# Questions tagged [quantum-gate]

For questions regarding usage, performance, implementation, application or theory related to quantum gates.

1,803 questions
Filter by
Sorted by
Tagged with
166 views

### Why does $H^2=X^2 =I$ not imply $H=X$?

If $HH = I$ and $XX =I$, then is $H=X$? One may argue that $HH = I = XX$ or, $HH = XX$ so, taking the square root, is $H = X$? This is absurd but how to disprove it?
1 vote
107 views

### Simulator able to run SwitchCase in qiskit?

I am trying to run my circuit (shown below) on a simulator. However, the execution of the circuits fails with (almost) all the simulators, raising error ...
278 views

### Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

I see here in Olivia DeMatteo's notes, she states: When we consider the action of the entire Clifford group on a single non-identity Pauli, it maps that Pauli to each of the $d^2 − 1$ other possible ...
1 vote
44 views

165 views

### Can the ancilla for Qiskit's mcx with mode="recursion" be dirty?

I think their implementation is close to the one from here which does use a dirty ancilla, but I just want to make sure
234 views

### What are the possible gates that I can use to vary input states before CNOT?

We know that the noise of the $\text{CNOT}$ gate varies depending on the input state before it. What are the gates that I can put in the place of the 4 identity gates above to change the input state ...
1k views

1 vote
187 views

### How to implement an unitary operator expressed as a linear combination of unitaries without qubits ancilla

Let's say that I know the decomposition of a unitary operator $\hat{A}$ in terms of other unitary operators $\{U_k\}_{k=0, \dots, M}$, i.e: $\hat{A} = \sum_k \alpha_k U_k$. I know how to implement in ...
134 views

### Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

There is another question asked on this on stack exchange but I did not find any answers there that fully answered the question. In Gottesman's paper "The Heisenberg Representation of Quantum ...
1k views

### What is the difference between quantum gates and quantum channels?

I'm not sure if this is a dumb question, since they seem to be very basic building blocks of quantum information theory; however, I can't seem to wrap my head around the difference between the two. As ...
345 views

### 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 ...
435 views

### Exotic transversal gate group for stabilizer code

What are examples of interesting $[[n,1,d]]$ or $[[n,2,d]]$ stabilizer codes, $d \geq 2$, whose group of transversal gates is not isomorphic to a subgroup of the Clifford group (on 1 and 2 ...
416 views

### How do you embed a POVM matrix in a Unitary?

In QuantumKatas Measurement Task 2.3 - Peres-Wootters Game, we are given 3 states A,B and C. We construct a POVM of these states. But how do we convert that POVM into a Unitary that we can apply. ...
430 views

### Realizing a swap gate using a commutator sequence and an auxiliary qudit

Say I have two qudits $1$ and $2$, each of which has Hilbert space of dimension $m$. Is it possible to introduce an auxiliary qudit $a$ (of any dimension $d_a\in \mathbb{Z}_{\geq 2}$) and find quantum ...
105 views

2k views

154 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 ...
587 views

### How many $\sqrt{X}$ are there?

I was reading this post and it got me thinking about square roots. Recall the Pauli $X$ gate $$X=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix};$$ How many unitary matrices $U$ are there ...
63 views

### time evolution of Hamiltonian to generate the Bell pair

Consider two different Hamiltonians: $$H_1(t) = ZZ + \alpha(t)X_1 + \beta(t)X_2\quad\text{ and }H_2(t) = XX + \alpha(t)Z_1 + \beta(t)Z_2\,,$$ where $\alpha(t)$ and $\beta(t)$ are time-dependent ...
1 vote
286 views

This is an incredibly basic question, but basically I'm really struggling to understand what the "addition modulo 2" is and why is it used in quantum computing. I've tried Wikipedia, endless ...
88 views

### Is the normaliser of the Clifford group the Clifford group itself?

The Clifford group is defined as the normaliser of the group of multi-qubit Pauli unitaries. What is the normaliser of the Clifford group? Since Paulis are themselves Cliffords, such a unitary in the ...
1 vote
180 views

### How to fix two flip-bit errors in a 3 qubit input

Aware of how a 1-bit flip can be fixed for 3-qubit input by having 2 Ancilla bits, encoding, and using the Toffoli gate to fix the error and decode. How can this idea be extended for a 2-bit flip (...
386 views

### Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not

In Exercise 10.40 of Nielsen and Chunang's textbook, the reader is supposed to construct an inductive proof of Theorem 10.6 that any Clifford gate can be made of Hadamard, phase and c-not. There it is ...
64 views

### How is $R_{ZZ}(\pi/2)$ gate maximally entangling?

IBM Qiskit website says $R_{ZZ}$ gate is maximally entangling at $\theta = \pi/2$. But acting $R_{ZZ}$ on two qubits initialized as $|00 \rangle$ still remains $|00 \rangle$ so it is separable in ...
45 views

### What is the difference between Gate.power() and Gate.repeat()?

Why are the gates a and b in this code not the same? a = UGate(0,0,0.9*np.pi).power(2) b = UGate(0,0,0.9*np.pi).repeat(2) I thought that unitary gates function ...
### 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}^{(...