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I'm trying to understand the three qubits bit-flip code. I use the book of Phillip Kaye An introduction to quantum computing.

In this book he introduce the three qubits bit-flip code with this circuit :

Three qubits bit-flip code from Phillip Kaye

But, I saw in Quantum Error Correction for Beginners (here), that they introduce the three qubits bit-flip code with an another circuit:

Three qubits bit-flip code from *Quantum Error Correction for Beginners*

Therefore, I ask myself, why do we need two more ancilla qubits in the second algorithm ? What do we earn with that ?

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Both circuits start by turning $|\psi\rangle = a |0\rangle + b|1\rangle$ into $|\psi\rangle = a |000\rangle + b|111\rangle$, copying the 0/1 information onto two additional qubits.

Note that, for any pair of qubits in this state, either both are in state $|0\rangle$, or both are in state $|1\rangle$. Any superposition must be expressed only in terms of states that satisfy this condition.

This gives us a way to check up on whether everything is okay. We make a measurement that checks whether the 0/1 states of two neighbouring qubits are the same or different. We want it to give us only this information and no more, to ensure that preserves superpositions.

Each ancilla in the second circuit is there to perform this exact measurement. The two cnots that act on the ancilla, as well as the measurement gate itself, are all part of the measurement. Th measurement outputs 0 when two qubits can be expressed as a superposition or mixture of $|00\rangle$ and $|11\rangle$. It outputs 1 if they must be expressed with $|01\rangle$ and $|10\rangle$.

Given this information, we can see if errors happened, and try to guess what they were. A corresponding correction operator can then be performed.

The first circuit seems like it could probably correct a bit flip on the first qubit. I'm not really sure what it is designed for, though.

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    $\begingroup$ What do you mean by : "It is not a scalable approach to error correction, since it has no measurements to remove the entropy introduced by noise." ? $\endgroup$
    – lufydad
    Jun 18, 2019 at 21:12
  • $\begingroup$ Errors introduce entropy you don't want, and error correction needs to remove it. Usually we do this using measurement to find out what errors occurred. But that sentence was admittedly a bit vague, so I removed it. $\endgroup$
    – James Wootton
    Jun 19, 2019 at 9:09
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The second circuit doesn't have to perform a quantum CCNOT operation (which in practice decomposes into a dozen other operations) and doesn't have to leave the code space (allowing it to be applied iteratively).

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The second circuit applies a stabilizer formalism. A three qubits bit-flip code has stabilizer generators $Z_1Z_2$ and $Z_2Z_3$. You can find the circuit in page 473 in Nielsen and Chuang's book - quantum computation and quantum information. I also have a question about the circuit of three qubits bit-flip code. I would be appreciated if you take a look at this.

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The question is "why do we need two more ancillary qubits in the second algorithm ?"

The answer is in the no-cloning theorem: "by the qbits nature, they cannot be copied you can only entangle them". And if you measure the transmitted qbits the system will collapse and their information will be lost. Therefore, that's why you need TWO ADDITIONAL qbits to know the kind of correction to be done.

Please read the same explanation from the book Quantum Computing With Silq Programming in chapter 11 section Understanding quantum error correction

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