10 - No String Attached

So far, those who followed the newsletter, know-how stabilizers of surface code, are encoding a quantum state with a very low error rate due to the topological properties of surface code. The no errors state is called the quiescent state.

But you probably noticed that we still did not say anything about how the state is corrected in the case of a physical error that happens to the qubits on the surface.

Today, we will see that in case of one error at a time.

So, what is an error? You might remember that in some probability (which we call "the physical error rate"), an X/Y/Z flip can happen to one of the qubits. In this case, **spoiler alert**, I claim that the stabilizers (measurements) next to the error are doing two things:

1. In case of coherent error, they vanish the error in some big probability. Or convert the error to a simple Pauli X/Y/Z error in a small probability.
2. They tell you: if the Pauli error is still there, where it is, and what its type is (this information is usually enough to correct it manually!)

Let's see it using a graphic example.

Suppose we start from a quiescent state, which is a superposition of an even amount of strings from the right edge to the left edge of the surface:

Notice that each stabilizer measurement (on the black dots ancilla qubits) will return +1 result since there are even amounts of |1? to each stabilizer.

In a case of an error, let's assume X error:

Which flips |0? to |1? and flips |1? to |0?:

2 stabilizers will prompt that they found the error because a measurement of ZZZZ stabilizer that finds if the amount of |1?s is even or odd will see an odd amount of |1?s:

Now, imagine that you do not know the error happened.

All you know is the measurements that are going to the classical computer that controls the error correction: the -1 results of the ZZZZ stabilizers (also XXXX, but we do not look at them for now), so all you see is:

It is very easy to see that probably what happened is a Pauli X operation between the two -1 results! (it could also happen from more than 1 error case, but we keep it to the next articles).

Since XX=I, we can operate with X again on this errored qubit and fix that error!

Wow! now you know how a stabilizer in surface code deals with an error!

In the next articles, we will:

• Learn what happens in a case of coherent error, which is a superposition of no error + error.
• Analyze the case of more than 1 error at a one-time frame and how it might prevent the stabilizer from doing its job well.

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