2
$\begingroup$

In multiples papers or online article on the QAOA algorithm (such as this one), I found notation for the Hamiltonian similar to this one :

$$ \sum_{ij} \frac{1}{2} (I-\sigma_i^z \sigma_j^z)$$

I don't know how to interpret the notation $\sigma_j^z$. I guess it has something to do with the Pauli matrix $\sigma_z$ but that's all I can understand.

$\endgroup$
7
$\begingroup$

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 $$ I^{\otimes(i-1)}\otimes\sigma^z\otimes I^{n-i} $$ if your system has $n$ qubits. A term such as $\sigma^z_i\sigma^z_j$ is then a product of two of these, which is equivalent to the tensor product of $\sigma^z$ on qubits $i$ and $j$ and identity everywhere else. The summation is usually taken over pairs $i,j$ which are nearest-neighbours on some underlying geometry of the qubits.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.