# Questions tagged [mathematics]

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### Decomposition of a $4 \times 4$ unitary matrix

I am currently studying the paper "Decomposition of unitary matrices and quantum gates (2012)" and referring to the textbook Quantum Computation and Quantum Information. Among the topics, I ...
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1 vote
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### Representing networks with qubits as edges

I am looking to take a classical non-negative real valued network and generalize it to the quantum case for processing. A network is given by an adjacency matrix, essentially edge weights $e_{ij}$ for ...
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### How to calculate the Haar measure for the middle SU(2), in an SU(3) factorization?

I read this blog https://pennylane.ai/qml/demos/tutorial_haar_measure#deguise2018 regarding a basic introduction to haar measure. In the "show me more math" section, they said $SU(3)$ can be ...
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### Simulating Sparse Hamiltonians: help understanding query complexity bounds

tl;dr: How can I show that $e^k/k^k$ is less than $\epsilon^2/2$ when $k=\Omega\left(\frac{\log(1/\epsilon)}{\log \log(1/\epsilon)}\right)$, where $k,\epsilon\in \mathbb{R}$ and > 0? Context: Berry ...
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### Given an observable $O$, what's the achievable maximum value of $\operatorname{Tr}(O\rho)$?

The maximum value of expectation value of an observable $O$ with respect to a density matrix $\rho$ can be computed by using Holder's inequality as follows: \begin{equation} \text{Tr}(O\rho) \leq \...
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### Are quantum channels bounded linear maps?

I've been reading about quantum channels from a couple of sources and have some doubts regarding some mathematical perspectives and properties of quantum channels. I've listed them below: It is known ...
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### What will be the new state after the outcome of the measurement of the state of the first qubit as ǀ0>? [closed]

Two qubits are prepared in a superposition state of the form: What will be the new state after the outcome of the measurement of the state of the first qubit as ǀ0>?
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### How to prove that CNOT and Rz gates are permutable?

How to prove that CNOT and Rz gates are permutable? I tried to equate their switch to zero and calculate it, but for this you need to multiply the matrices. But the 4x4 and 2x2 matrices cannot be ...
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### What are the expected measurement results in the diagram below? [closed]

I ask you to give a mathematical solution to this problem
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### Weakly transversal gates for the $[[5,1,3]]$ code

For the $[[5,1,3]]$ code https://en.wikipedia.org/wiki/Five-qubit_error_correcting_code $X^{\otimes 5}$ implements logical $X$ and $Z^{\otimes 5}$ implements logical $Z$. A less common gate ...
Consider the following quantity: $$f_O(|\psi\rangle) = \langle \psi | O | \psi \rangle$$ How would we study a perturbation on $|\psi\rangle$, given that it has to be a valid quantum state? What ...