# Quantum Control Landscapes for Generation of H and T Gates in an Open Qubit with Both Coherent and Environmental Drive

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## Abstract

**:**

## 1. Introduction

## 2. Environment-Assisted Control of a Qubit

## 3. Objective Functionals for the Gate Generation Problem

- The first set $\{{\rho}_{0}^{\left(1\right)},{\rho}_{0}^{\left(2\right)}\}$ corresponds to basis states $|0\rangle $ and $|1\rangle $ in $\mathcal{H}={\mathbb{C}}^{2}$:$${\rho}_{0}^{\left(1\right)}=|0\rangle \langle 0|,\phantom{\rule{2.em}{0ex}}{\rho}_{0}^{\left(2\right)}=|1\rangle \langle 1|.$$
- The second set corresponds to three states determining the implementation of the unitary operation among all dynamic maps [72]:$${\rho}_{0}^{\left(1\right)}=\left(\begin{array}{cc}2/3& 0\\ 0& 1/3\end{array}\right),\phantom{\rule{2.em}{0ex}}{\rho}_{0}^{\left(2\right)}=\left(\begin{array}{cc}1/2& 1/2\\ 1/2& 1/2\end{array}\right),\phantom{\rule{2.em}{0ex}}{\rho}_{0}^{\left(3\right)}=\left(\begin{array}{cc}1/2& 0\\ 0& 1/2\end{array}\right).$$We sometimes call the objective functional defined using this set as the GRK (Goerz–Reich–Koch)-type objective functional.
- The third set corresponds to four basis Hermitian matrices in the linear space in which the dynamic maps act:$${\rho}_{0}^{\left(1\right)}=|0\rangle \langle 0|=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right),\phantom{\rule{2.em}{0ex}}{\rho}_{0}^{\left(2\right)}=|1\rangle \langle 1|=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right),\phantom{\rule{0ex}{0ex}}{\rho}_{0}^{\left(3\right)}=|+\rangle \langle +|=\frac{1}{2}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right),\phantom{\rule{2.em}{0ex}}{\rho}_{0}^{\left(4\right)}=|i\rangle \langle i|=\frac{1}{2}\left(\begin{array}{cc}1& -i\\ i& 1\end{array}\right).$$

## 4. Gradient-Based Optimization Method

**both coherent and incoherent controls**was developed recently in [28] for general N-level open quantum systems, where the exact solution for a qubit was found. The implementation of this method for the considered one-qubit quantum system was performed in [69]. Here, we briefly present the concept and provide the basic expression for the gradient of our objective functionals.

## 5. Numeric Analysis of the Control Landscapes

`scipy.linalg.expm`for matrix exponential computation using the Padé approximation, and

`scipy.linalg.inv`for matrix’ inverse computation. The following values of the system parameters were used: the transition frequency $\omega =1$, the dipole moment $\mu =0.1$, the decoherence rate coefficient $\gamma =0.01$, and the regular partition of the time segment $[0,T]$ with $T=5$ into $M=10$ segments, such that each segment has the length $\Delta {t}_{k}=T/M=0.5$. The stopping parameter $\epsilon ={10}^{-5}$, the parameters of the adaptive scheme of the step length [69] were ${h}^{\left(0\right)}=1$, $c=1.1$, $d=0.5$, ${L}_{\mathrm{stuck}}=20$. Integral formulae were approximated using the trapezoidal rule with the number of partitions ${N}_{\mathrm{partition}}=20$.

## 6. Discussion and Open Problems

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

GKSL | Gorini–Kossakowski–Sudarshan–Lindblad; |

GRAPE | GRadient Ascent Pulse Engineering; |

GRK-type objective | objective functional for generating unitary gates under dissipative evolution where only the three initial density matrices as considered by M.Y. Goerz, D.M. Reich, and C.P. Koch in [72]. |

## Appendix A. Sets of States

#### Appendix A.1. First Set

#### Appendix A.2. Second Set

#### Appendix A.3. Third Set

## Appendix B. Parametrization and Property of the Functionals

**Proposition**

**A1.**

**Proof.**

## Appendix C. Gradient-Based Optimization Method

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**Figure 1.**Histograms describing distributions obtained by GRAPE values of the objective functionals for generating the H gate. The values obtained starting from $L=1000$ random initial conditions uniformly distributed in some hyper-rectangle. Left column: for two (

