bibkey: vandennest2004graphical authors: M. Van den Nest, J. Dehaene, B. De Moor year: 2004 title: “Graphical description of the action of local Clifford transformations on graph states” doi: 10.1103/PhysRevA.69.022316 url: https://arxiv.org/abs/quant-ph/0308151v2 claim: “Each stabilizer state is equivalent to a graph state under local Clifford operations; in the binary framework a generator matrix is brought to an invertible X-block by local Hadamard operations and its diagonal is cleared by local phase operations.” strata_touched:

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Van den Nest, Dehaene and De Moor, local Clifford reduction to graph states

M. Van den Nest, J. Dehaene and B. De Moor, Graphical description of the action of local Clifford transformations on graph states, Phys. Rev. A 69, 022316 (2004), arXiv:quant-ph/0308151v2.

Verified locator

DOI: 10.1103/PhysRevA.69.022316. Primary version: https://arxiv.org/abs/quant-ph/0308151v2. The TeX source localcliffgraph.tex of v2 supplies §III (“Reduction to graph states”) and Theorem 1.

Source statements (§III)

“Theorem 1: Each stabilizer state is equivalent to a graph state under local Clifford operations.”

Proof outline in the binary framework, for a generator matrix $S=\left[\begin{array}{c}Z\\ X\end{array}\right]$ whose columns span a self-orthogonal subspace for the symplectic form $P$: “The result is obtained by proving the existence of a local Clifford operation $Q\in C^l$ such that $QS=\left[\begin{array}{c}{Z}^{\prime }\\ X^{\prime }\end{array}\right]$ has an invertible lower block $X^{\prime }$. Then $S^{\prime }:=QS{X}^{\prime -1}=\left[\begin{array}{c}{Z}^{\prime }{X}^{\prime -1}\\ I\end{array}\right]$, where $Z^{\prime }{X}^{\prime -1}$ is symmetric from the property $S^{\prime T}P{S}^{\prime }=0$; furthermore, the diagonal entries of $Z^{\prime }{X}^{\prime -1}$ can be put to zero by additionally applying the operation $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ to the appropriate qubits”. The invertible lower block is reached by “a Hadamard transformation $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ on the qubits $k+1,\dots ,n$” after a basis change that splits $S$ by the rank $k$ of $X$.

Scope

The paper states the reduction for qubit stabilizer states and notes that it is a special case of a result of Schlingemann for $d$-level stabilizer codes. It gives an explicit construction of the Hadamard set from a rank decomposition of the generator matrix.