Questions tagged [eigenvalues-and-eigenvectors]
The eigenvalues-and-eigenvectors tag has no usage guidance.
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A Simple way of Encoding qubit in Stabilizer Codes with Stim
In Stim, is there a easy way to encode qubits into a code state $| 0_L \rangle$ in stabilizer codes, other than constructing an encoding quantum circuit? For example, in Steane's seven-qubit code, can ...
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Can a Hamiltonian of a tripartite system map an product state into a product state?
Suppose we have a finite dimensional Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$ and a Hamiltonian has the following form:
$$H = H_A \otimes I \otimes I + I \otimes H_B \...
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What is the Eigen Value corresponding to a Coin Toss when we make a measurement?
I am asking this question to all the Quantum Computing or Quantum Mechanics practitioners.
I have studied that when you measure a state, then you will get an eigenvalue for sure corresponding to ...
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How do I construct the eigenvectors of the Hadamard gate using Quantum Gates?
How do I use quantum gates offered on IBM quantum to construct either eigenvector of the Hadamard gate?
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Initialize data qubits in the first step of syndrome measurement
I saw many papers that say that in syndrome measurement of the stabilizer codes, such as surface codes and color codes, firstly, all data qubits are initialized to $|0\rangle$. What is the reason? I ...
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The matrix norm $||A||=\max_{\langle u|u\rangle=1}|\langle u|A|u\rangle|$ in the proof of Lieb's theorem
In Exercise A6.4, Appendix 6: Proof of Lieb’s theorem, Page 645, Quantum Computation and Quantum Information by Nielsen and Chuang, A matrix norm of $A$ is defined as $$||A||=\max_{\langle u|u\rangle=...
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What are examples of quantum maps with complex eigenvalues?
Chapter 6 of Michael Wolf's notes (MichaelWolf/QChannelLecture.pdf) discuss the structure of the spectrum of quantum maps and channels. However, it seems like the only explicit example given in the ...
glS♦
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Can we always simultaneously diagonalize $H_A \otimes \mathbb{1}$ and $\mathbb{1} \otimes H_B$?
Suppose we have systems $A$ and $B$ with respective Hamiltonians $H_A \otimes \mathbb{1}$ and $\mathbb{1} \otimes H_B$. These Hamiltonians commute, so they share the same eigenbasis and hence can be ...
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What does the notation $\sum_\lambda \lambda \left | \lambda \right > \left < \lambda \right |$ mean?
In many places, I see a form of representing a matrix with:
$A = \sum_\lambda \lambda \left | \lambda \right> \left < \lambda \right |$
Where $\lambda$ is an eigenvalue, and $\left | \lambda \...
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Changing the eigenvalues used in HHL algorithm
For an HHL algorithm implemented exactly as depicted in Figure 2 of this paper by Dutta et. al (https://arxiv.org/abs/1811.01726), how do I go about changing the eigenvalues that they use? Obviously, ...
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Qiskit implementation for projecting a hermitian operator and finding its eigenvalues
I'm brand new to quantum computing and have been learning Qiskit for a few weeks now. I am attempting to find the number of 0 eigenvalues of the following operator:
$$ T = PHP$$
where $P$ is a ...
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Solving Hamiltonian eigenvalue problem
I would like to solve an eigenvalue problem of a Hamiltonian. I was able to find the lowest eigenvalue by converting the Hamiltonian into a matrix and applying linear algebra eigenvalue techniques. ...
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What are the eigenstates of an operator?
Sorry if this is a silly question, I am new to quantum computing
I was just reading this article that talked about the eigenstates of an operator. And I wonder, how can we find those eigenstates for a ...
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Surface Code Eigenstates as Circles
I learned that logical $|0\rangle$ of surface code, is an eigenstate, where all stabilizers are +1 value, and since the z-stabilizer is enforcing an even amount of edged in each node, and the x-stab ...
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How to find the eigenstates of a general $2\times 2$ Hermitian matrix?
Given a measurement operator in the general Hemitian form
$$
M = \begin{pmatrix} z_1 & x+iy \\ x-iy & z_2\end{pmatrix},
$$
where $x,y,z_1,z_2 \in \mathbb{R}$, show that the eigenvalues are
$$
...
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Weak Schur sampling and state distinguishability
Consider the task of distinguishing between the following two $n$ qubit quantum states.
$$ \rho = \frac{\mathbb{I}}{2^{n}}.$$
$$ \sigma = \frac{1}{2^{n/2}}\sum_{x \in \{0, 1\}^{n/2}} |x\rangle\langle ...
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Why are all the eigenvalues of a "Hermitian block-encoding" equal to $\pm1$?
I was looking at the paper : https://arxiv.org/abs/2002.11649 and the eigenvalue discussion is not clear to me.
Block-encoding is a general technique to encode a nonunitary matrix on a quantum ...
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Interpretation of the unitaries involved in the eigenvalue decomposition of a density operator
If $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle \psi_{i}|$, this ensemble doesn't require $\langle \psi_{i}|\psi_{j}\rangle$=0. Given that $\rho$ is positive semi-definite, by the spectral theorem it ...
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What are the eigenvectors of the superoperator $[H,\cdot]$ with $H$ the Hamiltonian?
Let $\{A_\alpha\}$ be a set of hermitian operators and $\{\Pi(\varepsilon)\}$ a set of projectors on the (finite-dimensional) $\varepsilon$ subspace. Define $$A_\alpha(\Delta\varepsilon)=\sum_{\...
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How do I calculate the eigenvalues of the positive partial transpose of this two-qubit state?
How can I calculate the eigenvalues of $\rho^{T_{B}}$ (PPT) of the following state
$$
\rho =\frac{1}{2}|0\rangle\langle0|\otimes|+\rangle\langle+|+\frac{1}{2}|+\rangle\langle+|\otimes|1\rangle\langle1|...
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In the quantum phase estimation algorithm, why can't we directly compute the eigenvalue from the known eigenvector?
The Quantum Phase Estimation algorithm wants to approximate the phase $\varphi$ of an eigenvalue $\lambda = e^{2\pi i \varphi}$ of a unitary operator $U$.
Besides $U$ an eigenvector $x$ corresponding ...
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Using Qiskit VQE for solving an eigenvalue problem
I am a new user of Qiskit and I believe there is a simple answer to my question but I have had a very hard time finding a straightforward answer. I am trying to transform a given $3 \times 3$ (...
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How are the eigenvalues of $\rho=\frac12(|a\rangle\!\langle a| +|b\rangle\!\langle b|)$ derived?
Let's say I have a density matrix of the following form:
$$
\rho := \frac{1}{2} (|a \rangle \langle a| + |b \rangle \langle b|),
$$
where $|a\rangle$ and $|b\rangle$ are quantum states. I saw that ...
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