Questions tagged [matrix-representation]

For questions about matrix representations of quantum gates.

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19
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1answer
4k views

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:...
13
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3answers
5k views

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 ...
12
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2answers
3k views

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 ...
11
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1answer
490 views

Advantage of simulating sparse Hamiltonians

In @DaftWullie's answer to this question he showed how to represent in terms of quantum gates the matrix used as example in this article. However, I believe it to be unlikely to have such well ...
9
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2answers
552 views

How to check if a quantum circuit can be constructed for a given matrix representation?

Let's say I have a matrix representation, e.g. $$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$ How ...
9
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2answers
2k views

Is there a name for the 3-qubit gate that does NOT NOT NOTHING?

It seems to me that the "disjunction gate" (aside: is that its proper name?) can be thought of as the combination of three gates, G1, G2, and G3, where G2 is the CCNOT gate, and $G1 = G3 = ¬...
8
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2answers
4k views

What is the matrix of the iSwap gate?

Mostly I'm confused over whether the common convention is to use +$i$ or -$i$ along the anti-diagonal of the middle $2\times 2$ block.
8
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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&...
7
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3answers
205 views

How to efficiently calculate the inverse of a Kronecker product?

This is a follow-up question to a previous question I had, where the correct answer was to use the Kronecker product. Given, for example, a vector representing two qbits $$\begin{bmatrix}0 \\ 1 \\ 0 \...
7
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1answer
2k views

What are theta, phi and lambda in cu1(theta, ctl, tgt) and cu3(theta, phi, lam, ctl, tgt)? What are the rotation matrices being used?

I was reading the documentation for qiskit.QuantumCircuit and came across the functions cu1(theta, ctl, tgt) and ...
7
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3answers
114 views

$R_z$ gate representations

Why is the $R_z$ gate sometimes written as: $$ R_{z}\left(\theta\right)=\begin{pmatrix}1 & 0\\ 0 & e^{i\theta} \end{pmatrix}, $$ while other times it is written as: $$ R_{z}\left(\theta\...
7
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1answer
746 views

Solving linear systems represented by NxN matrices with N not power of 2

As far as I have seen, when it comes to solving linear systems of equations it is assumed to have a matrix with a number of rows and columns equal to a power of two, but what if it is not the case? ...
7
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2answers
634 views

Matrix representation of continuous-variable gates

In the introduction to continuous-variable quantum computing by Strawberry Fields (Xanadu), it lists the primary CV gates (rotation, displacement, squeezing, beamsplitter, cubic phase) along with ...
6
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2answers
2k views

What is the square root of the NOT gate? [duplicate]

I have encountered different matrix of operator "the Square Root of NOT gate". For example, the matrix is specified here: $\sqrt {NOT} = \frac{1}{2}\left( {\begin{array}{*{20}{c}} {1 + i}&...
6
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3answers
110 views

Decomposing gates resembling exponentiated members of desired gateset

Suppose I have access to a pretty typical gate set, for example $\{\text{CNOT}, \text{SWAP}, \text{R}_{x}, \text{R}_{y}, \text{R}_{z}, \text{CR}_x, \text{CR}_y, \text{CR}_z\}$ where $\text{CR}$ is a ...
6
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2answers
481 views

How to represent an $n$-qubit circuit in matrix form?

If a given quantum circuit has $n$ qubit inputs and a certain number of gates, how can we represent the whole circuit in matrix form? Here's an example: I am sorry, I am confused on how to express ...
6
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2answers
307 views

Is there a tool that can give you the unitary representing a quantum circuit from just a string?

Say I have a string representing the operations of a quantum circuit. I want to have the unitary operator representing it. Is there a tool for doing so in Python or else?
6
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1answer
658 views

How 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 $\sqrt{...
6
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1answer
132 views

What is the set of generators for the qutrit Clifford group?

According to this article, any Clifford gate, acting on $n$ qubits, can be generated by Hadamard, CNOT, and S gates. What are the set of generators for qutrit Cliffords?
6
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1answer
518 views

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?
6
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1answer
253 views

Is quantum computer equivalent to Turing machine with matrix multiplication oracle?

Since quantum computer with $n$ qubits is described by a $2^{n}\times2^{n}$ unitary matrix is it equivalent to an oracle that can do multiplication of large matrix and return $n$ numbers computed ...
5
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2answers
244 views

Representation of rotation operators $e^{-i\theta(I-Z_1\otimes Z_2 \otimes Z_3)}$ about arbitrary axis for $3$ qubits

I was wondering in how to interpret and represent the operator $e^{-i\theta(I-Z_1\otimes Z_2 \otimes Z_3)}$ for a 3 qubit system in a circuit using qiskit. I was thinking I could just perform an ...
5
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1answer
277 views

A basic question on circuits and matrix representation

I have several (rather basic) questions on matrix representation of circuits and I would be very grateful to anyone that could clear up my confusion, thank you in advance. 1) When reading circuit ...
5
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1answer
180 views

Equivalent determinant condition for Peres-Horodecki criteria

The Peres-Horodecki criteria for a 2*2 state states that if the smallest eigenvalue of the partial transpose of the state is negative, it is entangled, else it is separable. According to this paper (...
5
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1answer
300 views

Efficient implementation of the Clifford group for $n$ qubits

I'm looking for an efficient implementation of the Clifford group $\mathcal{C}_n$ of $n$ qubits. The Clifford group $\mathcal{C}_n$ has stucture $(2_+^{1+2n} \circ C_8).Sp(2,n)$, where $2_+^{1+2n}$ ...
4
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3answers
301 views

What is the matrix for the operator that implements a function to tell the parity of its argument?

