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When considering quantum error correction over depolarizing channels, the depolarizing probability $p$ such that an error of the kind $X,Y,Z$ will happen is used as a priori information in order to use it for determining the most likely error coset to have affected the quantum information $|\psi\rangle$ sent through such channel.

In this context, it is logical to use such a priori information for decoding, as it gives the information needed about the channel to decode with high probability of success. However, in the reality, such knowledge about the depolarizing channel will be an estimate $\hat{p}$ that will be used as a priori inforamation, as perfect knowledge of the channel is not possible. I was wondering then how such estimation of the depolarizing probability might be performed, and so which would be the error that such estimates might present.

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    $\begingroup$ Do you want to constrain the experimenter some more? Can we just keep repeating a measurement of passing a specially prepared state through the channel as many times as needed to get enough statistics? $\endgroup$ – AHusain Mar 11 at 22:40
  • $\begingroup$ Such questions are interesting indeed for the estimation problem I am stating here. Of course the number of tests that can be done in order to get the statistics of the channel are important. In reality I reckon that an small finite number of those would be done after some transmission rounds (in a similar fashion to the channel impulse response estimation in fading channels). However, I am not so concerned about the estimation being so realistic, but just some ideas/references about the topic. Consequently, I am interested in general about the topic. $\endgroup$ – Josu Etxezarreta Martinez Mar 12 at 8:35
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This probably only addresses half of your question...

Most of the time, estimates of the error probability are not so important. If you're using a non-degenerate error correcting code for a single logical qubit, all you need to know (effectively) is enough to resolve a choice between two different error corrections (i.e. you want to know that the number of errors is smaller than half the distance of the code with high probability). Small mistakes in estimation are not going to change that, unless you're right up at the boundary of whether it works or not (and then, whichever side of the boundary you're on, the 'with high probability' rider I just put on it is unlikely to hold. You want to know you're at least a few standard deviations away from that threshold).

The easiest example is the classical repetition code. At the time of decoding, if you see a bunch of 0s and 1s, which do you decode it to? You assume that the per-bit probability of error is less than 1/2, and you decode to whichever bit value you see more of. It doesn't matter if that error probability is 0.3 or 0.31, you proceed in the same way.

Where your estimate of the error probability does make a difference is if you're using error correcting codes out beyond the literal distance value of the code. The typical example of this is with the Toric Code. This is defined for $O(N^2)$ qubits on an $N\times N$ grid. The code has distance $N$, i.e. there are combinations of $N/2+1$ errors that would lead to a logical error when you do the syndrome measurement and correction. However, people still try to use the Toric Code when there are $\epsilon N^2$ errors, for some small enough $\epsilon$. That is based heavily on an assumption about the noise model and its parameters to determine which errors are most likely. I did some work at some point (not quite for depolarising noise) looking at the Toric code and how the error correction threshold changes depending on how good an estimate you have of the noise parameters. For example, if you have different rates of X and Z errors, but you assume they're the same rate, as compared to knowing what their rates are. See Figure 2 of that paper.

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  • $\begingroup$ I see what you are trying to state, but what I want to say is that when decoding Quantum Turbo codes with sum-product like algorithms, the estimate might be important as those algorithms are sensitive to the a priori information. Consequently, a model about the estimate is important to know to work with such decoding problems. $\endgroup$ – Josu Etxezarreta Martinez Mar 12 at 9:47

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