Quantum Computing Stack Exchange is a question and answer site for engineers, scientists, programmers, and computing professionals interested in quantum computing. It only takes a minute to sign up.
Sign up to join this community
I was reading about the Steane's code for QEC. However I could not find any implementation or circuit describing its creation. Can anyone share/explain how the Steane's code could be implemented.
The answer can be Circuit Diagram or Implementation in any Quantum Programming Language (Q#, Qiskit etc)
One way of defining the Steane code is via its stabilizers. There's a set of operators $\{K_n\}_{n=1}^6$ which all commute, such that a state in the code space is defined by being the $+1$ eigenstate of all these operators.
So, you can perform syndrome extraction simply by measuring the value of each stabilizer. This is a standard circuit,
(the $\sigma_1\otimes\ldots\otimes\sigma_n$ corresponds to a single $K_i$ term in this setting).
One very simple way that you can produce a codeword in the code space is simply to start with $|\psi\rangle$ as any state you want. Perform syndrome extraction and error correction on it, and the result must be a word in the space.
In fact, if you set $|\psi\rangle=|0000000\rangle$, you'll get the logical 0 state because this is a $+1$ eigenstate of $Z^{\otimes 7}$. Moreover, this state is already the $+1$ eigenstate of all the $Z$-type stabilizers so you don't have to measure any of those.
There do exist unitary encoding methods, but from the conceptual point of view, this is a very simple method that generalises to any stabilizer code.
