Questions tagged [unitarity]
For questions related to the unitarity (unitary evolution) of quantum systems, as applicable to quantum computing or quantum information.
197
questions
3
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1
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Why is the operator $M_a |x\rangle= |a \cdot x \pmod{N} \rangle $ unitary?
If $N\geq 2$, $a\in \mathbb{Z}_N$, and $a^r= 1$ for some $r$. Consider the operator $M_a$, which is related to order finding :
$M_a |x\rangle= |a \cdot x \pmod{N} \rangle $ if $x\in \mathbb{Z}_N$
What ...
- 51
3
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2
answers
122
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What do the values in a unitary matrix represent/what do they mean? How do you figure out what gate a unitary matrix represents?
I am trying to learn Qiskit on my own. I am struggling with unitary matrices. I understand what a unitary matrix is, and why a matrix is unitary. But, I don't understand what the values inside of the ...
- 31
1
vote
0
answers
30
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Why does unitary matrix acts only on input qubit state of a vector that is a result of add modulo 2?
Let $|\psi\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k \oplus b\rangle $ so that $|\psi'\rangle = \frac{1}{\sqrt{2}}[\sum_{k=2}^{1}(-1)^{ka}(U_{A}|k\rangle \otimes U_{B}|k \oplus b\rangle)]...
- 510
2
votes
1
answer
74
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Hadamard gate for qudit (dimesion 8)
I am using the code for generating the gates for qu8it
The resultant gate is not unitary (and so not hermitian too). I am a bit confused with the result. Is it possible for a quantum gate to be non-...
- 614
4
votes
2
answers
273
views
closeness between two unitaries on the bloch sphere
The fidelity between two (single-qubit) quantum states can be easily translated into the euclidean distance between the two states on the Bloch sphere (hilbert-schidmit distance). I'm curious if this ...
- 127
1
vote
0
answers
71
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How to modify the Hadamard test for a non-unitary operator
Assuming I am doing statevector simulations, I need to compute an inner product of the type
$$
X_b = \langle\psi | I_0^{\otimes (n-1)} \otimes X | \psi \rangle,
$$
where $\psi$ is a generic input ...
- 59
1
vote
3
answers
155
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Map $n$ qubit state with complex amplitudes to $n+1$ qubit state with real amplitudes
For the simplest case, consider a single qubit state $|\psi\rangle$, and assume access to a state preparation unitary $V$ satisfying
$$
V|0\rangle = |\psi\rangle
$$
and
$$
V|1\rangle = |\perp\rangle.
...
- 611
3
votes
2
answers
275
views
Sufficient conditions for a single-qubit unitary to be the identity
Say I have a unitary $U = e^{-iHt}$ where $H = \alpha X + Z$.
First, suppose $U = I$. Then it rotates a set of initial states to themselves. Say I'm working on a computational basis, then on the Bloch ...
- 127
2
votes
0
answers
35
views
Error in repeated applications of a quantum channel?
Suppose I have two quantum channels. Assume they they consist of $r\in \mathbb{Z}$ applications of unitaries, $U$ and $V$ respectively. Let the error between the channels acting on some state $\rho$ ...
- 131
6
votes
2
answers
104
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Conditions for entangling $A$ with $C$ via an interaction on $AB$
I have three qubits in subsystems $A$, $B$, $C$. System $A$ initially contains some state $\rho_A$, and $BC$ contains a bipartite pure state $|\psi\rangle_{BC}$. I apply a unitary operation $U$ acting ...
- 6,108
1
vote
1
answer
43
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Unitarity of a matrix in the EPR experiment
I'm having troubles in understanding a statement in Box 2.7 at page 113 in the Nielsen & Chuang.
Firstly, it assumed to be working with a two-qubits quantum system in state $|\psi\rangle = \frac{|...
