49 research outputs found
Towards a universal set of topologically protected gates for quantum computation with Pfaffian qubits
We review the topological quantum computation scheme of Das Sarma et al. from
the perspective of the conformal field theory for the two-dimensional critical
Ising model. This scheme originally used the monodromy properties of the
non-Abelian excitations in the Pfaffian quantum Hall state to construct
elementary qubits and execute logical NOT on them. We extend the scheme of Das
Sarma et al. by exploiting the explicit braiding transformations for the
Pfaffian wave functions containing 4 and 6 quasiholes to implement, for the
first time in this context, the single-qubit Hadamard and phase gates and the
two-qubit Controlled-NOT gate over Pfaffian qubits in a topologically protected
way. In more detail, we explicitly construct the unitary representations of the
braid groups B_4, B_6 and B_8 and use the elementary braid matrices to
implement one-, two- and three-qubit gates. We also propose to construct a
topologically protected Toffoli gate, in terms of a braid-group based
Controlled-Controlled-Z gate precursor. Finally we discuss some difficulties
arising in the embedding of the Clifford gates and address several important
questions about topological quantum computation in general.Comment: 57 pages, 26 EPS figures, Latex2e with elsart class package; v2: one
remark added and some misprints correcte
Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
Transfer matrices and matrix product operators play an ubiquitous role in the
field of many body physics. This paper gives an ideosyncratic overview of
applications, exact results and computational aspects of diagonalizing transfer
matrices and matrix product operators. The results in this paper are a mixture
of classic results, presented from the point of view of tensor networks, and of
new results. Topics discussed are exact solutions of transfer matrices in
equilibrium and non-equilibrium statistical physics, tensor network states,
matrix product operator algebras, and numerical matrix product state methods
for finding extremal eigenvectors of matrix product operators.Comment: Lecture notes from a course at Vienna Universit
Universality of Z3 parafermions via edge-mode interaction and quantum simulation of topological space evolution with Rydberg atoms
Parafermions are Zn generalizations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. Here we investigate the generation of quantum gates by allowing Z3 parafermions to interact in order to achieve universality. In particular, we study the form of the nontopological gate that arises through direct short-range interaction of the parafermion edge modes in a Z3 parafermion chain. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We illustrate how to access highly noncontextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3 topological states with Rydberg atom technology
Entanglement and Bell correlations in strongly correlated many-body quantum systems
During the past two decades, thanks to the mutual fertilization of the research in quantum information and condensed matter, new approaches based on purely quantum features without any classical analog turned out to be very useful in the characterization of many-body quantum systems (MBQS). A peculiar role is obviously played by the study of purely quantum correlations, manifesting in the “spooky” properties of entanglement and nonlocality (or Bell correlations), which ultimately discriminate classical from quantum regimes. It is, in fact, such kind of correlations that give rise to the plethora of intriguing emergent behaviors of MBQS, which cannot be reduced to a mere sum of the behaviors of individual components, the most important example being the quantum phase transitions. However, despite being indeed closely related concepts, entanglement and nonlocality are actually two different resources.
With regard to the entanglement, we will use it to characterize several instances of MBQS, to exactly locate and characterize quantum phase transitions in spin-lattices and interacting fermionic systems, to classify different gapped quantum phases according to their topological features and to provide a purely quantum signature of chaos in dynamical systems.
Our approach will be mainly numerical and for simulating the ground states of several one-dimensional lattice systems we draw heavily on the celebrated “density matrix renormalization group” (DMRG) algorithm in the “matrix product state” (MPS) ansatz. A MPS is a one-dimensional tensor network (TN) representation for quantum states and occupies a pivotal position in what we have gained in thinking MBQS from an entanglement perspective. In fact, the success of TNs states mainly relies on their fulfillment, by construction, of the so called “entanglement area law”. This is a feature shared by the ground states of gapped Hamiltonians with short-range interactions among the components and consists of a sub-extensive entanglement entropy, which grows only with the surface of the bipartition. This property translates in a reduced complexity of such systems, allowing affordable simulations, with an exponential reduction of computational costs. Besides the use of already existing TN-based algorithms, an effort will be done to develop a new one suitable for high-dimensional lattices.
