Questions tagged [quantum-gate]
For questions regarding usage, performance, implementation, application or theory related to quantum gates.
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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?
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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
...
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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 ...
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Build a query circuit that set the first bit to 0
This is a question from Chapter 2, 8(c) in Quantum Computing: Lecture notes by Ronald de Wolf.
Suppose we can make queries of the type $|i, b\rangle→ |i, b\oplus x_i\rangle$ to input $x\in\{0, 1\}^N, ...
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Forward propogation of rotated clifford gates in lattice surgery
In A Game of Surface Codes (arXiv), I have been trying to understand the process that is described for forward propogating rotated clifford gates through other rotated clifford gates, and either I am ...
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Custom Quantum Gate equivalence to backend basis gate set
Hello I am using Qiskit and I have defined a custom gate which implements the array shown in the image:
This array is unitary and the function R(theta) works by tuning a continious variable into 0 of ...
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Finding a unitary transformation to swap the control bit
Suppose we have a controlled-$U$ gate for some unitary $U$. The convention is that the control-bit is the first bit. I'm interested in finding unitary $V$ so $V^{\otimes 2}(cU)V^{\otimes 2}$ is the &...
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How would you write $ e^{i \frac{\pi}{4} \text{CNOT}}$ in ZX calculus?
How would you write $ e^{i \frac{\pi}{4} \text{CNOT}}$ operation in ZX calculus? Is there any simple method to write gates of the form $e^{i \frac{\pi}{4} \sigma_z^{(1)} \sigma_x^{(2)}}
$?
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IBM quantum composer possible bug?(state amplitudes of quantum circuit)
I have this quantum circuit:
and from the IBM quantum composer I get this state amplitudes:
I wanted to flip the target qubit and keep only its pure state however this is not what I am getting.I am ...
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How to realize the index shift operation in quantum circuit?
Can we realize the index shift in a quantum circuit as we do in a classic circuit?
$\forall$ $a,b,c \in \{0,1\}$, $\left|abc\right>\mapsto\left|bca\right>$
e.g. $\left|001\right>\mapsto \left|...
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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
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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 ...
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Given circuits preparing $|\psi\rangle$ and $|\phi\rangle$, what's a circuit preparing $|\psi\rangle+|\phi\rangle$?
Given a quantum circuit $C_1$ that generates a state $\vert\psi\rangle$ and another circuit $C_2$ that generates $\vert\phi\rangle$, is there a way to construct a circuit that outputs
$$\frac{1}{\sqrt{...
glS♦
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Small, distance 2 quantum codes with transversal Clifford group
Are there any small, distance 2 quantum codes with a transversal Clifford group? For example, the 7-qubit Steane code is distance 3 but has the full single-qubit Clifford group. Are there smaller ...
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Given a non-Clifford quantum circuit $U$, is it possible to construct a commuting Clifford circuit $C$?
Let a non-Clifford circuit $U$, say $$U = \prod_{i=1}^k e^{i \theta_i P_i} $$ for $P_i \in \{ I,X,Y,Z\}^{\otimes n} $ with $\theta_i \in \mathbb{R}$ be given. Is it possible to construct a non-trivial ...
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Why are rotation gates defined via the exponential?
Consider the standard definitions $R_Z(\theta) := e^{-i\frac{\theta}{2}Z}$ and $R_{ZX}(\theta) := e^{-i\frac{\theta}{2}XZ}$. My question is: why is the rotation gate defined in an exponential format? ...
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Why is the CNOT representation $e^{i\frac{\pi}{4}\left(I-Z\right)\otimes\left(I-X\right)}$ hardly found in books?
The CNOT gate is usually written as
$|0\rangle\langle0|\otimes I + 1\rangle\langle1|\otimes X$
(with $X,Y,Z$ being the Pauli Basis and $I$ the Identity).
I have yet to stumble across the ...
glS♦
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Coding a hamiltonian in qiskit
I have a hamiltonian of the form:
$H=\sum_{i=1}^N Z_i Z_{i+1}-Z_NZ_1$
And another one as:
$H=-\sum_{i=1}^N X_i$
I need it to it for N terms.