- 11
2
votes
2
answers
43
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Proof that two sets of quantum maps are equivalent only when they are related by a unitary transformation
I am trying to show that the two different quantum maps $\rho'=\sum_{\alpha} K_{\alpha} \rho K_{\alpha}^{\dagger}$ and $\rho''=\sum_{\beta} L_{\beta} \rho L_{\beta}^{\dagger}$ are equivalent i.e. $\...
- 237
2
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1
answer
58
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What's the reasoning behind writing the isometric representation of a channel?
I am reading about phase damping channel from Preskill's notes. He writes off the unitary representation of the channel as
Unitary representation. An isometric representation of the channel is
\begin{...
- 237
3
votes
1
answer
35
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On unitary matrix form suggested in the Elementary gates paper
In the Elementary gates for quantum computation paper by Barenco et al authors start their proofs by defining a generic form of 2x2 unitary matrix of $\mathbb{C}$ as follows:
Can you help me with the ...
- 133
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0
answers
88
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How to implement a PauliSparseOp object in qiskit if its non unitary?
I have a PauliSparseOp object whose matrix is non-unitary, is there a way I can implement this on qiskit? It is written in terms of sum of tensor products of I,X,Y and Z operators.
- 149
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votes
3
answers
138
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Finding the rotation angle $\theta$ of a 2x2 unitary matrix
We can represent a 2x2 unitary matrix as follows:
$$U = \cos(\theta)I - i \sin(\theta) \vec{n} \cdot \vec{\sigma},$$
where $\vec{n} \in \mathbb{R}^3$ and $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$...
- 2,359
4
votes
1
answer
64
views
Definition of quantum junta is not basis independent: isn't this a problem?
A quantum $k$-junta is defined as a unitary matrix $U$ acting on $n$ qubits which has the form $U = V \otimes \mathbb I$ where $V$ is a unitary acting some $k < n$ of the qubits. The fact that a ...
- 485
1
vote
1
answer
104
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What is the general unitary matrix for two- and three-qubit states?
As pointed out in the QisKit tutorial here, for one qubit there exists a general unitary (see the expression for it in the previous link). I wonder if there exists equally unambiguous expressions for ...
- 265
0
votes
0
answers
46
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Understanding notation regarding phase oracle continued
Since I am not able to comment on my post, I had to register an account and start a new post. Maybe my old post can be deleted, which is found : here
I am new to quantum computing. I cannot get my ...
- 41
0
votes
1
answer
116
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Are permutations of the Pauli strings unitary operations?
Consider the set of Pauli strings $P_N=\{\tau \}$,
composed out of tensor products of Pauli matrices $\sigma_i^\alpha$
acting on $N$ or qubits, e.g.
$\tau=\sigma^x_1 \otimes \mathbb{1}_2 \otimes \...
- 360
2
votes
0
answers
44
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Understanding notation regarding phase oracle [duplicate]
I am new to quantum computing. I cannot get my head around the following:
Consider a finite random variable $X : \Omega \rightarrow E$ on a probability space $(\Omega, 2^{\Omega}, P)$. Let $H_{\Omega}$...
- 41
0
votes
0
answers
35
views
Detect if a given binary number belongs to a certain subset with an unitary transformation
I want to create an operator $A$ which, given three binary numbers, $a_1$, $a_2,a_3$, will detect whether $a_1a_2a_3$ (as a binary number) is in certain set of numbers (for example, detect whether $...
- 101
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votes
1
answer
88
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Twirling of quantum states: Maximally entangled states
I have been reading the paper "Resource theory of unextendibility and non-asymptotic quantum capacity" (https://arxiv.org/pdf/1803.10710.pdf) by Kaur et.al, I have two questions ...
- 13
1
vote
1
answer
116
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How to write a two qubit state as "diagonal" in the basis of Pauli matrices?
Any two qubit density matrix can be written as
$$ \rho = \frac{1}{4} \sum_{n,m = 0}^{3} M_{nm} (\sigma_n \otimes \sigma_m), $$
where $\sigma_\mu$'s are the identity and Pauli matrices.
