Questions tagged [unitarity]
For questions related to the unitarity (unitary evolution) of quantum systems, as applicable to quantum computing or quantum information.
171
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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.
...
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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.
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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 (-)...
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2
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{\...
- 272
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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\...
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3
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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 ...
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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 ...
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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:
...
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226
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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 ...
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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 ...
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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 ...
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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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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:
$$\eta^\pm = \frac{1}{2}|001\rangle + \frac{1}{2}|...
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Unitary operations in a Quantum Neural Network
I'm currently reading Classification with Quantum Neural Networks on Near Term Processors and I'm having trouble with one of the calculations.
The system is composed of $n+1$ qubits, $n$ of those are ...
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Quantum Signal Operator and the unitary state preparation oracle? [closed]
I am looking into IL Chuang and GH Low's Hamiltonian Simulation with Qubitization paper.
I am very confused on the terminology and motivation behind definition 1.
I do not understand what the unitary ...
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How to find a circuit for a unitary operator $e^{-i s |v \rangle \langle v| t }$?
Let $|v \rangle$ be an eigenstate of an $n$-qubit and $2$-local Hamiltonian
$$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 \...
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Sample random unitary at a given distance from a given unitary
Is it possible?
I.e., what is the most natural procedure of such sampling? The sampling has to be 'uniform' in a vicinity (of radius $\epsilon$) of given $U$ (can I say "according to Haar measure ...
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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 ...
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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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Check that a channel implements a particular unitary
Consider a channel $C$ with Kraus operators $\{K_k\}$ and a unitary U.
How can I check that $C$ implements $U$ ?
One can write that their Choi matrices are equal i.e:
\begin{equation}
\sum_{i,j}|i\...
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Is there an inverse for Stinespring dilation?
Given a set of Kraus operators we can find a unitary that does the equivalent map on an extended space including the environment using Stinespring dilation. My question is how do we go about doing the ...
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Implementing Odd Permutations Without Ancilla Bit
The paper says that
The inversion $\alpha \mapsto \alpha^{-1} $ (where 0 is mapped to 0)
can be seen as a permutation on $\mathbb F_{256}$. This permutation is odd, while
quantum circuits with NOT, ...
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3
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Is effective quantum cloning possible, given that any classical function can be implemented as a quantum circuit?
As in Compiling a classical function to a quantum circuit in practice, as far as my understanding goes, it is known that any classical function can be implemented as a quantum circuit. So given $f(x)=...
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What are rotation angle and axis corresponding to a higher-dimensional unitary?
We know that a single-qubit Unitary can be defined as a single rotation of angle $\theta$ around some axis $\hat{n}$, together with a global phase $\alpha$ (see Nielsen & Chuang Eq. 4.9):
$$ U = e^...
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77
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Simulation of non hermitian operators with Qiskit
I am trying to simulate on a quantum computer a wavepaket evolution with a non unitary evolution operator (Hamiltonian with an absorbing (imaginary) potential for instance) and I found this post :
...
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Is there a general way to parametrize 2-qubit unitaries?
So in the single-qubit case, we can write any unitary operation as an instance of the following parametrized unitary:
$$U(\theta, \phi, \lambda) = \begin{bmatrix} \cos(\theta) & -e^{i\lambda}\sin(...
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Circuit for pre-factors +-i, -1
Setting
I have a (6 qubit) circuit which implements a unitary $U$.
Goal
I need the circuits which implement $-U, iU, -iU$.
Phase matters, because I later embed a controlled version of $\pm i U $ into ...
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Do unitary matrices acting on entangled states always give a quantum state?
I'm trying to understand what happens when Alice(Bob) apply a unitary to her(his) part of an entangled state. Let us consider the following unitary transformations:
$$U_1 = \frac{1}{\sqrt{2}}
\...
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3
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How are black-box oracles implemented in Hamiltonian simulation?
I am currently trying to decompose a hessian to a sum of unitaries $H=\sum a_i U_i$.
