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10 votes
Accepted

What is the intuition behind quantum t-designs?

The $t$ in $t$-design is essentially a measure of how good a job the set of gates does in terms of randomising a state (the larger t, the more random, with properly random requiring the infinite limit)...
DaftWullie's user avatar
  • 59.9k
7 votes
Accepted

Why is the orbit of a unitary t design a complex projective t design?

One of the equivalent definitions of a unitary t-design $\{U_i\} \subset \mathbb{U}(d)$ is that $$ \frac{1}{n}\sum_{i=1}^n (U_i^{\otimes t})M(U_i^{\otimes t})^\dagger = \int_{\mathbb{U}(d)} (U^{\...
Danylo Y's user avatar
  • 7,482
5 votes
Accepted

Which Clifford groups are 2-designs?

The confusion stems from the existence of incompatible definitions of the "Clifford group" in dimensions which are not prime. With your definition, the Clifford group is indeed a unitary 2-...
Markus Heinrich's user avatar
5 votes

How to sample from a unitary 2-design?

Unitary 2-designs without efficient sampling access are arguably not very useful. Indeed, if you go by the textbook definition, then a unitary design is nothing but a set of unitaries and there's ...
Markus Heinrich's user avatar
4 votes
Accepted

Are MUBs complex projective 3-designs?

There is a bound for the size $n$ of a complex projective $t$-design (see Eq. 2.5 of Roy and Scott paper): $$ n \ge \binom{d-1+⌊t/2⌋}{⌊t/2⌋}\binom{d-1+⌈t/2⌉}{⌈t/2⌉}. $$ So, for a 3-design, $n\ge \...
Danylo Y's user avatar
  • 7,482
4 votes
Accepted

At what depth and for what architecture are random quantum circuits $1$-designs?

To study unitary $t$-designs, we define the moment operator with respect to a probability measure $\nu$ as $$ M_t(\nu) := \int_{U(d)} U^{\otimes t} (\cdot) (U^{\otimes t})^\dagger d\nu(U) \simeq \int_{...
Markus Heinrich's user avatar
3 votes
Accepted

What are well-known orthogonal 2-designs, other than the real Clifford group?

The paper https://arxiv.org/abs/1810.02507 does the following "Relying on the main results of [GT], we classify all unitary t-groups for t ≥ 2 in any dimension d ≥ 2." In this answer I'm ...
Ian Gershon Teixeira's user avatar
3 votes
Accepted

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

An element $[v]\in\mathbb{CP}^{d-1}$ can be represented by $\pi(v) = |v\rangle\langle v|$, which is a linear operator on $\mathbb{C}^d$. A unitary action $(U,[v])\mapsto [Uv]$ on ${\Bbb CP}^{d-1}$ ...
Danylo Y's user avatar
  • 7,482
3 votes

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

Thinking about this a bit more: the point is that we're not using the common way to build linear operators commuting with the representation by taking averages over the group elements. That is, we're ...
glS's user avatar
  • 26k
3 votes

Approximating unitaries with elements from a t-design

It’s straightforward to see that a random element $U$ drawn from a design is, with very high probability, far away from any fixed unitary $V$, i.e. you can show that the distance between $U$ and $V$ (...
4xion's user avatar
  • 176
2 votes
Accepted

Why can unitary 2-designs be characterised via twirling superoperators?

Eq. 6 is equivalent to $$ \frac{1}{K}\sum_k \langle i|U_k^\dagger A U_kXU_k^\dagger BU_k |j\rangle = \int_{\mathbf U(D)} \langle i|U^\dagger A UXU^\dagger BU |j\rangle, $$ for all $i,j \in [D]$. If we ...
Danylo Y's user avatar
  • 7,482
1 vote

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

Maybe you can use approximate or Simplified Models Stabilizer Rank Methods: For circuits that predominantly contain Clifford gates and a small number of T gates, you can use techniques based on the ...
Bram's user avatar
  • 652
1 vote
Accepted

Simulating Large Quantum Systems with Single T-Gate in Qiskit: Memory Error Beyond Certain Qubit Threshold

On IBM Quantum platform, a universal simulator is limited by 32 qubits. There is also 64-qubit simulator Clifford+T which could be suitable for your problem. You can also run your circuit on 127-qubit ...
Martin Vesely's user avatar
1 vote

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

Figured out after posting that the reference I was thinking about, which was also the same I mentioned in this other question, was (Scott 2006), where they discuss the statement at hand at the ...
glS's user avatar
  • 26k
1 vote

Are MUBs complex projective 3-designs?

Another straightforward approach is to consider the alternative definition of (complex projective) $t$-designs in terms of frame potentials. Namely, as also mentioned e.g. in this other post, a set $X\...
glS's user avatar
  • 26k

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