While many useful results are available for the entanglement in many different contexts, less is known about the role of nonlocality. Formally, a state of a multi-party system is defined nonlocal if its correlations violate some “Bell inequality” (BI). The derivation of the BIs for systems consisting of many parties is a formidable task and only recently, a class of them, relevant for nontrivial states, has been proposed. In an important chapter of the thesis, we apply these BIs to fully characterize the phase transition of a long-range ferromagnetic Ising model, doing a comparison with entanglement-based results and then making one of the first efforts in the study of MBQS from a nonlocality perspective.Durante las dos Ăşltimas dĂ©cadas, gracias al enriquecimiento mutuo entre las investigaciones en informaciĂłn cuántica y materia condensada, se han desarrollado nuevos enfoques que han resultado muy Ăştiles en la caracterizaciĂłn de los sistemas cuánticos de muchos cuerpos (SCMC), basados en caracterĂsticas puramente cuánticas sin ningĂşn análogo clásico. El estudio de las correlaciones puramente cuánticas juega obviamente un papel fundamental. Estas correlaciones se manifiestan en las propiedades del entrelazamiento cuántico (“entanglement”) y no-localidad (o correlaciones de Bell), que en Ăşltima instancia discriminan los regĂmenes clásicos de los regĂmenes cuánticos. Este tipo de correlaciones son, de hecho, las que dan lugar a la plĂ©tora de comportamientos emergentes enigmáticos de los SCMC, que no pueden reducirse a una mera suma de los comportamientos de los componentes individuales, siendo el ejemplo más importante siendo las transiciones de fase cuánticas (TFC). Sin embargo, a pesar de ser conceptos estrechamente relacionados, el entrelazamiento y la no-localidad son en realidad dos recursos diferentes. Con respecto al entrelazamiento, lo utilizaremos para caracterizar varios ejemplos de SCMC, para localizar y caracterizar exactamente las TFC en retĂculos de espines y de sistemas de fermiones interactuantes, para clasificar las diferentes fases cuánticas de acuerdo con su topologĂa y para proporcionar una señal puramente cuántica del caos en los sistemas dinámicos. Nuestro enfoque será principalmente numĂ©rico y para simular los estados fundamentales de varios sistemas unidimensionales nos basamos en gran medida en el cĂ©lebre algoritmo “density matrix renormalization group” (DMRG), formulado en el ansatz de los “matrix product states” (MPS). Un MPS es un “retĂculos de tensores” (“tensor networks”, TN) unidimensional que representa estados cuánticos y ocupa una posiciĂłn central entre los mayores logros obtenidos al estudiar los SCMC desde la perspectiva del entrelazamiento cuántico. De hecho, el Ă©xito de los TN depende principalmente de su cumplimiento, por construcciĂłn, de una “ley del área” (“area-law”) de la entropĂa de entrelazamiento. Esta es una caracterĂstica compartida por los estados fundamentales de los Hamiltonianos con interacciones de corto alcance entre los componentes del sistema y con una brecha (“gap”) entre el estado fundamental y los niveles excitados, que consiste en una entropĂa de entrelazamiento subextensiva, que crece sĂłlo con la superficie de la biparticiĂłn. Esta propiedad se traduce en una menor complejidad de dichos sistemas, permitiendo simulaciones asequibles, con una reducciĂłn exponencial de los costes computacionales. Además del uso de los algoritmos ya existentes basados en TN, se desarrollará uno nuevo adecuado para sistemas en dimensiones altas. Si bien se dispone de muchos resultados Ăştiles para el entrelazamiento en muchos contextos diferentes, se sabe menos sobre el papel jugado por la no-localidad. Formalmente, un estado de un sistema compuesto de muchas partes, se define como no-local si sus correlaciones violan alguna “desigualdad de Bell” (“Bell inequality”, BI). La derivaciĂłn de dichas desigualdades para sistemas compuestos de muchas partes es un reto y sĂłlo recientemente se ha propuesto una clase de ellas, relevante para estados no triviales. En un capĂtulo importante de la tesis, aplicamos estas BIs para caracterizar completamente la transiciĂłn de fase de un modelo de Ising ferromagnĂ©tico con interacciones de largo alcance, haciendo una comparaciĂłn con los resultados basados en el entrelazamiento y luego haciendo uno de los primeros esfuerzos en el estudio de los SCMC desde una perspectiva de la no-localidad.Postprint (published version
Entanglement and Bell correlations in strongly correlated many-body quantum systems
During the past two decades, thanks to the mutual fertilization of the research in quantum information and condensed matter, new approaches based on purely quantum features without any classical analog turned out to be very useful in the characterization of many-body quantum systems (MBQS). A peculiar role is obviously played by the study of purely quantum correlations, manifesting in the “spooky” properties of entanglement and nonlocality (or Bell correlations), which ultimately discriminate classical from quantum regimes. It is, in fact, such kind of correlations that give rise to the plethora of intriguing emergent behaviors of MBQS, which cannot be reduced to a mere sum of the behaviors of individual components, the most important example being the quantum phase transitions. However, despite being indeed closely related concepts, entanglement and nonlocality are actually two different resources.
