# Questions tagged [mathematics]

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### How to derive the CNOT matrix for a 3-qubit system where the control & target qubits are not adjacent?

In a three-qubit system, it's easy to derive the CNOT operator when the control & target qubits are adjacent in significance - you just tensor the 2-bit CNOT operator with the identity matrix in ...
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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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### How is Grover's operator represented as a rotation matrix?

I have seen that it is possible to represent the Grover iterator as a rotation matrix $G$. My question is, how can you do that exactly? So we say that $|\psi\rangle$ is a superposition of the states ...
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### How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?

In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation: $$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$...
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### Purity of mixed states as a function of radial distance from origin of Bloch ball

@AHusain mentions here that the purity of a qubit state can be expressed as a function of the radius from the center of a Bloch sphere. The state corresponding to the origin is maximally mixed whereas ...
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### Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?

The $n$-fold Pauli operator set is defined as $G_n=\{I,X,Y,Z \}^{\otimes n}$, that is as the set containing all the possible tensor products between $n$ Pauli matrices. It is clear that the Pauli ...
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### How does one create the unitary sending $|0\rangle$ into a target quantum state?

The Hadamard gate allows us to construct an equal superposition of states. If one wants to construct an arbitrary superposition e.g. $\alpha\vert 0\rangle + \beta\vert 1\rangle + ..$, how does one ...
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### Grover algorithm for more than one element

I am currently working on the Grover algorithm again. In many lectures and documents, as well as books, I noticed that there is always talk of looking for a single element of $N$ elements. Now I read ...
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### Automatic compilation of quantum circuits

A recent question here asked how to compile the 4-qubit gate CCCZ (controlled-controlled-controlled-Z) into simple 1-qubit and 2-qubit gates, and the only answer given so far requires 63 gates! The ...
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### 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 ...
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### How can I calculate the inner product of two quantum registers of different sizes?

I found an algorithm that can compute the distance of two quantum states. It is based on a subroutine known as swap test (a fidelity estimator or inner product of two state, btw I don't understand ...
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### Is there an algorithm for determining if a given vector is separable or entangled?

I'm trying to understand if there is some sort of formula or procedural way to determine if a vector is separable or entangled – aka whether or not a vector of size $m$ could be represented by the ...
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### Minimum number of 2 qubit gates to build any unitary

Any unitary $U$ acting on $N$ qubits can be decomposed in a finite product $U=U_1U_2...U_n$ where every $U_i$ acts on only 2 qubits, for example through decomposition in CNOT, phase shifts and 1 qubit ...
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### Is the tensor product of two states commutative?

I'm reading "Quantum Computing Expained" of David McMahon, and encountered a confusing concept. In the beginning of Chapter 4, author described the tensor product as below: To construct a ...
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### What is a separable decomposition for the Werner state?

Consider the two-qubit Werner state, defined as $$\rho_z = z |\Psi_-\rangle\!\langle \Psi_-| + \frac{1-z}{4}I, \quad |\Psi_-\rangle\equiv\frac{1}{\sqrt2}(|00\rangle-|11\rangle),$$ for $z\ge0$. Using ...
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### When would I consider using an outer product of quantum states, to describe aspects of a quantum algorithm?

I know the inner product has a relationship to the angle between two vectors and I know it can be used to quantify the distance between two vectors. Similarly, what's an use case for the outer ...
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### How do you represent the output of a quantum gate in terms of its basis vectors?

I'm stuck while trying to understand the Hadamard Gate in a more linear algebra understanding. (I understand the algebraic way). This is because I want to program a simulation of a quantum computer. ...
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