In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A co...In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.展开更多
The classification of the reduced Abelian p-groups has been studied: Kaplansky proved that Ulm-Kaplansky invariants characterize the classification of countable groups; Kolettis extended this result to the direct sums...The classification of the reduced Abelian p-groups has been studied: Kaplansky proved that Ulm-Kaplansky invariants characterize the classification of countable groups; Kolettis extended this result to the direct sums of the countable groups; Parker and Walker further extended it to totally projective groups of length less than Ω_W; Hill proved that the greatest class of the p-groups which can be characterized by Ulm-Kaplansky invariants is the class of totally projective p-groups; Warfield have generalized this result to the simple presented modules in 1975.展开更多
Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter.They break the fermion-boson dichotomy and obey non-Abelian braiding statistics:their interchanges yield unitary o...Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter.They break the fermion-boson dichotomy and obey non-Abelian braiding statistics:their interchanges yield unitary operations,rather than merely a phase factor,in a space spanned by topologically degenerate wavefunctions.They are the building blocks of topological quantum computing.However,experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto,in spite of various theoretical proposals.Here,we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice.By implementing the ground states of the toric-code model with twists through quantum circuits,we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons,i.e.,the Ising anyons.In particular,we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type,and can be explored to encode topological logical qubits.Furthermore,we demonstrate how to implement both single-and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits.Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons,offering a new lens into the study of such peculiar quasiparticles.展开更多
The recent progress of the Majorana experiments paves a way for the future tests of non-Abelian braiding statistics and topologically protected quantum information processing.However,a deficient design in those tests ...The recent progress of the Majorana experiments paves a way for the future tests of non-Abelian braiding statistics and topologically protected quantum information processing.However,a deficient design in those tests could be very dangerous and reach false-positive conclusions.A careful theoretical analysis is necessary so as to develop loophole-free tests.We introduce a series of classical hidden variable models to capture certain key properties of Majorana system:non-locality,topologically non-triviality,and quantum interference.Those models could help us to classify the Majorana properties and to set up the boundaries and limitations of Majorana nonAbelian tests:fusion tests,braiding tests and test set with joint measurements.We find a hierarchy among those Majorana tests with increasing experimental complexity.展开更多
Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some nece...Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some necessary and sufficient conditions of determining ergodicity and sensitivity of the above additive CA were presented,respectively.A necessary condition for the positive expansivity of the above additive CA was given.The positive expansivity was proved to be preserved under the shift mappings for the general CA.The discussion was mainly based on the structure theorem of the finite abelian groups and the matrix associated with the global rule of the additive CA over the finite abelian p-groups.展开更多
Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,thi...Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.展开更多
文摘In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.
文摘The classification of the reduced Abelian p-groups has been studied: Kaplansky proved that Ulm-Kaplansky invariants characterize the classification of countable groups; Kolettis extended this result to the direct sums of the countable groups; Parker and Walker further extended it to totally projective groups of length less than Ω_W; Hill proved that the greatest class of the p-groups which can be characterized by Ulm-Kaplansky invariants is the class of totally projective p-groups; Warfield have generalized this result to the simple presented modules in 1975.
基金the National Natural Science Foundation of China(Grants Nos.92065204,12075128,T2225008,12174342,12274368,12274367,U20A2076,and 11725419)the Innovation Program for Quantum Science and Technology(Grant No.2021ZD0300200)+2 种基金the Zhejiang Province Key Research and Development Program(Grant No.2020C01019)supported by Tsinghua Universitythe Shanghai Qi Zhi Institute。
文摘Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter.They break the fermion-boson dichotomy and obey non-Abelian braiding statistics:their interchanges yield unitary operations,rather than merely a phase factor,in a space spanned by topologically degenerate wavefunctions.They are the building blocks of topological quantum computing.However,experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto,in spite of various theoretical proposals.Here,we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice.By implementing the ground states of the toric-code model with twists through quantum circuits,we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons,i.e.,the Ising anyons.In particular,we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type,and can be explored to encode topological logical qubits.Furthermore,we demonstrate how to implement both single-and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits.Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons,offering a new lens into the study of such peculiar quasiparticles.
基金supported by the National Natural Science Foundation of China(Grant No.11974198)the Innovation Program for Quantum Science and Technology(Grant No.2021ZD0302400)the Tsinghua University Initiative Scientific Research Program。
文摘The recent progress of the Majorana experiments paves a way for the future tests of non-Abelian braiding statistics and topologically protected quantum information processing.However,a deficient design in those tests could be very dangerous and reach false-positive conclusions.A careful theoretical analysis is necessary so as to develop loophole-free tests.We introduce a series of classical hidden variable models to capture certain key properties of Majorana system:non-locality,topologically non-triviality,and quantum interference.Those models could help us to classify the Majorana properties and to set up the boundaries and limitations of Majorana nonAbelian tests:fusion tests,braiding tests and test set with joint measurements.We find a hierarchy among those Majorana tests with increasing experimental complexity.
基金National Natural Science Foundation of China(No.11671258)。
文摘Some dynamical properties were discussed for additive cellular automata(CA)over finite abelian groups.These properties include surjection,ergodicity,sensitivity to initial conditions and positive expansivity.Some necessary and sufficient conditions of determining ergodicity and sensitivity of the above additive CA were presented,respectively.A necessary condition for the positive expansivity of the above additive CA was given.The positive expansivity was proved to be preserved under the shift mappings for the general CA.The discussion was mainly based on the structure theorem of the finite abelian groups and the matrix associated with the global rule of the additive CA over the finite abelian p-groups.
文摘Assume that G is a finite non-abelian p-group.If G has an abelian maximal subgroup whose number of Generators is at least n,then G is called an M_(n)-group.For p=2,M_(2)-groups have been classified.For odd prime p,this paper provides the isomorphism classification of M_(2)-groups,thereby achieving a complete classification of M_(2)-groups.
