Linked Questions

6 votes
2 answers
901 views

How do I decompose the given $4\times 4$ matrix in terms of Pauli matrices? [duplicate]

I have been working on a question where I have to decompose this matrix in terms of Pauli Matrices: \begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\...
Sire's user avatar
  • 63
1 vote
2 answers
230 views

How to write the $iSWAP$ unitary as a linear combination of tensor products between 1-qubit gates? [duplicate]

As far as I understood, it should always be possible to decompose any $n$-qubits unitary $W$ into a linear combination of tensor products between $n$ single-qubit unitaries $U_i$: $$W = \sum_k \...
SimoneGasperini's user avatar
14 votes
3 answers
2k views

Are spin-glass problems NP (-complete)?

It is well known that finding ground states for spin-glass systems (Ising, XY...) is NP-hard (at least as hard as the hardest NP-problems) so that they can be efficiently used to solve other NP ...
Wouter's user avatar
  • 291
20 votes
1 answer
4k views

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 ...
Josu Etxezarreta Martinez's user avatar
8 votes
1 answer
2k views

How are arbitrary $2\times 2$ matrices decomposed in the Pauli basis?

I read in this article (Apendix III p.8) that for $A\in \mathcal{M}_2$, since the normalized Pauli matrices $\{I,X,Y,Z\}/\sqrt{2}$ form an orthogonal matrix basis. $$A=\frac{Tr(AI)I+Tr(AX)X+Tr(AY)Y+...
lufydad's user avatar
  • 471
2 votes
1 answer
1k views

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 ...
mavzolej's user avatar
  • 1,921
3 votes
1 answer
378 views

What is the best way to write a tridiagonal matrix as a linear combination of Pauli matrices?

I'm looking for an algorithm to write an arbitrarily sized tridiagonal matrix as a linear combination of Pauli matrices. The tridiagonal matrix has the form, for example, \begin{pmatrix} 2 & -1 &...
Corey's user avatar
  • 127
1 vote
1 answer
286 views

How to express $n$-qubit Hermitian operator with Pauli matrices

How can we prove that all $n$-qubit Hermitian matrices can be written in terms of Pauli matrices $I$, $X$, $Y$, and $Z$ as $$ \sum_{W_k \in \{I, X, Y, Z\}} a_{W_1,\dots,W_n}W_{1}\otimes ... \otimes W_{...
qc6518's user avatar
  • 163
3 votes
1 answer
168 views

Relation between Jordan-Wigner transformation and Hilbert-Schmidt inner product

Given a fermionic Hamiltonian in a matrix form, we can write it as a sum over Kronecker products of Pauli matrices using the Hilbert-Schmidt inner product. However if the same Hamiltonian is given in ...
e-eight's user avatar
  • 302
2 votes
1 answer
161 views

How to determine an unknown quantum gate if we know all other gates in the circuit and the inputs and outputs? [duplicate]

Suppose we have a quantum circuit like this. All the gates are known except for one. For any input of q[0] and q[1], we know the corresponding output. I have provided the output state for four ...
wizzywizzy's user avatar
3 votes
1 answer
111 views

Confirming locality of a Hamiltonian through decomposition

I was trying to understand Trotterization. The given Hamiltonian is decomposed into a sum of $k$-local Hamiltonians which can be exponentiated in $O(1)$ gate complexity. After which the Trotter ...
Zee's user avatar
  • 361