Questions tagged [unitarity]

For questions related to the unitarity (unitary evolution) of quantum systems, as applicable to quantum computing or quantum information.

Filter by
Sorted by
Tagged with
0 votes
0 answers
12 views

Multimode unitary channel in terms of action on characteristic function

Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\...
Phil K.'s user avatar
1 vote
1 answer
61 views

Inner product as unitary operation

Inner products of two states $\psi$ and $\phi$ are usually performed at the end of a quantum algorithm where we measure the final state, e.g. using the swap test. However, this operation is not ...
Medulla Oblongata's user avatar
2 votes
1 answer
86 views

U(2) vs. SU(2) for single-qubit gates; ignoring global phases

So, while the only immediate restriction on an operator evolving a quantum state in time, is that it be unitary, in quantum computation/information, it is considered somewhat common knowledge that all ...
seba2390's user avatar
5 votes
5 answers
119 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 ...
am567's user avatar
  • 569
1 vote
1 answer
69 views

I would like to understand the meaning of applying permutation to a unitary matrix

$$U = \frac{1}{2} \begin{pmatrix} -1 & -1 & 1 & 1 \\\\ 1 & -1 & 1 & -1 \\\\ 1 & -1 & -1 & 1 \\\\ 1 & -1 & 1 & 1 \end{pmatrix}$$ $$P = \frac{1}{...
junghyunHa's user avatar
1 vote
1 answer
47 views

Simplification of a generic quantum state

We are given a generic 2-qubit density matrix $$\rho=\frac{1}{4}\left[I_4+\Sigma_i a_i \sigma_i \otimes I_2 + \Sigma_i b_i I_2 \otimes \sigma_i + \Sigma_{i,j} c_{ij} \sigma_i \otimes \sigma_j\right]$$ ...
Anindita Sarkar's user avatar
3 votes
1 answer
46 views

Can a generic 2-qubit state be unitarily converted into one of the form $I_2\otimes I_2+\lambda\sigma_z\otimes\sigma_z$?

Suppose I have a general 2-qubit state written in a basis consisting of tensor products of Pauli matrices: $\rho=\frac{1}{4}\left[I_2\otimes I_2+\Sigma_{i} a_i \sigma_i\otimes I_2+\Sigma_{i} b_i I_2\...
Anindita Sarkar's user avatar
1 vote
1 answer
45 views

conditions for two hermitians operators same up to unitary

Let $A$ and $B$ $2^n \times 2^n$ Hermitian matrices. What are sufficient and necessary conditions that they are equal up to some unitary, i.e. there exists $U$ such that $A = U B U^\dagger$? The first ...
Jon Megan's user avatar
  • 497
2 votes
0 answers
40 views

Unitary Matrix for Quantum Random Walk

I am re-working a project which aims to implement a very simple version of a quantum random walk. It actually is a simplified binary random walk on a cyclic graph. The idea is depicted in the ...
zuluratman's user avatar
5 votes
1 answer
35 views

Upper bound on $\Vert U_1 \otimes U_2 \otimes \cdots \otimes U_k - V_1 \otimes V_2 \otimes \cdots \otimes V_k \Vert$

Let $U_i$ and $V_i$ be unitaries that act on the same subsystems. Can we upper bound the difference between the tensor products of these unitaries, i.e. $\Vert U_1 \otimes U_2 \otimes \cdots \otimes ...
Mohan's user avatar
  • 161
1 vote
1 answer
127 views

Decomposing a $4 \times 4$ unitary matrix into 2-level unitary matrices

I am studying how to decompose a $4 \times 4$ unitary matrix into multiple 2-level unitary matrices. I have found a total of six 2-level unitary matrices, and they are as follows. At this point, I ...
junghyunHa's user avatar
2 votes
0 answers
35 views

Fisher information of parametric channel

Suppose $\Phi_\theta$ is a quantum channel whose action can be written for any state $\rho\in \mathcal S(\mathcal H_S)$ in the Stinespring representation as $\Phi_\theta(\rho)= \text{Tr}_E(U_\theta (\...
Quantastic's user avatar
1 vote
0 answers
36 views

Composing beam splitters

Let $a, b$ and $c$ be independent modes in a system $S$ and in environments $E_1$, $E_2$ respectively. Suppose $a$ goes through a beam-splitter characterized by a parameter $\theta$ coupling it to ...
Quantastic's user avatar
1 vote
1 answer
121 views

Is a linear combination of unitaries unitary?