Is it possible ...
- 325
0
votes
1
answer
41
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How to create a unitary matrix from a circuit
I've been trying to find a way to analytically find the unitary matrix of a circuit, but I cant find the resources to do so. How can I do so?
1
vote
1
answer
116
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How to prove that maximally entangled state remains maximally entangled under local unitaries?
We have a maximally entangled state $\phi$ of composite system $R$ and $Q$. Apply unitary $U$ to $\phi$ on the Q system. Now how to prove that $U \phi$ is also a maximally entangled state?
This ...
- 505
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vote
1
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174
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How use UnitaryGate to creat a CNOT gate?
This is my code using qiskit below. I am not familiar with Unitarygate, so I tried to creat a cnot-gate.
...
- 11
0
votes
0
answers
32
views
defining a unitary isometry
I am defining the coefficient unitary in full details but stuck. I tried many ways so that the cross terms gets cancelled and the diagonal terms has one.
- 1
0
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1
answer
43
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How to implement the -I matrix using Pauli gates
I'm trying to build a quantum walk circuit. I have the C0 matrix as follows
import numpy as np
C0 = np.array([[-1, 0], [0, -1]])
As we can see, it's the (-)...
- 547
2
votes
2
answers
504
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Arbitrary two qubit unitaries using SWAP gate instead of CNOT gate
An arbitrary two qubit gate can be constructed using local operations with CNOT gate. Are there other ways to implement these gates in this manner? In particular, can I decompose a two qubit gate in ...
- 325
0
votes
2
answers
70
views
Is there a criteria to ensure a one-qubit operator is exactly of the form $R_n(\theta)$ (i.e without a global phase $e^{i\alpha}$)?
Reading the Nielsen and Chuang, I saw that every unitary operator $U$ can be written as $e^{i\alpha} R_n(\theta)$ for some well chosen $n \in \mathbb{R}^3$ and $0 \leq \theta < 2\pi$.
I would like ...
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5
votes
1
answer
152
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Prove that maximally entangled states $|\Phi\rangle$ satisfy the identity $(U\otimes I)|\Phi\rangle=(I\otimes U^T)|\Phi\rangle$
The definition of maximally entangled state is
\begin{equation}
\vert \Phi \rangle = \frac{1}{\sqrt{d}} \sum_i \vert i \rangle \vert i \rangle,
\tag{1}
\end{equation}
where $d$ is the dimension of the ...
- 505
4
votes
1
answer
92
views
Is the function PU(2) and SO(3) induced by the Bloch sphere bijective?
I have difficulty understanding the fact that, as written in this reference,
every single-qubit
unitary corresponds to a unique rotation of R3 and vice versa.
If I understand well, this means there ...
- 81
4
votes
2
answers
116
views
How to check a given unitary evolution is correct in a real quantum computer in Qiskit?
For a given unitary, I want to know whether this unitary gate is correctly evolved in the circuit. In the simulator, I can use "statevector" to get the state vector to check the correctness ...
1
vote
1
answer
84
views
Does the Bell's state entanglement violate the reversibility property of unitary matrices?
I read unitary matrices are reversible, so when we apply a unitary operator $U$ on some input state and got an output state, then if we apply $U^\dagger$ (transpose conjugate) we get back the original ...
- 25
5
votes
2
answers
486
views
Doing non-unitary operations on quantum computer
So I am trying to implement non-unitary operations on Qiskit.
There is an option to perform conditional operations in Qiskit.
Suppose I prepare a qubit state in superposition. $|\psi\rangle=\sqrt{\...
- 332
3
votes
1
answer
52
views
Complementary channel of binary sum channel
This isn't strictly a quantum question but the idea of complementary channels is the following: Take any channel $N_{A\rightarrow B}$. Take it's Stinespring dilation (which is an isometry) $V_{A\...
