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Questions tagged [matrix-representation]

For questions about matrix representations of quantum gates.

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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 (...
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How do I write a tensor product of conditional gates in matrix form?

I am writing a program where I need to find the eigenstates of an operator that is a Kronecker product of conditional quantum gates. I am wondering how I would compute this product in matrix form as ...
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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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Matrix representation of multiple qubit gates (Hadamard transform on single wire)

I would like to know how the unitary matrix for this circuit looks like: I'm not sure but I would try something like this: First part: $\begin{pmatrix}1&0\\0&0\end{pmatrix}\otimes H_1=\...
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How to interpret the matrix representation of a quantum gate?

I am trying to understand how the quantum gates work, so I started with the simplest one, the Pauli X gate. I get that it turns $|0\rangle$ into $|1\rangle$ and $|1\rangle$ to $|0\rangle$. So my ...
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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 ...
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Error in showing $\operatorname{CPHASE}_{12}=\operatorname{CPHASE}_{21}$ in the matrix representation

I read that the relation $\operatorname{CPHASE}_{12}=\operatorname{CPHASE}_{21}$ in the matrix representation but when I tried to work it out I don't see how. $\operatorname{CPHASE}_{12}$ acts in the ...
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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\}...
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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 ...
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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 ...
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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 ...
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Nielsen & Chuang Exercise 2.2 - Matrix representations in different input and output basis [duplicate]

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 $...
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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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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 $2_+^{1+2n}$ ...
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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. ...
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Building a matrix corresponding to the teleportation circuit

I'm trying to build the matrix that corresponds to this quantum teleportation circuit, but it never works when I test it in the quirk simulator, I tried finding the matrix corresponding to every part ...
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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{...
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How to properly write the action of a quantum gate implementing an operator $U$ on the superposition of its eigenvectors?

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 ...
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What's the matrix representation of this 3 qubit circuit?

How do I calculate the matrix representation of this part of a teleportation circuit? It must be a matrix of dimension 8.
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Matrix representation and CX gate [duplicate]

I am having hard time figuring out how the CX (controlled-NOT) gate is represented in the matrix representation. I understood that tensor product and the identity matrix are the keys, and I ...
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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 ...
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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, ...
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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 ...
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$R_z$ 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\...
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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 (...
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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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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?
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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? ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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Nielsen & 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 ...
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Nielsen & Chuang Exercise 2.3 - “Matrix representation for operator products” [closed]

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 ...
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1answer
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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\...
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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?
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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:...