Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 324 / 1 / b / i / Solution

The matrix-element orthogonality theorem for irreducible unitary representations states

\sum_{g \in G} \overline{M_{α, ij} (g)} M_{β, k l} (g) = \frac{∣ G ∣}{d _{α}} δ_{α β} δ_{ik} δ_{j l},

and

\sum_{α} d_{α}^{2} = ∣ G ∣

. Consequently the normalized vectors

∣ α ij ⟩ = \frac{d _{α}}{∣ G ∣} \sum_{g \in G} \overline{M_{α, ij} (g)} ∣ g ⟩

form an orthonormal basis. The non-abelian quantum Fourier transform is the unitary basis change

∣ α ij ⟩ ⟼ ∣ α ⟩ ∣ i ⟩ ∣ j ⟩,

up to the harmless choice of transform direction and complex-conjugation convention.

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