Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0703278 · Physical Review A

Graphical description of the action of Clifford operators on stabilizer states

Matthew Elliott, Bryan Eastin, Carlton M. Caves

Published Apr 8, 2008• 22 citations
arXivHTMLPDF

Abstract

We introduce a graphical representation of stabilizer states and translate the action of Clifford operators on stabilizer states into graph operations on the corresponding stabilizer-state graphs. Our stabilizer graphs are constructed of solid and hollow nodes, with (undirected) edges between nodes and with loops and signs attached to individual nodes. We find that local Clifford transformations are completely described in terms of local complementation on nodes and along edges, loop complementation, and change of node type or sign. Additionally, we show that a small set of equivalence rules generates all graphs corresponding to a given stabilizer state; we do this by constructing an efficient procedure for testing the equality of any two stabilizer graphs.

DOI
10.1103/physreva.77.042307

Related papers

Graphical description of the action of local Clifford transformations on graph states

Maarten Van den Nest, Jeroen Dehaene, Bart De Moor

We translate the action of local Clifford operations on graph states into transformations on their associated graphs, i.e., we provide transformation rules, stated in purely graph theoretical terms, which completely characterize the evolution of graph states under local Clifford operations. As we will show, there is essentially one basic rule, successive application of which generates the orbit of any graph state under local unitary operations within the Clifford group.

arXiv:quant-ph/0308151•Physical Review A•Feb 24, 2004•335 citations

Graphical description of Pauli measurements on stabilizer states

Matthew Elliott, Bryan Eastin, Carlton M. Caves

We use a graphical representation of stabilizer states to describe, simply and efficiently, the effect of measurements of Pauli products on stabilizer states. This work complements our earlier work (2008 Phys. Rev. A 77 042307), which described in graphical terms the action of Clifford operations on stabilizer states.

Journal of Physics A Mathematical and Theoretical•Dec 10, 2009•13 citations

Graphical description of Pauli measurements on stabilizer states

Matthew Elliott, Bryan Eastin, Carlton M. Caves

We use a graphical representation of stabilizer states to describe, simply and efficiently, the effect of measurements of Pauli products on stabilizer states. This work complements our earlier work [Phys. Rev. A 77, 042307 (2008)], which described in graphical terms the action of Clifford operations on stabilizer states.

arXiv:0806.2651•Physical Review A•Jun 16, 2008•0 citations

Efficient algorithm to recognize the local Clifford equivalence of graph states

Maarten Van den Nest, Jeroen Dehaene, Bart De Moor

In Van den Nest et al. [Phys. Rev. A 69, 022316 (2004)] we presented a description of the action of local Clifford operations on graph states in terms of a graph transformation rule, known in graph theory as local complementation. It was shown that two graph states are equivalent under the local Clifford group if and only if there exists a sequence of local complementations which relates their associated graphs. In this Brief Report we report the existence of a polynomial time algorithm, published in A. Bouchet [Combinatorica 11, 315 (1991)], which decides whether two given graphs are related by a sequence of local complementations. Hence an efficient algorithm to detect local Clifford equivalence of graph states is obtained.

arXiv:quant-ph/0405023•Physical Review A•Sep 17, 2004•70 citations

Graph states and local unitary transformations beyond local Clifford operations

Nikoloz Tsimakuridze, Otfried Gühne

Abstract Graph states are quantum states that can be described by a stabilizer formalism and play an important role in quantum information processing. We consider the action of local unitary operations on graph states and hypergraph states. We focus on non-Clifford operations and find for certain transformations a graphical description in terms of weighted hypergraphs. This leads to the indentification of hypergraph states that are locally equivalent to graph states. Moreover, we present a systematic way to construct pairs of graph states which are equivalent under local unitary operations, but not equivalent under local Clifford operations. This generates counterexamples to a conjecture known as LU-LC conjecture. So far, the only counterexamples to this conjecture were found by random search. Our method reproduces the smallest known counterexample as a special case and provides a physical interpretation.

arXiv:1611.06938•Journal of Physics A Mathematical and Theoretical•Mar 20, 2017•28 citations

Graphical description of local Gaussian operations for continuous-variable weighted graph states

Jing Zhang

The form of a local Clifford [LC, also called local Gaussian (LG)] operation for the continuous-variable (CV) weighted graph states is presented in this paper, which is the counterpart of the LC operation of local complementation for qubit graph states. The property of the CV weighted graph states is shown, which can be expressed by the stabilizer formalism. It is distinctively different from the qubit weighted graph states, which cannot be expressed by the stabilizer formalism. The corresponding graph rule, stated in purely graph theoretical terms, is described, which completely characterizes the evolution of CV weighted graph states under this LC operation. This LC operation may be applied repeatedly on a CV weighted graph state, which can generate the infinite LC equivalent graph states of this graph state. This work is an important step to characterize the LC equivalence class of CV weighted graph states.

arXiv:0810.1343•Physical Review A•Nov 4, 2008•21 citations
We introduce a graphical representation of stabilizer states and translate the action of Clifford operators on stabilizer states into graph operations on the corresponding stabilizer-state graphs. Our stabilizer graphs are constructed of solid and hollow nodes, with (undirected) edges between nodes and with loops and signs attached to individual nodes. We find that local Clifford transformations are completely described in terms of local complementation on nodes and along edges, loop complementation, and change of node type or sign. Additionally, we show that a small set of equivalence rules generates all graphs corresponding to a given stabilizer state; we do this by constructing an efficient procedure for testing the equality of any two stabilizer graphs.
In Van den Nest et al. [Phys. Rev. A 69, 022316 (2004)] we presented a description of the action of local Clifford operations on graph states in terms of a graph transformation rule, known in graph theory as local complementation. It was shown that two graph states are equivalent under the local Clifford group if and only if there exists a sequence of local complementations which relates their associated graphs. In this Brief Report we report the existence of a polynomial time algorithm, published in A. Bouchet [Combinatorica 11, 315 (1991)], which decides whether two given graphs are related by a sequence of local complementations. Hence an efficient algorithm to detect local Clifford equivalence of graph states is obtained.
Abstract Graph states are quantum states that can be described by a stabilizer formalism and play an important role in quantum information processing. We consider the action of local unitary operations on graph states and hypergraph states. We focus on non-Clifford operations and find for certain transformations a graphical description in terms of weighted hypergraphs. This leads to the indentification of hypergraph states that are locally equivalent to graph states. Moreover, we present a systematic way to construct pairs of graph states which are equivalent under local unitary operations, but not equivalent under local Clifford operations. This generates counterexamples to a conjecture known as LU-LC conjecture. So far, the only counterexamples to this conjecture were found by random search. Our method reproduces the smallest known counterexample as a special case and provides a physical interpretation.
The form of a local Clifford [LC, also called local Gaussian (LG)] operation for the continuous-variable (CV) weighted graph states is presented in this paper, which is the counterpart of the LC operation of local complementation for qubit graph states. The property of the CV weighted graph states is shown, which can be expressed by the stabilizer formalism. It is distinctively different from the qubit weighted graph states, which cannot be expressed by the stabilizer formalism. The corresponding graph rule, stated in purely graph theoretical terms, is described, which completely characterizes the evolution of CV weighted graph states under this LC operation. This LC operation may be applied repeatedly on a CV weighted graph state, which can generate the infinite LC equivalent graph states of this graph state. This work is an important step to characterize the LC equivalence class of CV weighted graph states.