Research On Fonault-tolerant Quantum Computing Based On Error-Correcting Codes

At present,quantum computing is rapidly advancing and is expected to solve problems that are challenging for classical computers,such as large number factorization and complex path searching,due to its remarkable parallel processing capabilities.Based on quantum physics and quantum information science,quantum computing research is gradually expanding into multiple interdisciplinary fields.Furthermore,it has a significant impact on various industries,including cybersecurity,machine learning,artificial intelligence,and the medical industry.Quantum computing is faced with significant technical challenges due to the entanglement of quantum states and their disruption by the environment.Quantum error correction is necessary to eliminate noise and errors in the system,but it presents three main issues.Firstly,quantum error correction is complex due to the instability and difficulty in manipulating qubits compared to classical bits,as well as the possibility of phase errors,which significantly increases the complexity of error correction.Secondly,qubits require significant overhead in the encoding process,where redundant physical qubits are introduced to ensure the correctness of logical qubits.Additionally,more qubits are needed to assist in measuring coding blocks during the error correction process due to limitations such as non-cloning and superposition state collapse.Thirdly,quantum gates are incomplete,which increases the probability of failure in the whole system and further increases errors when billions of quantum gate operations are performed to achieve efficient quantum computing.Therefore,low overhead fault-tolerant quantum computing will be the trend for future development.To tackle the aforementioned issues,this thesis will delve deep into the hat problem and channel fidelity,based on the error correction properties of classical and quantum error-correcting codes.Additionally,we will explore a low-overhead,error-tolerant measurement calculation method by combining error proliferation and propagation,as well as the role of quantum gates,to reduce qubit overhead and correct errors introduced by incomplete quantum gates.The primary research results and innovations of this thesis are as follows:(1)Aming at the difficult problem of solving the winning probability in hat game,a scheme of the hat problem based on qubit-flip codes is proposed.Firstly,the BB84 protocol in quantum key distribution is used as the entry point to construct the hat game model,and the hat problem is reduced to the error correction problem of high-dimensional qubit-flip codes.Then,the hat game model is combined with the error correction properties of qubit-flip codes to propose a scheme on how to solve the winning probability in the hat problem.A theoretical proof of the feasibility of the proposed scheme is presented,and an example of the "seven prisoners puzzle" is analysed.Finally,the proposed scheme is compared with existing schemes to the hat problem,and it is found that the proposed scheme can be applied to more general hat problems,while improving the winning probability of the participants.(2)Aiming at the problem that the pre-shared entangled state is disturbed by storage errors,which reduces the channel fidelity,a channeloptimized communication scheme based on combination codes is proposed.Firstly,the quantum teleportation is used as the entry point to make the sender and receiver pre-share the Bell state before communication.Then,the storage error model and channel noise model are constructed respectively.The combination codes are constructed using qubit-flip codes and general stabilizer codes to complete the correction of storage error and channel noise error.Finally,the channel fidelity of the combination codes is simulated using Monte Carlo.Comparing the combination codes with the entanglement-assisted quantum error-correcting codes(EAQECC),it is found that the combination codes have certain advantages in terms of channel fidelity.(3)Aiming at the overhead problem of qubit and circuit depth in quantum error detection circuit,a low-overhead fault-tolerant proposed.First,the syndrome qubits are prepared as Bell states,and each generator of the stabilizer is split into X and Z type operators.The detection of quantum channel noise is accomplished by performing parallel measurement of the coding block using the method of categorical measurement of the generaters.Then,two flag qubits are added to the error detection circuit to detect quantum gate faults.Finally,a theoretical proof and an example analysis of the calculation method proposed in this thesis.It is found that the fault-tolerant measurement method proposed in this thesis has the advantage of low overhead compared to the common faulttolerant measurement methods,especially in terms of the number of auxiliary qubits and the quantum circuit depth.In addition,the syndrome qubits states used in this thesis are not only easy to prepare,but also can effectively reduce the proliferation and propagation of error.(4)Aiming at the problem that the correlated error caused by the quantum gate fault is difficult to be corrected,an error correction and faulttolerant calculation method based on high-dimensional quantum Hamming code is proposed.First,the error detection circuit of general highdimensional quantum stabilizer codes and the fault detection circuit with the addition of flag qubits are designed to achieve the detection of quantum channel noise and the determination of the type of errors introduced by quantum gate faults.Then,by adjusting the order of the quantum gate operations so that diff-erent correlated errors correspond to different syndromes,and the syndromes are used to identify correlated errors caused by different quantum gate failures.Further,the source of the error in the coding block is determined by comparing the probability of a quantum gate fault and channel noise to the occurrence of an error in the syndrome qubit.Finally,an example analysis of the quantum Hamming code[13,7,3]3 is carried out to further extend the proposed error correction and fault tolerance computation method for high-dimensional quantum Hamming codes to high-dimensional quantum CSS codes.Based on the aforementioned research,it has been discovered that this method utilizes only two auxiliary qubits,and the overhead cost of these auxiliary qubits is independent of both the weight and quantity of the generator.