I am a bit lost can anybody help. I tried looking for ...
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Generators for single qudit Clifford, $d=4$
The generators for single qubit Clifford are phase $ P $ and Hadamard $ H $. The generators for single qutrit Clifford can be found for example here.
What is a small/minimal generating set of matrices ...
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Approximating the concatenation of two approximate circuits
Suppose I have two quantum circuits $A_n,B_n$ that I have already found to approximate the operations $U,V$ within some error $\epsilon_n$ and each with an overall circuit depth $\ell_n$ using $n$ ...
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Is it possible to approximately compile Toffoli using H and CSWAP?
Question:
Given all controlled-SWAP (CSWAP) and Hadamard (H) gates on 3 qubits, is it possible to approximately compile the Toffoli (CCX) gate?
Discussion
From basic simulations, it appears that all $\...
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Equivalence checking of quantum circuits up to error
Suppose you are given two circuit descriptions $A$ and $B$ where by a circuit description I mean a sequence of gates (in the order they are applied) and the qubits they are applied on. (For the sake ...
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What are the possible non-entangling two-qubit gates?
The non-entangling gates in $ SU_4 $ contains the entire group of gates of the form
$
SU_2 \otimes SU_2.
$
It also contains
$$
\zeta_8 SWAP= \zeta_8 \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 &...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
...
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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 ...
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How is the computational power defined in Quatum Computing?
In classical data centers, the computational power is expressed as function of the number of executed tasks per second as follows :
$$P(\lambda)=P_{idle}+(P_{peak}-P_{idle})\cdot(\lambda/CPU_{capacity}...
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What are the Error Reduction Factors for Shor’s EC, Knill’s Gadget, Steane’s Gadget, and Flag Gadgets?
Similar to how cat state gadgets make the number of errors $p$ into $(1-p)^m$ what do the aforementioned gadgets do? If not the error reduction, what is the threshold?
Even if it is a link to a ...
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How many single and double qubit gates are required to create a uniform superposition of vertices of a Johnson Graph J(n,r)?
Can I in $\tilde{O}(r)$ number of gates (single and double qubit) create a uniform superposition of vertices of Johnson Graph $J(n,r)$? I would like to create a state $|\psi\rangle = \frac{1}{\sqrt{n \...
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Parameterized Gates [duplicate]
I am trying to create my own custom parameterized gate and then transpile it to the AerSimylatorGate. I have been able to create the virtual circuit but in the transpilation to the backend basis set ...
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Numerical values for the volume of qubits
This paper mentions the volume of qubit in equation (9) given by:
$$Q_0=U*C_1*T_o*n_p^{(2/3)}*(v_q)^{(2/3)}$$
where
$Q_0$ is the heat entering the chamber
$U$ is the heat transfer coefficient
$C_1$ ...
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Matrix representation of any conditioned gate
Is there an algorithm explaining how to represent any gate in the matrix form?
Suppose, the circuit is the following:
where
$$ U := e^{iA\pi/4} =
\begin{bmatrix}
0.35-0.85i & -0.35-0.15i \\
-0....
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How Does the Qiskit Reset Gate Affect Other Entangled Qubits
I am trying to understand how the reset gate in Qiskit affects qubits its entangled with. Consider the following circuit with qubits $q_0$ and $q_1$:
Where circuit240 takes $|0\rangle$ to $a|0\rangle ...
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Transpilation error: CircuitError: 'The amount of qubit(0)/clbit(0) arguments does not match the gate expectation (2).'
I am trying to implement a algoritm that reduces the number of qubits required for MAXCUT optimization. I need to create a custom gate that encodes a diagonal unitary matrix where the diagonal ...
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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
$...
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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 ...
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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 ...
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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 ...
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Modular Addition general explanation
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 ...
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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 ...
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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 (...
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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 ...
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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 ...
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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 ...
glS♦
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Link between quantum computing and Lie theory?
I know only little about Lie theory but I would like to learn more about its link to quantum computing.
Has someone got some references explaining it well?
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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}^{(...
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