Suppose you have a pure state $\vert\psi\rangle$. Consider the following operation. For unitaries $U_1$ and $U_2$, one can take complex numbers $\alpha, \beta$ where $|\alpha|^2 + |\beta|^2 = 1$ and ...
user1936752's user avatar
  • 2,913
4 votes
0 answers
68 views

A measure of entanglement created by a unitary operation

Let $U$ be a unitary matrix acting on a 3-qubit system. If there is no correlation among any pairs of the three qubits, the unitary operation can be represented as $U = U_1 \otimes U_2 \otimes U_3$, ...
user185671631's user avatar
3 votes
1 answer
99 views

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 ...
metaUser's user avatar
3 votes
2 answers
144 views

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 ...
shard's user avatar
  • 31
1 vote
0 answers
34 views

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)]...
Physkid's user avatar
  • 520
2 votes
1 answer
97 views

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-...
quest's user avatar
  • 636
4 votes
2 answers
327 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 ...
Hailey Han's user avatar
1 vote
0 answers
87 views

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 ...
francler's user avatar
  • 181
1 vote
3 answers
170 views

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. ...
Cuhrazatee's user avatar
3 votes
2 answers
297 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 ...
Hailey Han's user avatar
2 votes
0 answers
38 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 $\rho$ ...
Hans Schmuber's user avatar
6 votes
2 answers
112 views

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 ...
forky40's user avatar
  • 6,718
1 vote
1 answer
64 views

Why is the matrix obtained from the coefficients of orthogonal states unitary?

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{|...
orangonabbo's user avatar
2 votes
2 answers
50 views

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. $\...
Anindita Sarkar's user avatar
2 votes
1 answer
119 views

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{...
Anindita Sarkar's user avatar
3 votes
1 answer
42 views

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 ...
Grwlf's user avatar
  • 133
0 votes
0 answers
243 views

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.
Cheshta Joshi's user avatar
2 votes
3 answers
241 views

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 $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$...
MonteNero's user avatar
  • 2,481
4 votes
1 answer
78 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 ...
SescoMath's user avatar
  • 507
1 vote
1 answer
144 views

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 ...
user3116936's user avatar
0 votes
0 answers
60 views

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 ...
Simon's user avatar
  • 41
0 votes
1 answer
166 views

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 \...
Nichola's user avatar
  • 392
2 votes
0 answers
45 views

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 $H_{\Omega}$...
Simon's user avatar
  • 41
0 votes
0 answers
37 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 $...
Qwertuy's user avatar
  • 101
0 votes
1 answer
132 views

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 ...
Newuser7's user avatar
1 vote
1 answer
227 views

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 ...
Bard's user avatar
  • 327
0 votes
1 answer
77 views

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?
Thomas Mikhail's user avatar
1 vote
1 answer
169 views

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 ...
Michael.Andy's user avatar
1 vote
1 answer
266 views

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. ...
Telore's user avatar
  • 11
0 votes
1 answer
49 views

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 (-)...
Van Peer's user avatar
  • 577
2 votes
2 answers
641 views

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 ...
Bard's user avatar
  • 327
0 votes
2 answers
101 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 ...
user8622655's user avatar
5 votes
1 answer
183 views

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 ...
Michael.Andy's user avatar
4 votes
1 answer
103 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 ...
user8622655's user avatar
4 votes
2 answers
130 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 ...
Qingyuan Wang's user avatar
1 vote
1 answer
119 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 ...
Lalit Vinde's user avatar
5 votes
2 answers
837 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{\...
Chetan Waghela's user avatar

1
2 3 4 5