Questions tagged [clifford-hierarchy]

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Is there a closure property for the entire Clifford hierarchy?

TL;DR Is the entire Clifford hierarchy (as opposed to any one level), a group? Background. The Clifford hierarchy (on $n$ qubits), is a collection of nested subsets $\mathcal C^{(1)} \subset \mathcal ...
Niel de Beaudrap's user avatar
8 votes
3 answers
778 views

Universal Gate Set, Magic States, and costliness of the T gate

The usual universal gate set is $\mathcal{C} + T$ where $\mathcal{C}$ is the Clifford group and $T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix} $ is the $\pi/8$ rotation gate. In ...
Eric Kubischta's user avatar
8 votes
1 answer
442 views

Inverses and the Clifford Hierarchy

Elements of the (qubit) Clifford Hierarchy are unitary matrices. For a good definition of the Clifford Hierarchy see: Is there a closure property for the entire Clifford hierarchy? While a complete ...
Jonas Anderson's user avatar
7 votes
1 answer
224 views

Is this single qubit gate in the Clifford hierarchy?

For a single qubit, the Clifford hierarchy is defined to be $$ \mathcal C^{(k)} = \Bigl\{ U \in \mathbf U(2) \mathrel{\Big\vert} \forall P \in \mathcal C^{(1)} : U P U^\dagger \in \mathcal C^{(...
Ian Gershon Teixeira's user avatar
6 votes
1 answer
182 views

Is the Clifford hierarchy finite?

This question is inspired by Is the Clifford group finite? Which shows that that the Clifford group (the second level of the Clifford hierarchy) is finite. (finite meaning finite mod global phases) ...
Ian Gershon Teixeira's user avatar
4 votes
1 answer
156 views

Can every unitary be approximated by gates from the Clifford Hierarchy?

For $k > 1$, we recursively define $\mathcal C^{(k)}(n)$ as $$ \mathcal C^{(k)}(n) = \Bigl\{ U \in \mathbf U(2^n) \mathrel{\Big\vert} \forall P \in \mathcal C^{(1)}(n) : U P U^\dagger \in \...
Ian Gershon Teixeira's user avatar
4 votes
0 answers
103 views

Is the Clifford hierarchy particularly useful beyond the third-level?

It is well known that the first three levels of the Clifford hierarchy (over $n$-qubits) $C_1, C_2, C_3, \dots, C_n, \dots $ correspond to $$C_1 \equiv \text{Pauli group}$$ $$C_2 \equiv \text{Clifford ...
R.W's user avatar
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3 votes
2 answers
585 views

Where does the Clifford circuits stand in the complexity hierarchy?

So, it's well known that given magic states as inputs, one can perform any quantum computation using only Clifford gates, and it's also known that the running of Clifford gates on the zero-state can ...
Dudu Ponar's user avatar
3 votes
1 answer
147 views

Can Clifford gates be diagonalized using a gate from the third level of the Clifford hierarchy?

Is it always possible to diagonalize a Clifford gate $ g $ using a gate $ V $ from the third level $\mathcal{C}^{(3)}$ of the Clifford hierarchy? In other words can every Clifford gate be written as $...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
164 views

Which monomial matrices are in the Clifford hierarchy?

This is essentially a follow-up on the very interesting answer given here Is there a closure property for the entire Clifford hierarchy? I'm interested in sufficient conditions to conclude that a ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
115 views

Are all powers $g^m$ in the Clifford hierarchy if $g$ is?

It is already known that the Clifford hierarchy is not closed under arbitrary products, see this post which shows that the product $ THT $ is not in any level of the hierarchy. What about products of ...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
75 views

Diagonal gates in qubit Clifford hierarchy are generated by $ C^i Z^{1/2^j} $

Let $ \mathcal{C}^{(t)} $ denote the $ t $ level of the $ n $ qubit Clifford hierarchy. Let $ \mathcal{F}^{(t)} $ denote the collection of all diagonal gates in $ \mathcal{C}^{(t)} $. $ \mathcal{C}^{(...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
214 views

The Clifford hierarchy and $ e^{2 \pi i/2^k} $

Could someone give me an example of a gate in the Clifford hierarchy which cannot be written as $$ e^{i \theta} V $$ for some unitary $ V $ with entries in terms of $ \zeta_{2^k} $? If no such example ...
Ian Gershon Teixeira's user avatar
2 votes
0 answers
67 views

How many gates are in the $ k $ level of single qubit Clifford hierarchy?

Define the single qubit Clifford hierarchy recursively by $$ \mathcal{C}^1:=<iX,iZ> $$ the determinaint 1 subgroup of the Pauli group. Define the rest of the the hierarchy inductively by $$ \...
Ian Gershon Teixeira's user avatar
1 vote
1 answer
63 views

Is every diagonal gate whose non-zero entries are $2^k$th roots of unity in the two qubit Clifford hierarchy?

Does the two qubit Clifford hierarchy contain all diagonal gates whose entries are $ 2^k $ roots of unity? In particular, is it true that every $ 4 \times 4 $ diagonal matrix whose diagonal entries ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
20 views

Why is $\exp(i \pi / 2^\ell Y)$ in the $\ell$ level of the Clifford Hiearrchy?

I'm confused about the definition of the Clifford hierarchy and would like resources to learn more. For example, in https://arxiv.org/abs/1603.04230, they define the hierarchy as: However, in prior ...
C. Kang's user avatar
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