# Discrete Wigner Function Derivation of the Aaronson–Gottesman Tableau Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Discrete Wigner Function for Odd d Qudits

#### 2.1. Clifford Gates

#### 2.2. Wigner Functions of Stabilizer States

**Theorem**

**1.**

## 3. Wigner Stabilizer Algorithm for Odd d Qudits

#### 3.1. Stabilizer Representation

#### 3.2. Unitary Propagation

#### 3.3. Measurement

**Case 1:**Random Measurement

**Case 2:**Deterministic Measurement

**Lemma**

**1.**

**Proof.**

## 4. Aaronson–Gottesman Tableau Algorithm for Qubits ($d=2$)

#### 4.1. Stabilizer Representation

**Definition**

**1.**

#### 4.2. Unitary Propagation

**Definition**

**2.**

- (i)
- ${\widehat{g}}_{1}^{\prime}$, ${\widehat{g}}_{2}^{\prime}$, …, ${\widehat{g}}_{n}^{\prime}$ commute.
- (ii)
- Each destabilizer ${\widehat{g}}_{h}^{\prime}$ anti-commutes with the corresponding stabilizer ${\widehat{g}}_{h}$, and commutes with all other stabilizers.

#### 4.3. Measurement

**Rowsum**: To sum row i and j, first update the bits that represent operators by ${x}_{ik}\oplus {x}_{jk}$ and ${z}_{ik}\oplus {z}_{jk}$ for $k=1,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}n$. To calculate the resultant phase, Aaronson and Gottesman first defined the following function:

**Case 1:**Random Measurement

**Case 2:**Deterministic Measurement

## 5. Discussion

## 6. Example of Stabilizer Evolution

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The initial perpendicular manifolds ${{p}_{0}}_{j}$ and ${{q}_{0}}_{j}$ and the harmonically evolved perpendicular manifolds ${{p}_{t}}_{i}$ and ${{q}_{t}}_{i}$. Description of the various lengths and angles are given in the text in the proof of Lemma 1.

**Figure 2.**A decomposition of the two qutrit Wigner function into nine $3\times 3$ grids, where each $3\times 3$ grid denotes the value of the Wigner function at all ${{p}_{t}}_{1}$ and ${{q}_{t}}_{1}$ for a fixed value of ${{p}_{t}}_{2}$ and ${{q}_{t}}_{2}$ denoted by the external axes. This organization is used in Figure 3 below.

**Figure 3.**The Wigner function of two qutrits initially prepared in (

**a**) the state $\left|0\right.\u232a\otimes \left|0\right.\u232a$. (1) This is evolved under ${\widehat{F}}_{1}$ to produce (

**b**) $\frac{1}{\sqrt{3}}\left(\right)open="("\; close=")">\left|0\right.\u232a+\left|1\right.\u232a+\left|2\right.\u232a$. (2) Subsequently, this state is evolved under ${\widehat{C}}_{12}$ producing (

**c**) the Bell state $\frac{1}{\sqrt{3}}\left(\right)open="("\; close=")">\left|00\right.\u232a+\left|11\right.\u232a+\left|22\right.\u232a$. (3) Qutrit 1 is then measured producing the random outcome 1, which collapses qutrit 2 into the same state, so that (

**d**) $\left|1\right.\u232a\otimes \left|1\right.\u232a$ results. The black color indicates the Wigner function specified by the lowest n rows of ${\delta}_{{\mathsf{\Phi}}_{t}\mathit{x},{\mathit{r}}_{t}}$, and the gray color indicates the Wigner function specified by the highest n rows (${\mathit{q}}_{0}({\mathit{p}}_{t},{\mathit{q}}_{t})$ and ${\mathit{p}}_{0}({\mathit{p}}_{t},{\mathit{q}}_{t})$, respectively). The evolution and algorithmic implementation are explained in the text.

Gates | Input | Output |
---|---|---|

Hadamard | $\widehat{X}$ | $\widehat{Z}$ |

$\widehat{Z}$ | $\widehat{X}$ | |

phase | $\widehat{X}$ | $\widehat{Y}$ |

$\widehat{Z}$ | $\widehat{Z}$ | |

CNOT | $\widehat{X}\otimes \widehat{I}$ | $\widehat{X}\otimes \widehat{X}$ |

$\widehat{I}\otimes \widehat{X}$ | $\widehat{I}\otimes \widehat{X}$ | |

$\widehat{Z}\otimes \widehat{I}$ | $\widehat{Z}\otimes \widehat{I}$ | |

$\widehat{I}\otimes \widehat{Z}$ | $\widehat{Z}\otimes \widehat{Z}$ |

**Table 2.**Binary representation of the Pauli operators and the Pauli group phase used in their tableau representation.

${\mathit{x}}_{\mathit{ij}}$ | ${\mathit{z}}_{\mathit{ij}}$ | ${\widehat{\mathit{P}}}_{\mathit{j}}$ |
---|---|---|

0 | 0 | ${\widehat{I}}_{j}$ |

0 | 1 | ${\widehat{Z}}_{j}$ |

1 | 0 | ${\widehat{X}}_{j}$ |

1 | 1 | ${\widehat{Y}}_{j}$ |

${\mathit{r}}_{\mathit{i}}$ | Phase | |

0 | $+1$ | |

1 | $-1$ |

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**MDPI and ACS Style**

Kocia, L.; Huang, Y.; Love, P.
Discrete Wigner Function Derivation of the Aaronson–Gottesman Tableau Algorithm. *Entropy* **2017**, *19*, 353.
https://doi.org/10.3390/e19070353

**AMA Style**

Kocia L, Huang Y, Love P.
Discrete Wigner Function Derivation of the Aaronson–Gottesman Tableau Algorithm. *Entropy*. 2017; 19(7):353.
https://doi.org/10.3390/e19070353

**Chicago/Turabian Style**

Kocia, Lucas, Yifei Huang, and Peter Love.
2017. "Discrete Wigner Function Derivation of the Aaronson–Gottesman Tableau Algorithm" *Entropy* 19, no. 7: 353.
https://doi.org/10.3390/e19070353