Implementation Of A Universal Logical Gate Set Based On Quantum Reed-Muller Codes

By virtue of the unique properties of quantum states,the quantum computer has unparalleled computing power compared with the classical computer.However,the inherent sensitivity of quantum states to the environment makes it difficult to store and transmit quantum information.The use of quantum error-correcting codes can effectively solve this problem.Through the process of coding,error detection and error correction,the error can be corrected and the correctness of the quantum information can be ensured.But in the face of dynamic and complex operations,it is hoped that the quantum computer has a certain degree of accuracy for any operation,that is,to have a fault-tolerant universal logical gate set.In this regard,it has been proved theoretically that no quantum error-correcting code can directly support the fault-tolerant universal logical gate set.Therefore,this paper mainly focuses on how to make quantum Reed-Muller code provides fault-tolerant universal gate set,mainly including the method of conversion between adjacent Reed-Muller quantum codes and the method based on single quantum Reed-Muller code.This paper mainly includes the following aspects:Firstly,the basic theory of quantum error-correcting codes is introduced.The basic idea of quantum error-correcting codes,the quantum error correction condition and the quantum performance limit are introduced.The theoretical framework of the stabilizer code is introduced,the coding,error detection and error correction of the stabilizer code are studied,and the examples are illustrated in detail.Furthermore,fault tolerance is introduced.The related theories of fault-tolerant computing,three kinds of fault-tolerant measurement schemes are analyzed in detail.Finally,the concept and implementation scheme of a universal logical gate set are briefly described.Secondly,based on the research of the fault-tolerant conversion scheme of adjacent Reed-Muller quantum codes,a specific optimization scheme is proposed.Take into account the large resource of the previous scheme and analyze the relationship between stabilizers,the number of the measured stabilizers is reduced from the exponential level of the previous scheme m to the linear level;In addition,by splitting the stabilizer and using existing syndromes,it can further reduce the resource required by the scheme.When random single-qubit errors are introduced and can be determined by measurement,their influence on the syndromes should be analyzed and the syndromes should be modified in time.Then the appropriate fix operation is selected according to the modified syndromes,and the fix operation and the single-qubit error-correction operation are finally applied to complete the conversion,that is,“the single-step conversion procedure” can be used to do correction and complete the conversion;The overall resources required for the scheme are analyzed and compared with the resources required by other schemes.The results show that the scheme has a certain superiority in the computing situation with more T gate operations.Thirdly,the gauge-fixing method is applied to quantum Reed-Muller codes reasonably,and a fault-tolerant logical H gate based on single quantum Reed-Muller code is proposed.This also means that a universal logical gate set is realized combined with its natural fault-tolerant logical gates.In the same way,the effect of single-qubit errors on the syndromes is analyzed,and the whole scheme is more close to the practical application;and the syndrome measurement is simplified by splitting the stabilizer;the model based on the stabilizer code is established and it is easier to find the corresponding fix operation for the unsatisfied stabilizer in the gauge-fixing method;a simplified circuit diagram is drawn according to the implementation of the scheme,and a software is used to verify the schemes based on 15-qubit and 31-qubit quantum Reed-Muller codes to implement the fault-tolerant logical H gate.The results show the feasibility of the scheme.Finally,the contents of the full text are summarized,and the direction and key points for further research are clarified.