# Questions tagged [matrix-representation]

For questions about matrix representations of quantum gates.

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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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### Deriving a controlled Kraus operator from an uncontrolled Kraus operator

I have a Kraus operator $M$. $M$ is composed of a list of matrices $M_k$ satisfying $$\sum_{k} M_k^\dagger M_k = I$$ I would like to control the application of $M$ using a control qubit. This ...
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### Expressing a term of an $n$-qubit Hamiltonian in terms of Pauli operators

Consider a $2^n\times 2^n$ Hermitian matrix $M$ containing up to two non-zero elements, which are $1$ (so, either $M_{ii}=1$ for some $i$, or $M_{ij}=M_{ji} = 1$ for some $i$ and $j$). Each such ...
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### Generic matrix exponential in Q#

I am trying to find a way to implement a unitary transformation in Q# that implements e^(iA) where A is a square matrix. However, I only found ways to do this in Q# if A can be represented as a ...
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### Nielsen & Chuang Exercise 4.34 “Measuring an operator”

I need help with the exercise 4.34 from Nielsen & Chuang Book. I am supposed to get a matrix corresponding to the circuit. Thanks
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### Show how the Bell state arises from the circuit with Hadamard and CNOT, using matrix notation

I understand that starting with , we can get to $\vert \Phi^+ \rangle$. First, we start with $\vert Q_1 \rangle \otimes \vert Q_2 \rangle = \vert 0 \rangle \otimes \vert 0 \rangle$ and then applying ...
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### 2 qubit gate operation on multi qubit systems

Considering a 3 qubit system, what does the matrix operation will look like if I apply CNOT on qubit 1 and qubit 2 and then apply CNOT on qubit 1 and qubit 3?
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 \$...