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Questions tagged [t-designs]

For questions about quantum t-designs: probability distributions over states or unitaries that replicate specific properties of the Haar distribution.

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7 votes
1 answer

Which Clifford groups are 2-designs?

Let $ X $ be the $ q \times q $ shift matrix sending $ |y \rangle \mapsto |y+1 \rangle $ where the ket index $ y=0,\dots, q-1 $ is taken mod $ q $. Let $ Z $ be the diagonal $ q \times q $ clock ...
Ian Gershon Teixeira's user avatar
2 votes
2 answers

Why do averages of tensor products of projections give $\int_{{\Bbb CP}^{d-1}}d\mu(x)\pi(x)^{\otimes t}=\binom{d+t-1}{t}^{-1} \Pi_{\rm sym}^{(t)}$?

This a lemma used in (Scott 2006) when discussing complex projective t-designs. Let $\pi(x)\equiv|x\rangle\!\langle x|$ be the projection onto some pure state (represented as an element of the complex ...
glS's user avatar
  • 25.3k
1 vote
1 answer

Proof of equivalence between Welch-bound-based and frame-potential-based definitions of t-designs

Let $X\subset\mathbb{C}^d$ be a (finite, non-empty) set of unit vectors. A standard way to define $X$ being a spherical $t$-design, is to impose it saturates the Welch bounds for all $k\le t$. ...
glS's user avatar
  • 25.3k