$\newcommand{\qr}[1]{|#1\rangle}$ I gave myself the task of building an operator that implements the following function: $f(0) = 0$, $f(1) = 1$, $f(2) = 1$, $f(3) = 0$. I restricted myself to $x$ up ...
4
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3answers
293 views

How do I write the matrix for a CZ gate operating on nonadjacent qubits?

I'm working on a teleport protocol and I need to open the matrix of each operator, however, there's a CZ gate between q0 and q2 at the end of it and I don't know how to write the matrix for it and ...
4
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2answers
80 views

Is the column vector of a uniformly sampled random unitary matrix a uniformly sampled random state vector?

I am wondering if a random unitary matrix taken from a Haar measure (as in, it is uniformly sampled at random) can yield a uniformly sampled random state vector. In section 3 of this paper it says &...
4
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3answers
418 views

Representing qubit swap using linear algebra

I want to write matrix representation of qubit swap algorithm, but I seem to be stuck. Here is the circuit I am trying to calculate using linear algebra: Initially $q_0 = |0\rangle$ or $\begin{...
4
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1answer
505 views

Non-unitary matrix decomposition as a sum of unitary matrices

Several quantum algorithms that deals with linear algebra and matrices that are not necessarily unitary circumvent the problem of non-unitary matrices by requiring a decomposition of the non-unitary ...
4
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1answer
210 views

What 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 ...
4
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1answer
137 views

Differentiate between local and global unitaries

Just like we have the PPT, NPT criteria for checking if states can be written in tensor form or not, is there any criteria, given the matrix of a unitary acting on 2 qubits, to check if it is local or ...
4
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3answers
311 views

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 ...
4
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1answer
218 views

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 ...
4
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1answer
116 views

Single-qubit rotations on a subspace within two-qubit unitary

I would like to implement the operation $$ U(a,b) = \exp\left(i \frac{a}{2} (XX + YY) + i \frac{b}{2} (XY - YX) \right) $$ ($a,b \in \mathbb{R}$) without using Baker-Campbell-Hausdorf expansion, ...
4
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2answers
150 views

Creating a custom gate (given by a matrix) more efficiently

I have the following gate (as represented by its matrix) on $4$ qubits which I hope to use in an error-detection code $$M=\left( \begin{array}{cccccccccccccccc} 1 & 0 & 0 & 0 & 0 &...
4
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1answer
227 views

Difference between change of basis in bra-ket notation and matrix notation

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\...
4
votes
1answer
124 views

How to calculate tensor product for the magic square

The magic square game is a two-player pseudo-telepathy game that was presented by Padmanabhan Aravind, who built on work by Mermin. In the magic square we have ones in columns (odd number) and rows (...
4
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1answer
442 views

Projection operator on Time evolution Operator

From a 9×9 Hamiltonian lying 9D space, I choose a certain subspace of 4D for designing a two qubit gate. Now the original unitary time evolution operator also lies in 9D space and it's a 9×9 size ...
3
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2answers
980 views

Difference between 3 qubits, 2 qutrits & 1 six level qunit

What is the difference between 3 qubits, 2 qutrits and a 6th level qunit? Are they equivalent? Why / why not? Can 6 classical bits be super-densely coded into each?
3
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2answers
111 views

How do you represent one-qubit rotations in two registers as a $4\times 4$ unitary matrix?

Let's say you have a circuit that performs a Z-rotation in the first register, and a Y-rotation in second register. How can we express this "moment" in terms of a 4x4 matrix, i.e. a two-...
3
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2answers
66 views

How to find original matrix from eigenbasis and eigenvalues?

I'm not sure is it the right place to ask this but, I think it is better to ask here than Math Overflow. It is about how to find the matrix representation of an operator (for the CHSH test). What are ...
3
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3answers
198 views

How to spot the matrix representation of the quantum NOT operation

Applying the above construction to AND we get the map $(x1,x2,y) \rightarrow (x1,x2,y⊕(x1∧x2))$ for $x1,x2,y \in \{0,1\}$. The unitaryoperator which implements this is then simply the map $|x1〉|x2〉|y&...
3
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3answers
442 views

Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates

Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates and write down the corresponding circuit. Recall from Quantum Information Theory that $Z=HXH$. As $CNOT$ is a ...
3
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1answer
585 views

Quantum addition and modulo operation using gates

I have a matrix equation $X_{\text{new}}=AX_{\text{old}}$, where $A=\begin{bmatrix}1 & 1 & 1\\ 2 & 3 &2\\ 3&4&4 \end{bmatrix}\bmod 64$, and $X_{\text{old, new}}\in \{1,2,...64\}...
3
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1answer
1k views

Nielsen & 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 $|1\...
3
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1answer
132 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 ...
3
votes
1answer
964 views

How are the Pauli $X$ and $Z$ matrices expressed in bra-ket notation? [duplicate]

For example: $$\rm{X=\sigma_x=NOT=|0\rangle\langle 1|+|1\rangle\langle 0|=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}}$$ $$\rm{Z=\sigma_Z=signflip=|0\rangle\langle 0|-|1\rangle\langle 1|=\...
3
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2answers
149 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{...
3
votes
2answers
887 views

Square 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 ...