# Can quantum error correction work on any type of channel?

It says on wikipedia that quantum error correction can (at best) correct phase flips and bit flips. A popular form of representing a quantum channel is in its Kraus representation (scroll down to section 'pure channel' or just CRTL+F 'Kraus'). This kind of representation can yield a whole plethora of different possible noise models.

That being said, are there possible quantum channels that may exist (either right now or in the future) that induce noise that can not be corrected by current QEC methods? This question is somewhat related to this unanswered question.

• It's hard to imagine an error that cannot be corrected using quantum error correction provided there's enough auxiliary qubits. Mar 22 at 19:35
• can you elaborate more? I'm only aware of QEC that can correct X, Y, or Z transformations. If i'm not mistaken, a Kraus representation of a quantum channel can induce more than just X,Y,Z transformations. Mar 22 at 19:59
• Any Kraus operator $E_k$ can be written as a linear combination of the Pauli operators, so if the code corrects single-qubit $X$, $Y$ and $Z$ errors then it corrects all single-qubit errors. This is quite remarkable (the ability to correct a few discrete types of errors turns out to be sufficient to correct infinite continuum of possible errors), but it is not hard to prove. See page 434 in Nielsen & Chuang for details. Mar 22 at 20:43

For any quantum error correcting code, it is possible to construct a channel which introduces errors that the code cannot correct. However, the key point is that such channels are highly adversarial and not at all representative of any physically reasonable error mechanism.

An easy way to construct such adversarial noise is to build it from the logical operators. For example, denote the logical Pauli $$X$$ operator by $$\overline{X}$$, then

$$\mathcal{B}(\rho) = (1-p)\rho + p\overline{X}\rho\overline{X}$$

is the logical bit-flip channel which changes logical state $$|\overline{0}\rangle$$ into logical $$|\overline{1}\rangle$$ and vice versa with probability $$p$$. Note that whenever the logical state is flipped the code is helpless to correct it because both $$|\overline{0}\rangle$$ and $$|\overline{1}\rangle$$ belong to the code subspace and therefore their syndrome is trivial, i.e. indicative of no error.

More generally, any error that keeps the code subspace invariant, but does not act as identity cannot be corrected. For stabilizer codes this means that any error operator in $$N(S) - S$$ where $$S$$ is the stabilizer group of the code and $$N(S)$$ is the normalizer of $$S$$ cannot be corrected. Note that this is by design. If a quantum error correcting code treated logical operators as errors then they could not be used for quantum computation since error correction would interfere with logical gates.

The objective of quantum error correction is then to find codes such that the logical operators are extremely unlikely to be applied spontaneously by the noise. Generally speaking, this is achieved by choosing logical operators that require a very coordinated change across a large number of physical qubits that is highly unlikely to occur by chance without programmed intervention from the qubit control stack. It is not hard to see that we can find such codes by observing for example that we can find a stabilizer $$S$$ such that $$N(S)-S$$ contains operators of very high weight, i.e. high number of non-identity factors. The smallest weight $$d$$ of operators in $$N(S) - S$$ is called the code distance. A code with distance $$d$$ can correct up to $$\lfloor{\frac{d-1}{2}}\rfloor$$ single-qubit errors regardless of the specific single-qubit operators applied by the errors.

For more details including the necessary and sufficient conditions for a given quantum error correcting code to recover from a given noise operation, see the so-collect quantum error-correction conditions in theorem 10.1 on page 436 in Nielsen & Chuang and theorem 10.8 on page 466 for a version for stabilizer codes.