Quantum Error Correcting Codes

This thesis deals with quantum error correcting codes. In first two chapters necessary introduction to quantum computation and classical error correction is presented. Previous results on construction of quantum error correcting codes are presented in the third and fourth chapter. Mainly Calderbank-Steane-Shor (CSS) codes and stabilizer codes are discussed together with the introduction to coding, decoding and recovery circuits’ construction. Second part of the thesis presents our own results. We have concentrated our effort on the exploration of new CSS codes and examination of their usability. CSS codes are presented as a wide class of quantum codes in literature, but conditions for their practical construction are quite complicated. Well known CSS codes are Steane code correcting errors on a single qubit and [[23,1,7]] code derived from Golay code (correcting three errors). However, no CSS code encoding one logical qubit and correcting errors on up to two qubits is established in the field of quantum error correction. Such code would need to use at least seventeen encoding qubits. We present probabilistic algorithm searching for CSS codes. We used this algorithm and found [[19,1,5]] CSS code. It remains an open question whether 17 or 18 qubits will suffice to construct a CSS code correcting two arbitrary errors. In last two chapters of the thesis results of numerical and theoretical analysis of found [[19,1,5]] code are shown. The concept of fault tolerant quantum computation is used in the analysis. Found [[19,1,5]] code is compared to Steane code. It follows that [[19,1,5]] CSS code provides better results than Steane code, if fault rate of used quantum gates is below 2, 5.10−4. We have optimized fault tolerant error correction schema for [[19,1,5]] code using the analysis and numerical simulations of several potential architectures. Probability that [[19,1,5]] code would fail to protect encoded qubit is shown (theoretically and also experimentally) to be O(ξ), where ξ is probability of single gate failure. In the thesis also coding, decoding and recovery circuits for found [[19,1,5]] code are designed.