# Circuit for implementing Steane's code for Quantum Error Correction

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)

## 1 Answer

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.