Questions tagged [eigenvalues-and-eigenvectors]

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Geometrical difference between row and column vector [closed]

How to visualize/show geometrically the difference between a row (bra) and column (ket) vector within a coordinate system/plane; or to say a vector and its transpose in terms of quantum computing?
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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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3 votes
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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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6 votes
3 answers
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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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3 votes
1 answer
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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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