For questions about quantum t-designs: probability distributions over states or unitaries that replicate specific properties of the Haar distribution.
Questions tagged [t-designs]
21 questions
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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 is a specific case of "Quantum t-design."
I read the…
Blackwidow
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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 matrix sending $ |y \rangle \mapsto (e^{2 \pi i…
Ian Gershon Teixeira
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Why is the orbit of a unitary t design a complex projective t design?
The paper Qubit stabilizer states are complex projective 3-designs states in the final paragraph that "any orbit of a unitary t-design is a complex projective t-design." Using this fact one can take the simple proof that the Clifford group is a…
Ian Gershon Teixeira
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What are well-known orthogonal 2-designs, other than the real Clifford group?
The paper Real Randomized Benchmarking (arXiv) makes use of the fact that the real Clifford group is an orthogonal 2-design on $ n $ qubits in order to do randomized benchmarking (in other words, it uses that fact that the real Clifford group is a…
Ian Gershon Teixeira
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What's the relation between spherical and complex-projective t-designs?
A statement found for example in (Ambainis Emerson 2007) is
Quantum $t$-designs are related to $t$-designs of vectors on the unit sphere in $\mathbb{R}^N$, called spherical $t$-designs.
[...]
A spherical $t$-design in $\mathbb{R}^N$ can be…
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 elaborate a bit, consider the Haar measure over $U(d)$.…
Alex May
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Why does the definition of a unitary t-design use tensor products?
A t-design $X$ is defined by the following equation
$$\frac{1}{|X|} \sum_{U \in X} U^{\otimes t} \otimes\left(U^*\right)^{\otimes t}=\int_{U(d)} U^{\otimes t} \otimes\left(U^*\right)^{\otimes t} d U$$
where the right hand side uses the Haar measure…
user1936752
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Does Levy's lemma hold for unitary/spherical designs?
Let $\mathcal{H}$ be a $d$-dimensional Hilbert space equipped with the Haar measure. Levy's lemma says that, for an $L$ -Lipschitz function $f$ on $\mathcal{H}$, the probability that $f(x)$ for a randomly drawn unit vector $x$ deviates from its…
Banach space fan
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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 X} \mathbb{P}_x = \frac{\Pi_{\rm…
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 found mostly focus on utilizing Hamiltonian dynamics…
C. Kang
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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 projective space), denote with $\mu$ the uniform…
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 $t$ in the matrix elements of $U$ and at most $t$ in…
glS
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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, means that the density matrix of the ensemble is…
BlackHat18
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Simulating Large Quantum Systems with Single T-Gate in Qiskit: Memory Error Beyond Certain Qubit Threshold
I'm currently conducting experiments on unitary t-designs, utilizing random Clifford and T gates within the Qiskit framework. My goal is to simulate quantum circuits that involve the application of a single T-gate across various system sizes.…
Asim Sharma
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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 $k\le t$. Following the notation in…
glS
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