Threshold analysis with fault-tolerant operations for nonbinary quantum error correcting codes

Abstract

Quantum error correcting codes have been introduced to encode the data bits in extra redundant bits in order to accommodate errors and correct them. However, due to the delicate nature of the quantum states or faulty gate operations, there is a possibility of catastrophic spread of errors which might render the error correction techniques ineffective. Hence, in this thesis we concentrate on how various operations can be carried out fault-tolerantly so that the errors are not propagated in the same block. We prove universal fault-tolerance for nonbinary CSS codes. This thesis is focussed only on nonbinary quantum codes and all the results pertain to nonbinary codes. Efficient error detection and correction techniques using fault-tolerant techniques can help as long as we ensure that the gate error probability is below a certain threshold. The calculation of this threshold is therefore important to see if quantum computations are realizable or not, even with fault-tolerant operations. We derive an expression to compute the gate error threshold for nonbinary quantum codes and test this result for different classes of codes, to get codes with best threshold results.