# The merit of quantum error correction codes

We know that word error rate (WER) rather than qubit error rate (QER) is used to evaluate the performance of quantum Turbo codes and quantum LDPC codes. In classical coding theory, when we are computing the WER, we can either calculate the erroneous whole codeword rate, or only calculate the erroneous information part rate (e.g., the system codes). In QECCs, I find that most people always calculate the erroneous whole codeword rate rather than the erroneous information part rate when computing the WER. I want to know that can we calculate the erroneous information part rate when computing the WER, just like the system codes in classical coding theory? Is it possible to do this? Thanks.

• I come from the MathOverflow. – Bruce Fang Jul 23 '19 at 1:22
• I didn't mention there, but it is also usual policy to only keep the question in one of the two sites and delete the other. – AHusain Jul 23 '19 at 8:42
• Could you give any reference to what you are stating? I mean some paper that speaks about the WER as you say. – Josu Etxezarreta Martinez Jul 23 '19 at 18:50
• Many refrerences, e.g., "Sparse-Graph Codes for Quantum Error Correction", TIT 50.10 (2004): 2315-2330. "Quantum serial turbo codes." TIT 55.6 (2009): 2776-2798. – Bruce Fang Jul 24 '19 at 2:13
• In my opinion, the recovery of quantum states after the decoding may need a wholely correct codeword. – Bruce Fang Jul 24 '19 at 2:16

From my own experience working with QLDPC codes and what I have gleamed from the literature, using the erroneous whole codeword rate for the computation of the WER is the de facto procedure to evaluate quantum LDPC code performance. This is broached with relative simplicity in "Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies" (https://doi.org/10.1109/ACCESS.2015.2503267). I'm sceptical with regard to it being the sole metric for Quantum Turbo code performance assessment, as I'm fairly certain to have seen publications where the erroneous information (the erroneous content of the pre-encoded message) was used to study the quality of Quantum Turbo codes. Perhaps skimming through existing papers related to the latter codes might help in providing better insight than what I've been able to muster in this answer.

• Thanks. In quantum settings, I think, if you can not correct the whole codeword, then it maybe impossible to recover the quantum information to the pre-encoded states. But in classical coding theory, system codes can do this. Is it right? – Bruce Fang Jul 24 '19 at 8:01
• In QLDPC decoding, at least within a classical framework of quantum code simulation, the aim is to estimate the most likely error pattern to which an $n$-qubit block is subjected to when being transmitted through a quantum channel. This error pattern (usually represented by a length $2n$ vector $P = (P_z|P_x)$) is obtained by running a syndrome-based version of the SPA algorithm over the Factor graph associated to the QLDPC code in question. Thus, the WER is computed by comparing the estimated pattern $\hat{P}$ (obtained via syndrome SPA decoding) to the original error pattern $P$. – Patrick Fuentes Jul 24 '19 at 10:03
• This decoding stratagem differs from the classical schemes as it is adapted to the information that is obtained from quantum channels. Given the nature of QLDPC decoding, it seems logical that in the absence of a correct codeword it will be impossible to correctly recover the quantum information of the pre-encoded states. In other words, it doesn't seem plausible to obtain the original pre-encoded states when the recovered codeword is erroneous. Note that this is just conjectural, although I can't think of an instance in which the notion wouldn't hold. – Patrick Fuentes Jul 24 '19 at 10:11
• I agree with your opinonion. The erroneous codeword may lead to serious error propagation when doing the recovery. But @Josu Etxezarreta Martinez said that when doing the syndrome measurements directly, then partly recovery is possible. – Bruce Fang Jul 25 '19 at 2:22

To add some more insight regarding to the answer given before, I think that you confused when understanding the Word Error Rate as defined in Quantum serial turbo codes. In such paper, the authors describe the QBER as the fraction of logical qubits that have errors after the decoding, while saying that the WER is the probability that at least one qubit in the block is incorrectly decoded. In such description you undestood that such block refers to the whole codeword, but the authors are referring to the logical qubit block, that is, the block of information qubits.

Such thing can be seen more clearly when they describe the decoding algorithm in the paper, as such turbo decoder estimates the most probable logical error coset $$\hat{\mathcal{L}}$$ that affects to the information qubits rather than the most probable physical error, that would be $$\hat{\mathcal{P}}$$. Consequently, in the QTC paradigm at least, the WER refers to the figure of merit that you are looking for, and not to a WER that considers the estimation of the channel error, or the error that would affect the whole codeword sent through the depolarizing channel.

• Maybe I did not understand the refference titled "Quantum serial turbo codes" very well. But in the quantum settings, if you can not get the whole codeword length errors, then how can you recover the quantum information? Thanks. – Bruce Fang Jul 24 '19 at 9:09
• The thing here is the fact that when doing syndrome measurements in the ancilla qubits, you can do so directly or indirectly. When measuring such qubits, you destroy them, and so in the end you obtain the noisy logical qubits and the syndrome, and you infer the error on such information qubit from the syndrome. By measuring the syndrome indirectly, the codeword is not partly destroyed, and so form such syndrome you can infer the whole codeword error. It depend on the inference you are doing from the syndrome. – Josu Etxezarreta Martinez Jul 24 '19 at 9:49
• That sounds amazing. I have been learning QECCs for some time. This is the first time heard from you that the sydrome measurements can be done directly or indirectly (I only know the indirectly measurements before as you saied). But I sitll want to know that if we do the syndrome measurements directly, can we still get the full syndrome? Can you provide some references about the direct syndrome measurements? Thanks. – Bruce Fang Jul 25 '19 at 1:36