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The quantum Fourier transform (QFT) is sometimes said to be the source of various exponential quantum speed-ups.
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Since 𝔤 = d 1 ⋯ d m \mathfrak{g}=d_{1}\cdots d_{m} , one has log 2 𝔤 = ∑ i = 1 m log 2 d i \log_{2}\mathfrak{g}=\sum_{i=1}^{m}\log_{2}d_{i} . Since d i ≥ 2 d_{i}\geq 2 one has log 2 d i ≥ 1 \log_{2}d_{i}\geq 1 . This implies that log 2 𝔤 ≥ m \log_{2}\mathfrak{g}\geq m
Cited in the paper.
Consider G = 𝐙 2 m G=\mathbf{Z}_{2}^{m} . For every bit string a = a 1 ⋯ a m ∈ G a=a_{1}\cdots a_{m}\in G one finds X ( a ) = X a 1 ⊗ ⋯ ⊗ X a m X(a)=X^{a_{1}}\otimes\dots\otimes X^{a_{m}} and Z ( a ) = Z a 1 ⊗ ⋯ ⊗ Z a m Z(a)=Z^{a_{1}}\otimes\dots\otimes Z^{a_{m}} , where X X and Z Z represent the standard σ x \sigma_{x} and σ z \sigma_{z} Pauli matrices, respectively
Cited in the paper.
Here c x cx is the element in 𝐙 d 2 \mathbf{Z}_{d_{2}} given by c + ⋯ + c c+\cdots+c with x x summands
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