- 2,577
2
votes
3
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197
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How many parameters do we need to characterize a pure state?
Suppose I have a pure qubit. I can think of starting with the state $\vert 0\rangle$ and apply some unitary to it. Such a unitary has three parameters according to this link. In $d$ dimensions, the ...
- 23
3
votes
1
answer
55
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Why is it safe to ignore the phase factor when working with unitary operations? (and potentially elsewhere?)
After not understanding the explanation of the no-cloning theorem proof in my lecture notes I turned to Wikipedia, this explanation made more sense to me however it had an extra phase factor that is ...
- 129
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votes
1
answer
202
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calculating the unitary of a circuit using Qiskit's simulator
i am trying to verify qiskit's get_unitary() result. this is my code:
...
- 37
3
votes
2
answers
283
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How does a quantum system identify hermitian and unitary matrices?
I am a beginner in quantum computing. I know that multiplying a state $|u\rangle$ with a hermitian matrix $M$ yields spectral decomposition and multiplying $|u\rangle$ with a unitary matrix yield an ...
- 37
7
votes
1
answer
98
views
If a quantum algorithm requires a measurement, how can we use that as a subroutine in another quantum algorithm?
Some algorithms (like period finding), use one or more measurement step. The post measurement state is then acted upon by another set of gates to complete the algorithm.
If I imagine this as blackbox ...
- 71
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votes
1
answer
145
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When proving the Solovay-Kitaev theorem, why do we consider a small neighborhood $S_\epsilon$ of the identity?
There are number of points I haven't understood or am confused in the proof of Solovay-Kitaev theorem. The proof I'm reading in given in the Appendix 3 of Neilson and Chuang's book, Quantum ...
- 311
1
vote
2
answers
108
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How to check if a mapping is unitary?
In the case of the No-cloning theorem, it is argued that a unitary $U$ that is capable of performing coping does not exist. Specifically, for any two unknown states $|\psi_1\rangle$ and $|\psi_2\...
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votes
2
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128
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How to perform the measurements on a quantum circuit in W state basis?
I need to perform the measurements on a quantum circuit in the basis $\{ \eta^\pm,\zeta^\pm \} $. Where $ \eta^\pm,\zeta^\pm $ are given as follows:
$$\eta^\pm = \frac{1}{2}|001\rangle + \frac{1}{2}|...
- 53
1
vote
1
answer
53
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Unitary operations in a Quantum Neural Network
I'm currently reading Classification with Quantum Neural Networks on Near Term Processors and I'm having trouble with one of the calculations.
The system is composed of $n+1$ qubits, $n$ of those are ...
- 93
2
votes
0
answers
64
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Quantum Signal Operator and the unitary state preparation oracle? [closed]
I am looking into IL Chuang and GH Low's Hamiltonian Simulation with Qubitization paper.
I am very confused on the terminology and motivation behind definition 1.
I do not understand what the unitary ...
- 383
2
votes
2
answers
224
views
How to find a circuit for a unitary operator $e^{-i s |v \rangle \langle v| t }$?
Let $|v \rangle$ be an eigenstate of an $n$-qubit and $2$-local Hamiltonian
$$H = \sum_{i=1}^n \left (X_i + a_i Z_i \right) + \sum_{(i,j)} b_{i,j} Z_i Z_j,$$
where $\sigma_i = I \otimes \cdots \...
- 2,359
2
votes
2
answers
187
views
Understanding different forms of an arbitrary Unitary transformation in $\mathcal{H}_2$
I'm working to have a greater understanding of the arbitrary unitary transformation matrix when working in the context of the Bloch sphere. At this time I have found several equivalent ...
- 452
2
votes
1
answer
192
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What is the tensor product expression for the following quantum circuit? [duplicate]
Qiskit generates the following matrix for this 3-qubit CNOT circuit.
Can anyone explain how do we get this mathematically ?
This is the Quantum Circuit
This is the Output of Unitary Simulator
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