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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Density Matrix for a Quantum Circuit with Clifford Gates and a $T$ Gate in Qiskit

I am trying to analyze the impact of a single $T$ gate within a quantum circuit that primarily consists of Clifford gates. My goal is to understand the $T$ gate's role in $T$-design and Anti-...
1 vote
1 answer
49 views

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 ...
4 votes
2 answers
119 views

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 ...
6 votes
1 answer
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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 ...
7 votes
1 answer
131 views

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 ...
1 vote
1 answer
30 views

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 $k\le t$. ...
3 votes
1 answer
132 views

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 ...
1 vote
1 answer
50 views

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 ...
21 votes
1 answer
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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 ...
2 votes
1 answer
170 views

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 $...
5 votes
1 answer
279 views

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 ...
2 votes
2 answers
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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 ...
1 vote
0 answers
163 views

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\...
5 votes
1 answer
212 views

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 ...
1 vote
0 answers
83 views

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 $...
2 votes
1 answer
312 views

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, ...