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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Are MUBs complex projective 3-designs?
Consider a finite subset $X\subset\mathbb{CP}^{d-1}$ of $d$-dimensional pure states.
Following e.g. (Roy and Scott 2007), we say that $X$ is a complex projective $t$-design if
$$\frac1{|X|}\sum_{x\in ...
glS♦
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Proof of equivalence between Welch-bound-based and frame-potential-based definitions of t-designs
Let $X\subset\mathbb{C}^d$ be a (finite, non-empty) set of unit vectors. A standard way to define $X$ being a spherical $t$-design, is to impose it saturates the Welch bounds for all $k\le t$. ...
glS♦
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How to sample from a unitary 2-design?
How do we actually go about sampling from a unitary 2-design? Because the size of the 2-design grows quickly with the number of qubits, it seems challenging to sample.
Some of the references I've ...
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Werner Twirling Channel - How to Retrieve Prefactors?
In Watrous' Theory of Quantum Information, Example 7.25 discusses the Werner Twirling Channel:
$$\Xi(X) = \int (U \otimes U) X (U \otimes U)^* \mathrm{d}\eta(U)$$
where $\eta$ denotes the Haar measure ...
glS♦
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What is the intuition behind quantum t-designs?
I started reading about Randomized Benchmarking (this paper, arxiv version) and came across "unitary 2 design."
After some googling, I found that the Clifford group being a unitary 2 design ...
glS♦
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Why can unitary 2-designs be characterised via twirling superoperators?
In (Dankert et al. 2009), the authors define a unitary t-design as a finite set of unitaries $\{U_k\}_{k=1}^K\subset \mathbf U(D)$ such that for all polynomials $P_{(t,t)}(U)$ of "degree at most $...
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What are well-known orthogonal 2-designs, other than the real Clifford group?
The paper Real Randomized Benchmarking
https://quantum-journal.org/papers/q-2018-08-22-85/
https://arxiv.org/abs/1801.06121
makes use of the fact that the real Clifford group is an orthogonal 2-design ...
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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 ...
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How to compute Haar average over the unitary group of a ratio of homogeneous polynomials?
I am interested in the following Haar average over the unitary group:
$\mathbb{E}_U\Big[\frac{tr(U^{\otimes p}|j\rangle\langle j|(U^\dagger)^{\otimes p}\rho \otimes \sigma ...)}{tr(U^{\otimes q}|j\...
glS♦
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Approximating unitaries with elements from a t-design
(This is basically a reference request)
I am wondering if there are any results out there on to what accuracy a given unitary can be approximated with an element drawn from a t-design.
To ...
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Optimality of the SWAP test versus weak Schur sampling for testing unitarily invariant properties
Consider the following setting.
I am either given the density matrix $|\psi\rangle \langle \psi|^{\otimes k}$ or the density matrix $\frac{\mathbb{I}^{\otimes k}}{2^{nk}}$, where $\mathbb{I}$ is the $...
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At what depth and for what architecture are random quantum circuits $1$-designs?
I was confused about something related to quantum $1$ designs.
Let us recap two facts we know about random circuit ensembles that form a $1$ design.
$1$ design, for a quantum circuit over $n$ qubits, ...
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