15 votes
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Can arbitrary matrices be decomposed using the Pauli basis?

For any matrix $A$ we can write $$ A =\sum_{i,j,k,l}h_{ijkl}\cdot \frac{1}{4}\sigma_i\otimes\sigma_j\otimes\sigma_k\otimes\sigma_l, $$ where $$ h_{ijkl} = \frac{1}{4}\text{Tr}\big((\sigma_i\otimes\...
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14 votes

How to construct matrix of regular and "flipped" 2-qubit CNOT?

$$ CNOT = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix} $$But what does this matrix ...
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14 votes

Can arbitrary matrices be decomposed using the Pauli basis?

$\newcommand{\bs}[1]{{\boldsymbol #1}} \newcommand{\tildebssigma}{\tilde{\bs\sigma}} \newcommand{\bssigma}{{\bs\sigma}}$Yes, products of Pauli matrices form a basis for the set of Hermitian matrices (...
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12 votes
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How to construct matrix of regular and "flipped" 2-qubit CNOT?

The CNOT gate is a 2-qubit gate, and consequently, its operation cannot be expressed by the tensor product of two one-qubit gates as the example you gave with the Hadamard gates. An easy way to check ...
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12 votes
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Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?

Yes, the set of tensor products of all possible $n$ Pauli operators (including $I$) form an orthogonal basis for the vector space of $2^n \times 2^n$ complex matrices. To see this first we notice ...
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  • 886
11 votes
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How to compute the average value $\langle X_1 Z_2\rangle$ for a two-qubit system?

I suggest two different ways of trying to solve this, which will give you experience of different bits of the formulation of Quantum Information Theory. I'll give examples that are closely related to ...
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11 votes
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How is the ground state of a Hamiltonian defined?

A couple of points: The ground state is by definition the eigenvector associated with the minimum valued eigenvalue. Lets consider the Pauli Z matrix as you have. First, \begin{align*} Z = \begin{...
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  • 505
11 votes
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How can I decompose a matrix in terms of Pauli matrices?

I call this the "Paulinomial decomposition" as you are writing the matrix $H$ as a polynomial of Pauli matrices: $H=a_{XX}X_1X_2 + a_{XY}X_1Y_2 +a_{XZ}X_1Z_2 + a_{XI}X_1 + a_{YY}Y_1Y_2 + \...
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  • 12.1k
10 votes
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How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis?

The Pauli matrices form an orthogonal basis of $\mathcal{M}_2$, this vector space can be endowed with a scalar product called the Hilbert-Schmidt inner product $$ \langle A,B\rangle=\mathrm{Tr}(A^\...
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10 votes
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Qiskit CNOT-gate matrix mixup?

Qiskit uses "little endian" bit ordering. That means, if A and B are $2 \times 2$ unitary matrices then $B \otimes A$ (note the order) is equivalent to applying $A$ to first qubit and $B$ to ...
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10 votes
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Why isn't $Ry(\pi/2)$ gate equivalent to Hadamard gate?

While Hadamard gate is defined as $$ H= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, $$ $y$-rotation by $\pi/2$ leads to gate $$ Ry(\pi/2)= \frac{1}{\sqrt{2}} \begin{...
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9 votes
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Obtaining gate $e^{-i\Delta t Z}$ from elementary gates

One way order to perform Z rotations by arbitrary angles is to approximate them with a sequence of Hadamard and T gates. If you need the approximation to have maximum error $\epsilon$, there are known ...
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9 votes
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How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?

First of all, note that the statement, as written, is wrong (or rather, it is correct only as long as the "$\equiv$" symbol is taken to mean "equal up to a phase"). An easy way to see it is by ...
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8 votes
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What is the difference between the action of $Z$ and $\exp(-i Z t)$ on a state?

Effectively, the Z operation (represented by the Pauli $Z$ matrix) applies a rotation about the $Z$-axis. As you note, rotations can also be written in the form $e^{...
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7 votes

Can there be multiple energy eigenstates corresponding to the same eigenvalue of a Hamiltonian (Pauli-X)?

The quantum states that differ by a global phase (i.e., by a complex number multiple which has absolute value of 1) are considered the same quantum state, since they can not be distinguished using any ...
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What does the notation $\sigma_j^z$ mean for Pauli matrices?

