Study On Fault-tolerant Extended-rectangle Constructions For Quantum CNOT Based On Quantum Error-correction Codes

Quantum error correction plays an important role in the protection of data transmission andstorage. In addition, it can be used to protecting the information in dynamic quantumcomputation. When the basic error rate of quantum circuit is lower than a threshold, thefault-tolerant quantum computation can be implemented with any accuracy using quantum errorcorrecting and quantum fault-tolerant components. CNOT is an important quantum componentfor quantum computation. This thesis will study the fault-tolerant extended-rectangleconstructions for quantum CNOT based on quantum error-correction codes.Firstly, the thesis studies the fault-tolerant extended-rectangle for quantum CNOT withquantum low density parity-check (quantum LDPC) codes. Quantum LDPC codes, as well astheir corresponding classical one, have some excellent properties. First, the construction ofquantum LDPC codes based on CSS and entanglement-assisted are analysed. The simulationresults show that these codes have a good error-correction. Then, the application of quantumLDPC code in quantum fault-tolerant computation is discussed by the combination of quantumLDPC code with Shor-EC. A fault-tolerant extended-rectangle construction for quantum CNOTis proposed, and (16,4) quantum LDPC codes are used as example to construct the fault-tolerantextended-rectangle for quantum CNOT. The analysis shows that the threshold of quantum CNOTconstructed by (16,4) quantum LDPC is higher than that by (23,1,7) Golay code, similar to thatby (49,1,9) quantum code, and the overhead of physical qubits is less than that by (49,1,9)quantum code.Furthermore, the thesis studies the fault-tolerant quantum CNOT constructed by (49,1,9)quantum code with the overlap method in ancilla state preparation. The ancilla state preparationcircuit is normally complicated in the construction of fault-tolerant extended-rectangle forquntum CNOT. The overlap method is the optimization of the traditional Latin rectangle methodby the two physical quantum bits controlled with the same control bit, and it can simplify theencoded ancilla preparation state circuit. The construction of the fault-tolerantextended-rectangle for quantum CNOT is presented with the overlap method to prepare theencoded ancilla state of fault-tolerant Steane-EC using (49,1,9) quantum code. The result showsthat the construction is optimized and the number of CNOT needed is much less than that withthe traditional Latin rectangle method.