# A non-commuting stabilizer formalism

## Abstract

We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = e{sup iπ/4} and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.

- Authors:

- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching (Germany)
- Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5 (Canada)
- (Germany)

- Publication Date:

- OSTI Identifier:
- 22403143

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ABELIAN ANYONS; HAMILTONIANS; MANY-BODY PROBLEM; QUANTUM ENTANGLEMENT; QUANTUM STATES; TOPOLOGY

### Citation Formats

```
Ni, Xiaotong, Van den Nest, Maarten, Buerschaper, Oliver, and Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin.
```*A non-commuting stabilizer formalism*. United States: N. p., 2015.
Web. doi:10.1063/1.4920923.

```
Ni, Xiaotong, Van den Nest, Maarten, Buerschaper, Oliver, & Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin.
```*A non-commuting stabilizer formalism*. United States. doi:10.1063/1.4920923.

```
Ni, Xiaotong, Van den Nest, Maarten, Buerschaper, Oliver, and Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin. Fri .
"A non-commuting stabilizer formalism". United States. doi:10.1063/1.4920923.
```

```
@article{osti_22403143,
```

title = {A non-commuting stabilizer formalism},

author = {Ni, Xiaotong and Van den Nest, Maarten and Buerschaper, Oliver and Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin},

abstractNote = {We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = e{sup iπ/4} and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.},

doi = {10.1063/1.4920923},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 5,

volume = 56,

place = {United States},

year = {2015},

month = {5}

}