# Questions tagged [matrix-representation]

For questions about matrix representations of quantum gates.

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### Can arbitrary matrices be decomposed using the Pauli basis? [duplicate]

Is it possible to decompose a hermitian and unitrary matrix $A$ into the sum of the Pauli matrix Kronecker products? For example, I have a matrix 16x16 and want it to be decomposed into something ...
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### How to interpret a quantum circuit as a matrix?

If a circuit takes more than one qubit as its input and has quantum gates which take different numbers of qubits as their input, how would we interpret this circuit as a matrix? Here is a toy example:...
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### How to construct matrix of regular and "flipped" 2-qubit CNOT?

When constructing the matrices for the two CNOT based on the target and control qubit, I can use reasoning: "If $q_0$==$|0\rangle$, everything simply passes through", resulting in an Identity matrix ...
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### Can we process infinite matrices with a quantum computer?

Can we process infinite matrices with a quantum computer? If then, how can we do that?
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### What applications does the quantum gate [(i,1),(1,i)] have?

I've been working through the great introduction to quantum computing on Quantum Country. One exercise there is to find a possible quantum gate matrix that is not the $X,I$ or $H$ matrix. I thought ...
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### Simple proof that $(U \otimes V)(|x\rangle \otimes |y\rangle) = U|x\rangle \otimes V|y\rangle$?

This transformation comes up a lot during symbolic manipulation of quantum operations on state vectors. It's the reason why, for instance, $(X\otimes \mathbb{I}_2)|00\rangle = |10\rangle$ - it lets us ...
In matrix notation, say I have the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. It is currently represented in the computational basis $\{\begin{bmatrix} 1 \\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1\... 0answers 204 views ### Clock matrix vs matrix clock In the process of research leading up to my previous question, I found out about matrix, vector & logical clocks. The citation in the aforementioned question mentions clock and shift matrices. ... 1answer 106 views ### How does the graphical notation used to denote doubly-controlled gates work?$\qquad\qquad$What is the difference between solid and hollow? How to express the corresponding matrix of these figures? In addition, if they are not adjacent, what should be done in the middle of ... 1answer 454 views ### What's the matrix representation of this 3-qubit CZ circuit? How do I calculate the matrix representation of this part of a teleportation circuit? It must be a matrix of dimension 8. 2answers 463 views ### Could the Hadamard gate have been constructed differently with similar characteristics? Say we had a Hadamard-like gate with the -1 in the first entry instead of the last. Let's call it$H^1$. $$H = \begin{bmatrix}1&1\\1&-1\end{bmatrix}$$ $$H^1 = \begin{bmatrix}-1&1\\1&... 1answer 129 views ### Can we understand multi-qubit gates in terms of rotation groups? I'm trying to reconcile (i) the statement that swapping two subsystems constitutes a rotation by 2\pi and (ii) the angle that is implied by the Hermitian generator of a SWAP gate. I haven't tracked ... 2answers 148 views ### What is the matrix representation for n-qubit gates? Let's say I have more than one qbits |0\rangle|1\rangle and I want to perform a H on both of them. I know the matrix representation for the Hadamard on a single qbit is$$\frac{1}{\sqrt{2}}\begin{... 1answer 308 views ### Solving a circuit implementing a two-level unitary operation The circuit below implements the following two-level unitary transformation:$\tilde{U}$is a unitary matrix:$\tilde{U} = \left[\begin{matrix} a & c \\ b & d \end{matrix}\right]$where$a, ...
Hadamard gate matrix is: $$\frac{1}{\sqrt2}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}$$ The matrix for $|0\rangle$ is: $$\begin{bmatrix}1 \\ 0\end{bmatrix}$$ I am unable to understand, how ...