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Questions tagged [unitarity]

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

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Getting a Hermitian operator from a Quantum circuit, and taking its expectation value

I have a circuit $C$, which acting on a state $|\psi \rangle$ is equivalent to a Unitary $U$ acting on the state: $$C(|\psi \rangle) = U|\psi \rangle$$ Now, this circuit is just a Hamiltonian ...
Soumyadeep sarma's user avatar
1 vote
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Definition of a quantum gate

A quantum gate is usually defined as a unitary transformation, like the definition found in "Mathematics of Quantum Mechanics" by Scherer. According to this definition, can we consider a ...
Josh's user avatar
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Why does it matter that Schmidt number is invariant under unitary transformations?

I am reading Nielsen & Chuang and they say this: "The bases $|i_A\rangle$ and $|i_B\rangle$ are called the Schmidt bases for A and B, respectively, and the number of non-zero values $\...
researcher101's user avatar
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Reasoning behind unitary freedom in the ensemble for density matrices theorem

Although my question has the same title of a different question, it is not a duplicate. I am asking a different question. I don't care why it made it into the book. Here is a theorem from Nielsen &...
researcher101's user avatar
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Alternative algorithm for quantum phase estimation problem

The Quantum Phase estimation problem is stated below: Input: Given $U$ as a unitary operator acting on an m-qubit register. If $| \psi \rangle$ is an eigenvector of $U$, then U$| \psi\rangle$ = $e^{ ...
Manish Kumar's user avatar
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A maximally distant unitary from a parameterized circuit

I have a parameterized circuit that includes a certain number of rotation gates, each parameterized by an angle. By sampling over these angles, I can obtain various unitaries. What are the unitaries ...
trurl's user avatar
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Axis and Angle of rotation of $\frac{1}{\sqrt{2}}\begin{bmatrix}-i&-1\\1&i\end{bmatrix}$

I have made use of the following formulas, \begin{align} \theta&=2\cos^{-1}\bigg(\frac{e^{-i\alpha}Tr(X)}{2}\bigg)\\ n_i&=\frac{e^{-i\alpha}Tr(X\sigma_x)}{2\sin\theta/2}\\ e^{i\alpha}&=\...
Sooraj S's user avatar
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How to represent unitary evolution in Python?

How to represent $U(t)$ (a unitary operator) in a code? Is there any package available for that in Python?
saaru darshini's user avatar
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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
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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
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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
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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 ...
am567's user avatar
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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
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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
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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
2 votes
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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
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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
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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
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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
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39 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
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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
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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 ...
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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
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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 ...
metaUser's user avatar
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2 answers
146 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
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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)]...
Physkid's user avatar
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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-...
quest's user avatar
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4 votes
2 answers
381 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
94 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
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3 answers
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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. ...
Cuhrazatee's user avatar
3 votes
2 answers
302 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
1 answer
48 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
115 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
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1 vote
1 answer
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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
55 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
159 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
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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.
Cheshta Joshi's user avatar
2 votes
3 answers
309 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
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4 votes
1 answer
81 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
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1 vote
1 answer
162 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
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0 answers
78 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
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1 answer
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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 \...
Nichola's user avatar
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2 votes
0 answers
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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 $H_{\Omega}$...
Simon's user avatar
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0 answers
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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
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1 answer
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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 ...
Newuser7's user avatar
1 vote
1 answer
273 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
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0 votes
1 answer
131 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
198 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
291 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
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