Quantum error correction is a fundamental aspect of quantum computation. I have read some material about "Quantum Channel" and "Quantum error correction".

I have known the formula of gate error which has close relationship with decoherence time. However, no one tell me what is the certain probability of each error is reliable or what is the exact possibility in real environment.

For example, in depolarizing channel, each error can be approximated into pauli error, which is X,Y,Z error with their own possibility $p_x$,$p_y$ and $p_z$. So is that a certain scale of possibility (just for example, $1\%<p_x<10\%$) of each error is reliable or something close to that?

  • $\begingroup$ If the environment/noise is approximately constant, then you can do quantum process tomography to figure out the dynamics and then tailor your error correcting code accordingly. For example, if by process tomography you know that dephasing noise is the dominant source of decoherence in your system, then you can use a code optimized for that. $\endgroup$ – keisuke.akira Aug 2 at 9:09
  • $\begingroup$ As a small note, the idea of the depolarizing channel is that $p_{x} = p_{y} = p_{z} (= \frac{p}{3})$, so its a Pauli channel where the probabilities for an $X$, $Y$ and $Z$ flip are all equal (and often taken as $\frac{p}{3}$). $\endgroup$ – JSdJ Aug 3 at 9:46

As I explained in my answer on a previous question of yours, the depolarizing channel is not really 'physical' - actual quantum systems don't really behave that way.

So for simulations where you, for instance, investigate the performance of some code against the depolarizing channel, it doesn't really matter what the exact value of $p$ is in your simulations. Moreover, what is much more interesting is to perform the simulation for a range of values of $p$, and see how the performance of the code depends on $p$. As you may be aware, for higher $p$, a code may do more bad than good (i.e. it creates more errors than it can correct); for a low $p$ the code will actually do some good, and thus an interesting value for $p$ might be the point where the code starts to add benefit - this value is often referred to as the pseudo threshold. This excellent answer elaborates a bit on why we call it the pseudo threshold, and not just threshold.

Now, if you want to simulate actual systems more closely, as I explained in my answer linked above, it's 'better' to use the amplitude damping and dephasing channel. The error rates are mostly expressed in the damping time $T_{1}$ and dephasing time $T_{2}$; what exactly these values are is highly dependent on what system you are simulating.

For instance, the transmon qubits from IBM have a typical $T_{1}$ of $~50 \mu s$ and a $T_{2}$ of $~60 \mu s$. As another example, the quantum dot based semiconductor qubits of QuTech's Quantum Inspire have a typical $T_{1}$ of $> 20 ms$ and a $T_{2}$ of $> 6 \mu s$.

As you can see, these values vary quite a bit; one of the main pillars of current physical qubit systems development is to increase these characteristic times.

Note that this is just one part of the story; $T_{1}$ and $T_{2}$ say only something about what happens to your qubits when they are idle. The operations that you perform on the qubits are faulty themselves as well, and in a proper simulation of a QECC this has to be taken into account. Errors in operations can be seen as the perfect operation followed by some error in the form of a quantum channel, but what channel is applicable here is another question entirely and out of the scope for this answer; of course you're welcome to ask a separate question about this!

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    $\begingroup$ thanks a lot~~~~~ $\endgroup$ – Henry_Fordham Aug 5 at 13:27
  • $\begingroup$ @JSdJ on Qiskit device noise models, they have both depolarizing errors as well as T1,T2 thermal relaxation errors. Your answer suggests that T1, T2 cause depolarizing errors, so why does qiskit noise model have depolarizing error? $\endgroup$ – Eesh Starryn Sep 1 at 11:49
  • $\begingroup$ @EeshStarryn The depolarizing error channel is not the same as the amplitude damping (T1) or dephasing (T2) channel - you might have misunderstood my answer. "Depolarizing errors" is to me a bit of an ambiguous term; if you stick to "depolarizing channel/noise" it's more obvious that it's not the same as either the amplitude damping or dephasing channel. $\endgroup$ – JSdJ Sep 1 at 12:34

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