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The stabilizer formalism is a scheme, generalizing well-known techniques developed by Gottesman [quant-ph/9705052] in the case of qubits, to efficiently simulate a class of transformations ("stabilizer circuits", which include the quantum Fourier transform and highly entangling operations) on standard basis states of d-dimensional qudits.
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This differs from τ = − e x p ( i π / d ) \tau=-\mathop{exp}\nolimits(i\pi/d) in Appleby [ 19 ] : these have similar features, but differ by a sign for d d even. We choose τ \tau so that Y = i † Z † X † Y=i^{\dagger}Z^{\dagger}X^{\dagger} is a Weyl operator for d = 2 d=2
Cited in the paper.
Note that the + 1 +1 -eigenstates of an operator τ − 2 ϕ W 𝐯 \tau^{-2\phi}W_{\bm{\mathbf{v}}} can also be described as τ 2 ϕ \tau^{2\phi} -eigenstates of W 𝐯 W_{\bm{\mathbf{v}}} . Phase coefficients may thus be used to denote powers of τ 2 \tau^{2} as eigenvalues, describing a stabilized space as an intersection of the corresponding eigenspaces of the Weyl operators
Cited in the paper.
Note that in the case of d d even, the Weyl operators W 𝐯 W_{\bm{\mathbf{v}}} for 𝐯 ∈ ℤ D 2 n \bm{\mathbf{v}}\in\mathbb{Z}_{D}^{2n} are not linearly independent; then we must show that such a map Φ \Phi is well-defined. However, by the discussion following Lemma II
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2011
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