**a**), three (

**c**), and four (

**e**) matrices. Right column: the Frobenius norm ${F}_{H}$ with optimized controls for two (

**b**), three (

**d**), and four (

**f**) matrices.

**Figure 2.**Histograms describing distributions obtained by GRAPE values of the objective functionals for generating the T gate. The values obtained starting from $L=1000$ random initial conditions uniformly distributed in some hyper-rectangle. Left column: for two (

**a**), three (

**c**), and four (

**e**) matrices. Right column: the Frobenius norm ${F}_{T}$ with optimized controls for two (

**b**), three (

**d**), and four (

**f**) matrices. Two separate peaks are shown in green and red colors.

**Figure 3.**Plots of all 1000 coherent (

**a**,

**c**) and incoherent (

**b**,

**d**) controls obtained by GRAPE optimization for generation of the T gate. Sub-plots (

**a**,

**b**): for the objective functional ${F}_{T,3}$. Sub-plots (

**c**,

**d**): for the objective functional ${F}_{T,4}$. Green (resp., red) color shows all controls leading to the left (resp., right) peak on the corresponding histograms in Figure 2. Both coherent and incoherent controls are clearly combined in two groups.

**Table 1.**Centers (${C}_{1}$, ${C}_{2}$) and widths (${W}_{1},{W}_{2}$) of the obtained distributions for objective functionals defined with number of matrices $j=2,3,4$. For cases with two peaks, the first is the peak shown in green in Figure 3.

Func. | ${\mathit{C}}_{1}$ | ${\mathit{W}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{W}}_{2}$ | |
---|---|---|---|---|---|

$j=2$ | ${F}_{H,2}$ | 1.601 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $4.227\times {10}^{-5}$ | - | - |

${F}_{H}$ | $5.611$ | $4.627\times {10}^{-1}$ | - | - | |

$j=3$ | ${F}_{H,3}$ | $3.484\times {10}^{-4}$ | $1.276\times {10}^{-5}$ | - | - |

${F}_{H}$ | $2.657\times {10}^{-2}$ | $1.979\times {10}^{-3}$ | - | - | |

$j=4$ | ${F}_{H,4}$ | $7.525\times {10}^{-4}$ | $2.317\times {10}^{-5}$ | - | - |

${F}_{H}$ | $5.183\times {10}^{-3}$ | $1.527\times {10}^{-4}$ | - | - | |

$j=2$ | ${F}_{T,2}$ | $2.374\times {10}^{-3}$ | $1.236\times {10}^{-5}$ | - | - |

${F}_{T}$ | $4.795\times {10}^{-1}$ | $1.346\times {10}^{-2}$ | - | - | |

$j=3$ | ${F}_{T,3}$ | $5.964\times {10}^{-4}$ | $6.720\times {10}^{-6}$ | $9.495\times {10}^{-4}$ | $1.821\times {10}^{-5}$ |

${F}_{T}$ | $1.111\times {10}^{-2}$ | $6.866\times {10}^{-4}$ | $1.091\times {10}^{-2}$ | $3.094\times {10}^{-4}$ | |

$j=4$ | ${F}_{T,4}$ | $1.317\times {10}^{-3}$ | $1.592\times {10}^{-5}$ | $1.624\times {10}^{-3}$ | $1.998\times {10}^{-5}$ |

${F}_{T}$ | $6.718\times {10}^{-3}$ | $2.592\times {10}^{-5}$ | $6.599\times {10}^{-3}$ | $2.735\times {10}^{-5}$ |

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**MDPI and ACS Style**

Petruhanov, V.N.; Pechen, A.N.
Quantum Control Landscapes for Generation of *H* and *T* Gates in an Open Qubit with Both Coherent and Environmental Drive. *Photonics* **2023**, *10*, 1200.
https://doi.org/10.3390/photonics10111200

**AMA Style**

Petruhanov VN, Pechen AN.
Quantum Control Landscapes for Generation of *H* and *T* Gates in an Open Qubit with Both Coherent and Environmental Drive. *Photonics*. 2023; 10(11):1200.
https://doi.org/10.3390/photonics10111200

**Chicago/Turabian Style**

Petruhanov, Vadim N., and Alexander N. Pechen.
2023. "Quantum Control Landscapes for Generation of *H* and *T* Gates in an Open Qubit with Both Coherent and Environmental Drive" *Photonics* 10, no. 11: 1200.
https://doi.org/10.3390/photonics10111200