What does the notation $\sigma_j^z$ mean for Pauli matrices?



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.

nathan raynal

Posted 2020-04-01T18:20:11.923

Reputation: 463



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.


Posted 2020-04-01T18:20:11.923

Reputation: 35 722