What $\sigma^z_i$ means is that you've got a Pauli-$Z$ applied to qubit $i$, and nothing else on the other qubits (i.e. the identity). So, you could expand it as $$ I^{\otimes(i-1)}\otimes\sigma^z\...
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7 votes
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How do I create an inverse identity gate?

As a general rule, you wouldn't bother constructing this: it is just a global phase that has no observable consequence. If you really insist on doing this, introduce an ancilla qubit in the $|1\rangle$...
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7 votes

How to create the state $\vert 0 \rangle+i \vert 1 \rangle$ using elementary gates?

Starting with the state $|\psi_0 \rangle = |0\rangle$, and we want to get to the state $|\psi_f \rangle = \dfrac{|0\rangle + i|1\rangle}{\sqrt{2}}$ then we must realize that we need to create some ...
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  • 12.7k
7 votes

Can we write Pauli-Y gate without even complex part?

$\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} = i \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $ There is no reason to factor out the $i$, it just make thing more cumbersome. I think ...
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7 votes
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Can we write Pauli-Y gate without even complex part?

As noted by @KAJ226 in another answer, the global phase factor $i$ is unobservable and can be ignored, unless we are considering a controlled gate in which case the phase factor $i$ becomes a relative ...
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7 votes
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How to prove the fundamental equation in the theory of angular momentum $\sum_{l=x,y,z}\langle J_l^2\rangle\le\frac{N(N+2)}{4}$?

We can transform the left hand side as follows $$ \begin{align} \sum_{l=x,y,z}\langle J_l^2\rangle &= \sum_{l=x,y,z}\langle\psi|J_l^2|\psi\rangle \\ &= \langle\psi|\left(\sum_{l=x,y,z}J_l^2\...
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7 votes
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Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

Conjugating a non-identity Pauli operator $P$ by Clifford operators yields all non-identity Pauli operators, including $P$, with equal frequency. Proof Let $G_n$ and $C_n$ denote the $n$-qubit Pauli ...
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7 votes

What is (formally) a transversal operator?

Based on informal conversations: there is no actually agreed upon definition of a transversal operator. People use it to mean different things. Typically it refers to either the operation being fast ...
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7 votes
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Calculate $\sqrt[4]{X}$ for the Pauli $X$ gate

There are a couple of ways that you might do this calculation directly. I'd start by finding the spectral decomposition of $X$: $$ X=|+\rangle\langle +|-|-\rangle\langle -|=e^{i0}|+\rangle\langle +|+e^...
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6 votes
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Shorthand notation for the sign flip gate

A unitary $U$ and $e^{i\phi}U$, which differs from it by a phase, act exact identically on any quantum state. Thus, they should really be considered the "same" unitary in terms of their action. ...
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6 votes

Definition of the Pauli group and the Clifford group

The difference in definitions is from either taking the unitary group or the projective unitary group. That accounts for the constant prefactors of $\pm i$ that are missing. In lieu of a tikz ...
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  • 3,553
6 votes

How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?

For questions like this, the conventional physics notation is easier to work with than the QIT gate notation. Define $\vec \sigma = (\sigma_1,\sigma_2,\sigma_3)$ to represent the three Pauli matrices ...
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6 votes

How to get specific state applying $e^{-i\phi \sigma_2/2}$ to $|0\rangle$ or $|1\rangle$?

A matrix function $f(A)$ for normal matrix $A$ is defined as follows \begin{equation} f(A)=\sum_{i=1}^{n}f(\lambda_i)v_iv_i^T \end{equation} where $\lambda_{i}$ is an eigenvalue and $v_{i}$ is ...
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6 votes
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How do I decompose the given $4\times 4$ matrix in terms of Pauli matrices?

I'm really not sure what the truth table is trying to represent but let me present a solution that (at least I think) is fairly simple. First we note that the Pauli matrices together with the identity ...
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  • 4,212
6 votes
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Why can any density operator be written this way? (quantum tomography)

From linear algebra, if $v_1, \dots, v_n$ is a basis of the vector space $V$ then every vector $v\in V$ can be written as a linear combination $$ v = a_1 v_1 + \dots + a_n v_n\tag1 $$ where the ...
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