Citation: Calderbank, A. Robert, and Peter W. Shor. "Good quantum error-correcting codes exist." Physical Review A 54.2 (1996): 1098.
Web: https://arxiv.org/abs/quant-ph/9512032
Tags: Computer-scientific, Foundational, Error-correcting-code, Information-theory
This is the paper which introduced CSS (Calderbank-Shor-Steane) codes. The work by Steane was done independently of Calderbank and Shor, and was published around the same time:
> Steane, Andrew M. "Error correcting codes in quantum theory." Physical Review Letters 77.5 (1996): 793.
These codes work by taking two orthogonal classical error correcting codes, and making a new quantum code whose Tanner graph is the hypergraph product of the Tanner graphs of the two classical codes. One of the classical codes has its parity checks become X-checks, and the other classical code has its parity checks become Z-checks. In a CSS code, all of the stabilizers are Pauli. Moreover, all of the checks either include all Xs or all Zs - there are no mixed terms or Ys. The best example of CSS codes is the toric code, which is a hypergraph product of two repetition codes. A more modern treatment of CSS codes can be found here:
> Tillich, Jean-Pierre, and Gilles Zémor. "Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength." IEEE Transactions on Information Theory 60.2 (2013): 1193-1202.
A key thing to remember is that finding good quantum codes is easy and was done in this original work by Shor and Calderbank. The hard part is doing it while using low density checks, which is required by the locality of physics.
As with many of these LDPC papers, the authors are very concerned with the implications for quantum information theory. The last section of this paper is a discussion of noise in quantum channels. This is the paper where the idea of a noisy quantum channel was introduced, and the important first observations were made.