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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 ...
1k views

Prove that any Hermitian Matrix is a real linear combination of Pauli operators [duplicate]

This is an important result in Quantum Computing because it means that the Hamiltonian of a Quantum System can be encoded as a sequence of real numbers and their corresponding Pauli Operator. How do ...
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51 views

Generalized set of Pauli elements for a basis for the linear transformations on the vector space [duplicate]

I have been doing some practice problems from "Gentle introduction to Quantum Computing". I am a little bit lost with this one: The generalized Pauli group $\mathcal G_n$ is defined by all elements ...
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7k views

Circuit construction for Hamiltonian simulation

I would like to know how to design a quantum circuit that given a Hermitian matrix $\hat{H}$ and time $t$, maps $|\psi\rangle$ to $e^{i\hat{H}t} |\psi\rangle$. Thank you for your answer.
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946 views

Computing variance under the action of a unitary operator

I wish to calculate the expectation and variance for an observable on a particular qubit of a multi qubit quantum state. I'm using a quantum computing simulation library which allows me to apply ...
• 151
583 views

Decomposing Hamiltonian into qubit model representation

One of the main applications of VQE is its application to find the approximation to the ground state energy (smallest eigenvalue of the Hamiltonian) for a particular molecule through an iterative ...
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249 views

Is there an efficient algorithm for decomposing an arbitrary Hamiltonian into Pauli strings?

Basically the title. If I have a $2^N\times 2^N$ Hamiltonian $H$ of random numbers (we can take the Hamiltonian as normalized if we want) and $N$ is an integer, is there an efficient way of writing  ...
400 views

VQE: Can I build a non-hermitian Hamiltonian with just Pauli matrices?

From the VQE paper they claim that a Hamiltonian can be expressed as a polynomial series of pauli operators (equation 1). While coding up VQE from scratch I made a function which would allow me to ...
155 views

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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1 vote
Given a state in bra-ket notation as $|\psi\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. What is the density matrix of this state written using Pauli's spin operator?
I am currently on a set of lecture notes which says that for a state vector $| \psi \rangle_{AB}$ describing a tensor product state, its density operator $| \psi \rangle \langle \psi |_{AB}$ can be ...