Questions tagged [matrix-representation]
For questions about matrix representations of quantum gates.
187 questions
4 votes
2 answers
232 views
Eigenvalues 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 ...
- 803
1 vote
0 answers
30 views
Why 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 ...
- 11
0 votes
2 answers
139 views
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 ...
- 155
1 vote
0 answers
82 views
Criterion 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) ...
- 111
0 votes
0 answers
36 views
How 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 ...
2 votes
1 answer
252 views
How are recursion and loops handled in quantum algorithms?
Quantum computing is Turing complete which means there should be a quantum equivalent for any classical program. From my understanding, it is possible to express any quantum program using a quantum ...
- 163
3 votes
1 answer
113 views
Is there a general circuit implementation for the commutation matrix?
The commutation matrix $K^{(r,m)}$ is defined by $ K^{(r,m)} = \sum_{i=1}^r \sum_{j=1}^m (\boldsymbol{e}_{r,i} \boldsymbol{e}_{m,j}^T) \otimes (\boldsymbol{e}_{m,j} \boldsymbol{e}_{r,i}^T) $ It also ...
- 621
1 vote
1 answer
127 views
Assuming one exists, what is the general algorithm of converting an uncontrolled gate into a controlled one?
A.S.: I'm an amateur, don't be too pedantic. In a multi-qubit system, it is very easy to pin-point a certain one qubit and apply a given uncontrolled single-qubit gate to it. For example, if I want to ...
- 113
1 vote
2 answers
303 views
Can I efficiently decompose this flexible unitary matrix into 1 and 2 qubit gates?
Consider the unitary matrix $A \in \mathbb{R}^{n^2\times n^2}$ which has only the first $n$ rows explicitly defined, with the remaining rows having some flexibility. $A$ can be written in block form ...
- 621
1 vote
0 answers
78 views
Implementation of the CSD Decomposition for a 4x4 matrix
I am currently reading Volume I of "Principles of Quantum Computation and Information" by Benenti-Casati-Strini. There is a section in chapter 3 which covers the decomposition of any ...
- 11
3 votes
2 answers
639 views
Constructing a quantum circuit for a "simultaneous four qubit rotational swap"
I am looking to construct a circuit that implements the following unitary matrix: $$ \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &...
- 33
-5 votes
1 answer
170 views
Why is the matrix representation of CNOT gate valid
$\begin{bmatrix}1&0&0&0\newline 0&1&0&0\newline 0&0&0&1\newline 0&0&1&0\end{bmatrix}$ Obviously, given any $4\times 1$ matrix, it fixes the first two ...
- 99
0 votes
1 answer
173 views
How to understand the matrix representation of the Toffoli produced by qiskit?
I would like to ask about the results of running the Toffoli gate matrix and circuit diagram on colab Possible reasons for inconsistent results with the following correct matrix results. I would also ...
- 21
2 votes
1 answer
301 views
Constructing a block unitary from non-unitary matrices
Background: I have a function $f(s_i, s_f, x)$ where $s_i \in \{0,1,2,3\}; \quad x,s_f \in \{0,1\}$ which is defined as: $$ f(s_i, s_f, x) = \begin{cases} 1, & \text{if } (s_i, s_f, x) \in\{(0,0,0)...
- 55
1 vote
0 answers
61 views
Find $U$, such that $(G+U | -G-2U)$is Clifford
Given a square matrix: $ \begin{equation} G=\frac{\sqrt{2}}{4}\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 ...
- 39