In recent work pertaining to digital quantum computations—the quantum parallel to classical computations—algebraic concepts are being introduced as a resource. This goes from an extensive use of group theory (finite groups such as Paulis and Cliffords, free groups with relations, group covariance in generalized quantum measurements, etc.), of geometry (e.g., finite geometries for modeling quantum commutation, entanglement, and contextuality) and of topology for adapting quantum error correction to nonlocality. Further, topological order and braids are being investigated for quantum computing in 2D (in anyons) and in 3D (with 3-manifolds).
Would you contribute to this topic?
Give some hints or a full paper at MDPI Quantum Reports, there is a special issue lounched about this topic
https://www.mdpi.com/journal/quantumrep/special_issues/quantumrep_GroupsGeometryandTopologyforQuantumComputations
Thanks in advance.
Michel Planat
Mathematics must come first.In real applications ,the science of mathematics comes first.Category theory ,for instance,provides us with unexpected results.Why should first creating a photon and then destroying one give a different result than first destroying one and then creating one? Category theory answers that the phenomenaon is not based on physical postulates , rather it is related to a fact about finite sets. The Feynman diagrams arise naturally from the theory of finite sets equipped with extra stuff.In the year 1736 ,Euler invented the theory of graphs , which is a branch of topology.Nowadays,software design leans on graph theory,e.g. designing a code using binary trees :implementing an expression in a digital computer.We can view a quantum circuit as a graph .It will be complicated if the quantum computer is built with much a larger number of qubits.Using the rules of graph theory we can reduce the complexity of the quantum circuit. Even we can distribute the ingredients of such a circuit among the nodes of a tree and innovate suitable rules to traverse the tree in order to get the same result.One of the quantum mysteries is that there exist processes by which one can learn the results of a computation without actually performing the computation , provided the possibility of performing that computation is available , even though the computation itself is not performed. To know about that possibility , we have to resort to the many worlds interpretation of quantum physics.As the quantum computer makes copies of itself in many universes ,we have to apply the inverse of topological rules of glueing universes ,and take hold of one of these universes to discover the feasibility of the computation.
Also, quantum computers need error correcting process. But this error correcting process is more complicated than classical one. Hence, for that reason, some quantum codes are constructed by using algebraic tools. For instance, stabilizer codes are constructed by the help of stabilizer groups. Here, errors are detected by checking change of eigenvalues. Moreover, normalizer and centralizer of the stabilizer group play an important role in detection and correction of an error. The following is a very useful paper about this topic.
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