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It is a proven and well-known fact that length-4 cycles are detrimental to the performance of classical LDPC codes. This is due to the fact that such short cycles impair the decoding algorithm (Sum Product or Belief Propagation over the code Tanner Graph), reducing its convergence threshold and negatively impacting its performance. For this reason, numerous algorithms and LDPC code construction techniques that minimize the presence of these short cycles (i.e maximize the code girth) have been devised.

In the Quantum realm, Quantum LDPC codes are designed based on the Stabilizer formalism, which allows us to construct QLDPC codes by making use of existing classical LDPC codes, albeit following specific criteria to select the aforementioned existing classical codes. They are also decoded based on implementations of the SPA or BPA over the associated Tanner graphs. Therefore, it is intuitive to assume, without considering the specifics of a quantum framework, that length-4 cycles will also have a negative impact on the performance of QLDPC codes based solely on what is known from classical communications.

Let us now consider how QLDPC codes, or their quantum parity check matrices more precisely, are constructed (I will employ the CSS construction method as it is the most common): $$ H_Q = (H_z|H_x) = \begin{pmatrix} H_z^{'} &0 \\ 0 &H_x^{'} \end{pmatrix}, $$ where $ H_z = \begin{pmatrix} H_z^{'} \\ 0 \end{pmatrix}$ and $ H_x = \begin{pmatrix} 0 \\ H_x^{'} \end{pmatrix}$. In the construction shown above, $H_z'$ and $H_x'$ are the parity check matrices of two classical LDPC codes $C_1$ and $C_2$, chosen so that $C_1^\perp \in C_2$ is fulfilled, where $C_1^\perp$ is the dual of the classical LDPC code $C_1$.

In essence, the Quantum PCM of a QLDPC code is constructed based on the classical PCMs of two classical LDPC codes. Thus, we are faced with a "double whammy"-like scenario:

  • On the one hand, the QPCM is built from two different parity check matrices that will inevitably have some length-4 cycles.

  • In addition, the symplectic representation employed in Quantum error correction automatically induces more length-4 cycles. This stems from the fact that the generators of the code must commute in order to satisfy the symplectic criterion, which ensures the resulting code is applicable in a quantum environment.

Based on what I have mentioned up to this point, it would seem reasonable to assume that minimizing or even eliminating all the length-4 cycles from the PCM of a QLDPC code would undoubtedly improve performance. However, based on the work I am conducting on QLDPC codes and results I have garnered, it appears as if less length-4 cycles is not always better. In other words, there seems to be a "sweet spot" value of length-4 cyles at which performance of QLDPC codes peaks and increasing or decreasing the amount of cycles at that point worsens said performance. I am not entirely certain as to why this happens, but I believe an argument could be made for the presence of at least a minimum number of short cycles in the QPCM of a QLDPC code (they might aid in spreading information over the entire decoding graph which could then be exploited by the decoding algorithm due to its iterative nature).

I am trying to ascertain if pursuing methods to determine or establish a threshold of length-4 cycles that optimizes code performance is a worth-while research endeavour. I would be greatful for insights, opinions or experiences others may have had in this field. I would also be thankful if I could be directed towads publications that may discuss this topic (I recall reading a paper that mentioned the importance of having length-4 cycles in decoding graphs by I have not been able to find it).

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