# 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

- Gottesman, D. The Heisenberg Representation of Quantum Computers. arXiv, 1998; arXiv:quant-ph/9807006. [Google Scholar]
- Aaronson, S.; Gottesman, D. Improved simulation of stabilizer circuits. Phys. Rev. A
**2004**, 70, 052328. [Google Scholar] [CrossRef] - Gottesman, D. Fault-tolerant quantum computation with higher-dimensional systems. In Quantum Computing and Quantum Communications; Springer: Heidelberg, Germany, 1999; pp. 302–313. [Google Scholar]
- Mari, A.; Eisert, J. Positive Wigner functions render classical simulation of quantum computation efficient. Phys. Rev. Lett.
**2012**, 109, 230503. [Google Scholar] [CrossRef] [PubMed] - Howard, M.; Wallman, J.; Veitch, V.; Emerson, J. Contextuality supplies the `magic’ for quantum computation. Nature
**2014**, 510, 351–355. [Google Scholar] [CrossRef] [PubMed] - Wootters, W.K. A Wigner-function formulation of finite-state quantum mechanics. Ann. Phys.
**1987**, 176, 1–21. [Google Scholar] [CrossRef] - Gross, D. Hudson’s theorem for finite-dimensional quantum systems. J. Math. Phys.
**2006**, 47, 122107. [Google Scholar] [CrossRef] - Veitch, V.; Ferrie, C.; Gross, D.; Emerson, J. Negative quasi-probability as a resource for quantum computation. New J. Phys.
**2012**, 14, 113011. [Google Scholar] [CrossRef] - Veitch, V.; Wiebe, N.; Ferrie, C.; Emerson, J. Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation. New J. Phys.
**2013**, 15, 013037. [Google Scholar] [CrossRef] - Kocia, L.; Love, P. Semiclassical Formulation of Gottesman–Knill and Universal Quantum Computation. arXiv, 2016; arXiv:1612.05649. [Google Scholar]
- Koh, D.E.; Penney, M.D.; Spekkens, R.W. Computing quopit Clifford circuit amplitudes by the sum-over-paths technique. arXiv, 2017; arXiv:1702.03316. [Google Scholar]
- De Beaudrap, N. A linearized stabilizer formalism for systems of finite dimension. arXiv, 2011; arXiv:1102.3354. [Google Scholar]
- Yoder, T.J. A Generalization of the Stabilizer Formalism for Simulating Arbitrary Quantum Circuits. 2012. Available online: https://pdfs.semanticscholar.org/b200/efe1709d07ffc1b5b7bd90e61c09e2729bdf.pdf (accessed on 6 July 2017).
- Anders, S.; Briegel, H.J. Fast simulation of stabilizer circuits using a graph-state representation. Phys. Rev. A
**2006**, 73, 022334. [Google Scholar] [CrossRef] - Bengtsson, I.; Zyczkowski, K. On discrete structures in finite Hilbert spaces. arXiv, 2017; arXiv:1701.07902. [Google Scholar]
- Mermin, N.D. Hidden variables and the two theorems of John Bell. Rev. Mod. Phys.
**1993**, 65, 803. [Google Scholar] [CrossRef] - Raussendorf, R.; Browne, D.E.; Delfosse, N.; Okay, C.; Bermejo-Vega, J. Contextuality as a resource for qubit quantum computation. arXiv, 2015; arXiv:1511.08506. [Google Scholar]
- Kocia, L.; Love, P. Discrete Wigner Formalism for Qubits and the Non-Contextuality of Clifford Operations on Qubit Stabilizer States. arXiv, 2017; arXiv:1705.08869. [Google Scholar]

**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**) $\frac{1}{\sqrt{3}}\left(\left|0\right.\u232a+\left|1\right.\u232a+\left|2\right.\u232a\right)\otimes \left|0\right.\u232a$. (2) Subsequently, this state is evolved under ${\widehat{C}}_{12}$ producing (

**c**) the Bell state $\frac{1}{\sqrt{3}}\left(\left|00\right.\u232a+\left|11\right.\u232a+\left|22\right.\u232a\right)$. (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 | $-1$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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