Dephasing noise and error correction

Question

It is my impression that $ Z $ errors are the dominant source of noise in many platforms used for quantum computing. For example, Ultrahigh Error Threshold for Surface Codes with Biased Noise says that noise biased towards dephasing "is common in many quantum architectures, including superconducting qubits, quantum dots, and trapped ions."

Is that because $ Z $ errors tend to occur on a shorter time scale than other types of errors and thus accumulate more quickly?

I was thinking about this in the context of the Steane syndrome extraction and error recovery procedure for CSS codes that is described for example in chapter 12 section 3 "Steane Error Correction and Measurement" of Gottesman's book Surviving as a Quantum Computer in a Classical World

There are two steps, one that extracts the syndrome with respect to $ X $ type stabilizers to figure out what $ Z $ Pauli correction to apply and a second step that extracts the syndrome with respect to $ Z $ type stabilizers and to figure out what $ X $ Pauli correction to apply. Gottesman points out that it is arbitrary wether you want to extract syndromes for $ X $ type stabilizers first or extract syndromes for $ Z $ type stabilizers first, but some choice must be made.

My question is the following: if it is true that in certain hardware $ Z $ type errors occur on a short timescale and $ X $ type errors only occur on a longer timescale then would it be true that the correct approach is to extract the syndrome for $ Z $ type stabilizers (the ones that detect $ X $ errors) 2nd because we only expect $ X $ type errors to occur later in the process.

I guess what I'm asking is when we are designing error correction gadgets like this for hardware where dephasing dominates then is it a reasonable approximation to assume that $ X $ errors only "happen on a longer time scale" and so we can assume no $ X $ errors (or very few $ X $ errors) occur until many time steps into the error correction protocol?

Answer 1

It is generally true that for a given qubit, you can define some axis of the Bloch sphere that is more noisy than any other axis of the Bloch sphere. It is also true that there are certain qubits, e.g. cat qubits, where this asymmetry is engineered to be extremely large.

However, there is a subtlety. Consider applying an $X$ gate to a qubit with strongly $Z$-biased noise. Naively, you would generate this rotation gate by applying a Hamiltonian $X$ for some length of time: $$ X \sim e^{-i\pi X/2} $$ Now, we have to ask the question: What if a $Z$ error occurs halfway through this gate? After all, Z errors can occur at any time. We have $$ X'=e^{-iX\pi/4}Ze^{-iX\pi/4}\sim Ye^{i\pi X/2} $$ which says that a $Z$ error halfway through an $X$ gate becomes a $Y$ error if you commute it to the end of the gate. $Z$ errors occuring earlier or later in the gate will become superpositions of $Y$ and $Z$ errors. In any case, we see that physical noise bias for the qubits isn't enough, since unitary operators can turn $Z$ errors into other Pauli errors!

The key to realizing biased-noise error correction is, you need to be able to implement all the gates for error correction without converting $Z$ errors into $X$ or $Y$ errors. This is not always possible! While you don't need an $X$ gate to do error correction, you do need a $CX$ gate, and the $CX$ gate runs into the same problem I outlined for the $X$ gate above. The reason why people talk about biased noise for 2-cats in particular, and rarely for any other qubits, is that 2-cats have a bias-preserving $CX$ gate that does not convert $Z$ errors into any other error.

But for general qubits, this will almost certainly not be true: Whatever noise bias your qubits have while idling will get mixed by the gates you apply.