The papers
VQLS and
Black-box Hamiltonian Simulation state that it can be done, but requires the use of an oracle ...
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How to convert between little/big-endian unitary forms in Braket?
As noted in this post, the Amazon Braket unitary calculation method as_unitary has been deprecated (#325) as it uses little-endian qubit order. The new, big-endian method is to_unitary. Here's a code ...
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Mapping $| y \rangle$ to $(-1)^{x \cdot y}| y \rangle$
I was checking some QC lecture notes by Ronald de Wolf and I came across this exercise that I can't solve.
Page 27 (pdf page 35), question 5, part b link: https://homepages.cwi.nl/~rdewolf/qcnotes.pdf
...
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Find unitary $U$ that transforms state $|\phi \rangle$ to state $|\phi' \rangle$ knowing those states differ only by relative phase
Given:
$|\phi \rangle = \sum_n c_n |a_n \rangle $
$|\phi' \rangle = \sum_n c'_n |a_n \rangle $
such that $\forall n: c_n = c'_n \lor c_n = -c'_n $, where $|a_n \rangle $ is a canonical basis,
and ...
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Rotation angles of unitary operator
Given a complex unitary $2*2$ matrix $A$ that represents some quantum gate on a single qubit.
What is the formula to extract to $\theta_X, \theta_Y, \theta_Z $ rotations around each one of the axes in ...
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What role does Landauer's principle play in quantum reversibility?
In section 3.2.5 of Nielsen and Chuang (starting page 153) they talk about Landauer’s principle, where they discuss the lower bound on the thermodynamic cost of erasing information.
In irreversible ...
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Is the Eastin-Knill Theorem incorrect?
I am reading through this paper (the Eastin-Knill Theorem) and there is a step in the proof of the main theorem that I do not understand.
Let $Q$ be a composite quantum system supporting a quantum ...
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Quantum channels that commute with any unitary channel
Consider a quantum channel $\Phi$ that maps from density operators $\mathcal{S}(\mathcal{H}_A)$ to itself, that commutes with any unitary channel $\mathcal{U}$ on $\mathcal{S}(\mathcal{H}_A)$, i.e. $\...
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Redundant parameters in $2\times 2$ unitary operators?
A complex $n \times n$ unitary operator has $n^2$ free real parameters.
For example, a $2 \times 2$ unitary matrix can be parametrized as
\begin{equation}
\begin{pmatrix}
e^{i(\alpha - \beta/2 - \...
- 71
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Using rotation gates instead of Grover
I have a conceptual question about Grover's algorithm. In the textbook case, we always assume to have an oracle that singles out the correct states by giving them a negative phase. Then, we use phase ...
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376
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How to generate statevector evolution?
I have created the following 2-qubit circuit in qiskit:
...
12
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1
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How can classical bits be copied if qubits cannot be copied?
The no-cloning theorem of quantum mechanics tells us there can be no general quantum circuit that can copy arbitrary qubit states, i.e. a quantum gate or circuit cannot send $|0\rangle |\psi\rangle\...
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Minimal Circuit Decomposition for a 3 qubit gate
I have this unitary matrix and i need to find the decomposition with the small number of c-not.
I tried to use Quantum Shannon Decomposition but the simple form of the matrix make me think that there ...
- 21
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332
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Showing that two unitary matrices are equal up to a global phase
Let $U$ and $V$ be two $d × d$ unitary matrices, representing two reversible quantum processes
on a $d$-dimensional quantum system. We say that the two processes “act in the same way”
on the state $|ψ\...
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Quantum Linear Algebra [closed]
[![Question][1]][1]
Find a 4 x 4 unitary matrix U such that U = eiA. (Possibly up to multiplying by a unit scalar, U is a matrix seen in the course.) Verify your calculation by showing how if U were ...
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2
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What is a Haar random quantum state?
Can somebody please explain me what is a Haar random state? I am not able to find any friendly resource to read about it.
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