# Questions tagged [linear-algebra]

For questions about vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc.

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### For stabilizer codes, why does the error syndrome not depend on the codeword?

While reading through some lecture notes on quantum error correction, I read the statement: "In particular, the syndrome doesn’t depend on the specific codeword, only on the Pauli error." I'...
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### Is every pure 1-qubit state an eigenstate of $aX + bY + cZ$?

As stated in the question, I have seen this claim made that a pure state can be written as an eigenstate of $aX + bY + cZ$ for some $a,b,c$ where $X,Y,Z$ are Pauli matrices. Why is this true and what ...
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### Do all Hermiticity-preserving maps generate completely positive maps?

I am confused about what kinds of maps are valid infinitesimal generators of completely positive maps. I know that any Markovian completely positive map can be written in the form $e^{t \mathcal{L}}$, ...
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### How to approximate the time-dependent Hamiltonian in quantum adiabatic theory by the non time-dependent Hamiltonian?

Recently, I am learning how to solve the linear equation $A\left | x \right \rangle =\left | b \right \rangle$ using quantum adiabatic theory. In the solving process, people usually need to set the ...
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### How to find the $+1$ eigenvectors of the stabilizers for the Shor code

I am currently working through chapter $3$ of "Stabilizer Codes and Quantum Error Correction" (Daniel Gottesman's thesis). I would like to know the general method for finding the $+1$ ...
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### Variational Quantum Linear Solver

I'm studying quantum computing right now and trying to implement variational quantum linear solver to solve a system of linear equations. From what I have understood from the paper written by Carlos ...
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### Commutation of $XX$ and $ZZ$ operators

It is known that the Pauli operators $XX$ and $ZZ$ commute. Consider the state $\vert{++}\rangle$ which is an eigenstate of $XX$. But we also know that $$ZZ\vert{++}\rangle = \vert{--}\rangle$$ so ...
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### How to systematically find the kernel of a channel from its Kraus operators?

A quantum channel is a completely positive trace-preserving map. Given a quantum state $\rho$ and channel $N$, let the output be $N(\rho)$. Given the Kraus operators of the channel, how can one find ...
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### Is $\rho = \sum_{j} p_j|n_j\rangle\langle n_j|$ a valid construction for any mixed state?

I have a mixed state $\rho$ and its hamiltonian $H$. Firstly, I find the eigenvalues $\{p_j\}$ of $\rho$, and orthonormal basis of $H$. I write $\rho$ in terms of $H$'s eigenstates and $\rho$'s ...
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### Can quantum computers help to solve questions of general relativity theory?

My question is rather straightforward: Can quantum computers be used to solve problem within general relativity theory? To put more context. As GR is based on solution of rather complicated systems of ...
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