# Questions tagged [t-designs]

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

16 questions
Filter by
Sorted by
Tagged with
31 views

### 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
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 ...
143 views

### 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 ...
• 3,762
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 ...
• 3,762
1 vote
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 $k\le t$. ...
• 24.8k
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 ...
• 1,716
1 vote
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 ...
• 105
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 ...
• 24.8k
170 views

1 vote
83 views

• 1,313
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 ...
• 201