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### 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\...
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1 vote
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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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### 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 &...
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1 vote
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### 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_{...
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### 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 ...
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### 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 ...
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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 ...