Building my own QFT matrix how do I continue?
I am interested in problems which are isomorphic to non abelian groups.Currently I'm working with problems which are isomorphic to the symmetric group and for simplicity I took the S3 subgroup.For S3...
View ArticleEigenvalues of Permutation matrix diagonalizable by a Clifford matrix
For an arbitrary $n$-qubit permutation matrix, $P$, if there exists an $n$-qubit Clifford matrix, $C$, such that $CPC^{-1}$ is diagonal, what are the restrictions on the eigenvalues (or entries) of...
View ArticleWhy are Quantum Query Gates $U_f$ always permutation matrices? [duplicate]
I was reading through some material by IBM Quantum and it said the following in a section about Query Gates:Here, our assumption is that $x \in \Sigma^{n}$ and $y \in \Sigma^{m}$ are arbitrary strings....
View ArticleWhat is the matrix for a SWAP operation on two qubits?
Say we want to swap qubits $a$, $b$ in the same register, where $a,b \in \left \{ 0, 1,\cdots, n-1 \right \}$. What would be the corresponding matrix.For those interested, I'm curious about this matrix...
View ArticleNielsen & Chuang Exercise 2.4 - “Matrix representation for identity” [closed]
Reproduced from Exercise 2.4 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):Show that the identity operator on a vector space $V$ has a matrix...
View ArticleNielsen & Chuang Exercise 2.3 - “Matrix representation for operator products”...
Reproduced from Exercise 2.3 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):Suppose $A$ is a linear operator from vector space $V$ to vector space $W$,...
View ArticleNielsen & Chuang Exercise 2.2 - “Matrix representations: example” [closed]
Reproduced from Exercise 2.2 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):Suppose $V$ is a vector space with basis vectors $|0\rangle$ and...
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Projection operator on Time evolution Operator
From a 9×9 Hamiltonian, I choose a certain subspace of 4D for designing a two qubit gate. Now the original unitary time evolution operator is also a 9×9 matrix. For the action of the unitary time...
View ArticleCan we process infinite matrices with a quantum computer?
Can we process infinite matrices with a quantum computer? If so, how can we do that?
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Is there a matrix expression for this quantum gate?
This gate depends on the value of $f(x)$. If $f(x)=1$, an $X$ gate is applied to the 2nd qubit. If $f(x)=0$, an identity is applied.In both cases, the 2-qubit transformation is unitary. But I don't...
View ArticleSimple proof that $(U \otimes V)(|x\rangle \otimes |y\rangle) = U|x\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...
View ArticleHow to properly write the action of a quantum gate implementing an operator...
Let's say, that we are in the possession of a quantum gate, that is implementing the action of such an operator$$ \hat{U}|u \rangle = e^{2 \pi i \phi}|u\rangle $$Moreover, let's say, that this operator...
View ArticleClock 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....
View ArticleHow can we be sure that for every $A$, $A^\dagger A$ has a positive square root?
In the Polar Decomposition section in Nielsen and Chuang (page 78 in the 2002 edition),there is a claim that any matrix $A$ will have a decomposition $UJ$ where $J$ is positive and is equal to...
View ArticleCriterion for periodic MPS representation
Given a state $|\psi\rangle$ on a Hilbert Space $\mathcal{H}=V\otimes \ldots\otimes V$.Is there any criterion that one can check whether it can be written exactly as an Matrix Product State (MPS) with...
View ArticleNielsen & Chuang Exercise 2.2 - Matrix representations in different input and...
This is a question in the Nielsen and Chuang textbook (Exercise 2.2).Suppose $V$ is a vector space with basis $|0\rangle$ and $|1\rangle$ and $A$ is a linear operator from $V \to V$ such that...
View ArticleDifferentiate between local and global unitaries
Just like we have the PPT, NPT criteria for checking if states are separable, i.e., can be written in tensor form or not, given the matrix of a unitary acting on 2 qubits is there any criterion to...
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How are multi-qubit gates extended into larger registers? [duplicate]
Implementing a single-qubit gate in a multi-qubit register is relatively easy. For example, this gate:This is equivalent to $I \otimes H \otimes I$. If the $H$ gate was on the first bit, it would be $H...
View ArticleSquare root of CNOT and spectral decomposition of the Hadamard gate
I'm trying to compute the spectral decomposition of the Hadamard gate but I'm making a mistake somewhere.Note: I believe (though I may be wrong so correct me if I am) that spectral decomposition is a...
View ArticleHow to calculate Minimum Residual Contangle for Entanglement
I used expectation values of Heisenberg Langevin equations to construct a 6 × 6 covariance matrix between three qubits.Now, I am confused in the formula for minimum Residual contangle. I have...
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