Questions tagged [matrix-representation]

For questions about matrix representations of quantum gates.

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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 ...
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74 views

Syndrome extraction operator as matrix?

I am trying to understand how to achieve the syndrome extraction operator matrix in quantum repetition code (if it even exists). It is given that the syndrome is defined here (page 4) as: [perform]...
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3answers
157 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{...
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3answers
71 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 ...
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1answer
54 views

How do I apply a controlled gate to specific qbits in the register?

Say, I have a specific scheme, where I need to specify inputs for controlled R logical gate, which here is $$ R(\theta)=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 &...
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Controlled phase shift gate

How does a controlled R gate look like (matrixwise)? And how to generate CCR, CCCR and so on?
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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 \...
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2answers
58 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{...
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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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358 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&...
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1answer
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Can I find the states of individual qubits in a quantum register using only linear algebra?

Say I have a quantum register consisting of two qubits like this $\left| -,0\right>$ which as a vector would be $\frac{1}{\sqrt{2}}(1, 0, -1, 0)$. If I only started with this vector, would it ...
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1answer
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Why is $Z_1Z_2$ spanned by this set ? Surely it's too small?

In the context of stabilizer codes my lecturer writes that $Z_1Z_2$ is spanned by $\{|000\rangle,|001\rangle,|110\rangle\, |111 \rangle \}$. But I don't see how this spans the matrix as it's given by ...
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158 views

Trace preserving condition in Choi's thorem

Choi's theorem states that any completely positive map $\Phi(\cdot) : C^*_{n\times n} \rightarrow C^*_{m \times m}$ can be expressed as $\Phi(\rho) = \sum_{j=1}^r F_j^\dagger \rho F_j$, for some $n \...
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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 ...
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407 views

Controlled Z gate acting on 3 qubits in matrix form

For a controlled Z gate $CZ_{1,2,3}$ acting on 3 qubits, which of the following is correct? If it is the first one then what is the difference between that and a CZ gate acting on qubits 1 and 3? $$I ...
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1answer
81 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 (...
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2answers
87 views

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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1answer
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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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1answer
130 views

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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1answer
89 views

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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336 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 ...
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1answer
53 views

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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1answer
259 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\}...
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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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1answer
104 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 ...
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147 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 ...
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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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1answer
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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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2answers
75 views

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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242 views

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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285 views

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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3answers
240 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 ...
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1answer
111 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, ...
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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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1answer
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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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232 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?
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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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1answer
618 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? ...
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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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2answers
375 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 ...
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1answer
928 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 ...
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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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1answer
300 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 ...
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1answer
195 views

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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1answer
317 views

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 ...