisy Pulse Sequences for Single Qubit Operations in Semiconductor Spin Qubits. *Entropy* **2019**, *21*, 1042.
https://doi.org/10.3390/e21111042

**AMA Style**

Ferraro E, De Michielis M.
Bandwidth-Limited and Noisy Pulse Sequences for Single Qubit Operations in Semiconductor Spin Qubits. *Entropy*. 2019; 21(11):1042.
https://doi.org/10.3390/e21111042

**Chicago/Turabian Style**

Ferraro, Elena, and Marco De Michielis.
2019. "Bandwidth-Limited and Noisy Pulse Sequences for Single Qubit Operations in Semiconductor Spin Qubits" *Entropy* 21, no. 11: 1042.
https://doi.org/10.3390/e21111042