Abstract
Fault-tolerant quantum computation relies on quantum codes that suppress logical error rates while keeping physical-qubit overhead manageable. Concatenated codes provide a systematic route to fault tolerance, but fully concatenated families offer only discrete choices of block size and distance. This thesis studies partially concatenated codes based on the C4/C6 architecture as a way to obtain finer tradeoffs between protection and resource cost.
We first analyze the fully concatenated C6 and C4/C6 families using stabilizer and logical operator methods. To treat nonuniform concatenation structures, we introduce the minimum logical-operator weight (MLOW) to interpret the minimal weights of corresponding logical operators. For a partially concatenated code constructed from different code blocks, the MLOW recursion shows that the resulting distance is determined by the two lower-level blocks with the smallest relevant logical-operator weights. This gives a method for computing the parameters of both fully and partially concatenated C4/C6 codes recursively. Examples such as [[10, 2, 3]], [[28, 2, 6]], and [[34, 2, 7]] show how partial concatenation fills distance gaps between neighboring fully concatenated codes.
We also evaluate these codes under a code-capacity depolarization channel, where depolarizing noise is applied only to data qubits, with ideal measurements. The results show that logical error rates for partial concatenation codes under this code-capacity model depend on the codes’ distance, which can be calculated by the recursive method. Finally, we construct fault-tolerant circuits for logical states preparation, logical gates, logical measurements, and Knill’s error detection/ correction teleportation. The decoding process can be applied fault-tolerantly by using post-selection to reject the decoding when errors are detected. We then perform circuit-level simulations for C4 and C6 decoding circuits with post-selection. Thus, the partially concatenated C4/C6 codes can be prepared fault-tolerantly and can be used for fault-tolerant quantum computation.