Research On Quantum Error Correcting Codes Based On Chamon Models

Quantum information mainly includes research orientations such as quantum communication and quantum computing.The research content of quantum communication includes quantum teleportation,quantum cryptography and quantum information technology;Quantum computing mainly studies the theoretical scheme and physical realization of quantum computers.Among them,the theoretical scheme mainly studies quantum circuits,quantum algorithms and quantum error correction codes,and the physical implementation mainly studies the construction of quantum computers.It has made quantum computing widely concerned that the contradiction between the increasing data scale and the bottleneck period in the development of classical computing power.However,the actual physical system inevitably interacts with the environment when performing quantum computing.These interactions,interfering with quantum information processing,are called quantum noise.In order to solve the problem of quantum noise,quantum error correction theory came into being.Quantum error correction theory mainly refers to quantum error correction codes.Quantum error correction theory mainly refers to quantum error correction codes,among which stabilizer codes are a important type of coding method in quantum error correction codes,and their research contents are mostly focus on the CSS codes and surface codes.The research idea of the former comes from classical error correction codes,and the latter comes from the protection of quantum information by topological properties in planar codes.A natural question is whether exists a kind of stabilzer code that are neither planar codes or belong non-CSS codes still can be fault-tolerant for quantum computation.The starting point of this paper is to study the unique topological properties in non-CSS cubic codes,and use topological properties to protect the processed quantum information and realize quantum error correction computing.The main research content is to construct non-CSS three-dimensional codes based on the Chamon model,which includes:(1)Study hthe feasibility of Chamon model as quantum error correcting code,and consider the structions of logical qubits at any scale.The state preparation problems of logical states is completed,focusing on topological properties during the process of research.(2)Consider the operation of the quantum universal gate of the stereo code of any scale to realize universal quantum computing.(3)Propose a unique decoding method and form an error correction algorithm.First,use the properties of cubic codes to eliminate errors,and then correct errors through logical operations.(4)The error correction algorithm can improve the decoding success rate by means of greedy algorithm preprocessing.According to the specific noise model,the error correction algorithm is used to numerically simulate the logical error rate of the cubic codes and calculate the threshold.The innovations of this paper are as follows:(1)The construction method of logical qubits in Chamon model of arbitrary scale is proposed and the state preparation problems of logical states is completed.The definitions of the half-plane operator and the square half-plane operator are given,and it is proved that they constitute the logical Pauli operators of the logical qubit space respectively.(2)The upper bound of code distance of Chamon model at any scale is analyzed.For given qubit number,when the greatest common divisor of the triaxial degree is 1,the code distance is the largest,and the explicit expression of the code distance in this case is obtained.The explicit expression of code distance under cubic Chamon model is also derived.This paper analyzes the relationship between the number of physical qubits,the number of logical qubits and the code distance.There is a balance between these parameters.Generally speaking,when the number of physical qubits is given,the larger the number of logical qubits,the smaller the code distance.The asymptotic properties are as follows:when the number of physical qubits N=O(n3)is given,the maximum number of logical qubits can be O(n),and the code distance is O(n);If the number of sacrificed logical qubits is O(1),the code distance can reach O(n2).(3)The error correction algorithm of Chamon model is party proposed.The error correction algorithm is an error correction algorithm based on the unique topological properties of the Chamon model.In the implementation process,the idea of recovering errors first and then searching randomly is used.The step of recovering the error is the core content of the algorithm.Due to the complex excitation form,the Chamon model cannot recover the errors according to the local measurement results,so the global stabilizer measurement results are considered,and the complex excitation form of the Chamon model is used to eliminate the errors.(4)An improved scheme of the error correction algorithm is proposed.The error correction ability of the error correction algorithm can be improved by the greedy algorithm,and the decoding success rate and threshold are improved in numerical experiments.The source of the improvement space is determined by the excitation form of the Chamon model.When one qubit is wrong,there will be four measurement errors in the adjacent stabilizer.Therefore,the preprocessing of the greedy algorithm can improve the error correction ability of the algorithm when the physical qubit error distribution is sparse.This idea is expected to be extended to other complex high-dimensional error-correcting codes.