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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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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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
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 ...
Hailey Han's user avatar
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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 ...
francler's user avatar
1 vote
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
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 ...
Hailey Han's user avatar
2 votes
0 answers
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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
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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 ...
forky40's user avatar
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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{|...
orangonabbo's user avatar
2 votes
2 answers
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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. $\...
Anindita Sarkar's user avatar
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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{...
Anindita Sarkar's user avatar
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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 ...
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
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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 $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$...
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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 ...
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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 ...
user3116936's user avatar
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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 ...
Simon's user avatar
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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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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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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 $...
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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
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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 ...
Bard's user avatar
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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?
Thomas Mikhail's user avatar
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1 answer
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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 ...
Michael.Andy's user avatar
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1 answer
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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. ...
Telore's user avatar
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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.
sultana's user avatar
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1 answer
43 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
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2 votes
2 answers
504 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
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2 answers
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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
152 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
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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
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 ...
Qingyuan Wang's user avatar
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 ...
Lalit Vinde's user avatar
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{\...
Chetan Waghela's user avatar
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\...
user1936752's user avatar
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2 votes
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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 ...
Jimbo's user avatar
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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 ...
redpanda2236's user avatar
0 votes
1 answer
202 views

calculating the unitary of a circuit using Qiskit's simulator

i am trying to verify qiskit's get_unitary() result. this is my code: ...
mohaddese's user avatar
3 votes
2 answers
283 views

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 ...
Jayakumar's user avatar
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 ...
ssj009's user avatar
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5 votes
1 answer
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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 ...
madeel's user avatar
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2 answers
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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\...
MonteNero's user avatar
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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}|...
Devesh's user avatar
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1 vote
1 answer
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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 ...
Kilian's user avatar
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2 votes
0 answers
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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 ...
Loic Stoic's user avatar
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 \...
MonteNero's user avatar
  • 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 ...
PGibbon's user avatar
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2 votes
1 answer
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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
Adityashu's user avatar