With regard to the entanglement, we will use it to characterize several instances of MBQS, to exactly locate and characterize quantum phase transitions in spin-lattices and interacting fermionic systems, to classify different gapped quantum phases according to their topological features and to provide a purely quantum signature of chaos in dynamical systems.
Our approach will be mainly numerical and for simulating the ground states of several one-dimensional lattice systems we draw heavily on the celebrated “density matrix renormalization group” (DMRG) algorithm in the “matrix product state” (MPS) ansatz. A MPS is a one-dimensional tensor network (TN) representation for quantum states and occupies a pivotal position in what we have gained in thinking MBQS from an entanglement perspective. In fact, the success of TNs states mainly relies on their fulfillment, by construction, of the so called “entanglement area law”. This is a feature shared by the ground states of gapped Hamiltonians with short-range interactions among the components and consists of a sub-extensive entanglement entropy, which grows only with the surface of the bipartition. This property translates in a reduced complexity of such systems, allowing affordable simulations, with an exponential reduction of computational costs. Besides the use of already existing TN-based algorithms, an effort will be done to develop a new one suitable for high-dimensional lattices.
While many useful results are available for the entanglement in many different contexts, less is known about the role of nonlocality. Formally, a state of a multi-party system is defined nonlocal if its correlations violate some “Bell inequality” (BI). The derivation of the BIs for systems consisting of many parties is a formidable task and only recently, a class of them, relevant for nontrivial states, has been proposed. In an important chapter of the thesis, we apply these BIs to fully characterize the phase transition of a long-range ferromagnetic Ising model, doing a comparison with entanglement-based results and then making one of the first efforts in the study of MBQS from a nonlocality perspective.Durante las dos Ăşltimas dĂ©cadas, gracias al enriquecimiento mutuo entre las investigaciones en informaciĂłn cuántica y materia condensada, se han desarrollado nuevos enfoques que han resultado muy Ăştiles en la caracterizaciĂłn de los sistemas cuánticos de muchos cuerpos (SCMC), basados en caracterĂsticas puramente cuánticas sin ningĂşn análogo clásico. El estudio de las correlaciones puramente cuánticas juega obviamente un papel fundamental. Estas correlaciones se manifiestan en las propiedades del entrelazamiento cuántico (“entanglement”) y no-localidad (o correlaciones de Bell), que en Ăşltima instancia discriminan los regĂmenes clásicos de los regĂmenes cuánticos. Este tipo de correlaciones son, de hecho, las que dan lugar a la plĂ©tora de comportamientos emergentes enigmáticos de los SCMC, que no pueden reducirse a una mera suma de los comportamientos de los componentes individuales, siendo el ejemplo más importante siendo las transiciones de fase cuánticas (TFC). Sin embargo, a pesar de ser conceptos estrechamente relacionados, el entrelazamiento y la no-localidad son en realidad dos recursos diferentes. Con respecto al entrelazamiento, lo utilizaremos para caracterizar varios ejemplos de SCMC, para localizar y caracterizar exactamente las TFC en retĂculos de espines y de sistemas de fermiones interactuantes, para clasificar las diferentes fases cuánticas de acuerdo con su topologĂa y para proporcionar una señal puramente cuántica del caos en los sistemas dinámicos. Nuestro enfoque será principalmente numĂ©rico y para simular los estados fundamentales de varios sistemas unidimensionales nos basamos en gran medida en el cĂ©lebre algoritmo “density matrix renormalization group” (DMRG), formulado en el ansatz de los “matrix product states” (MPS). Un MPS es un “retĂculos de tensores” (“tensor networks”, TN) unidimensional que representa estados cuánticos y ocupa una posiciĂłn central entre los mayores logros obtenidos al estudiar los SCMC desde la perspectiva del entrelazamiento cuántico. De hecho, el Ă©xito de los TN depende principalmente de su cumplimiento, por construcciĂłn, de una “ley del área” (“area-law”) de la entropĂa de entrelazamiento. Esta es una caracterĂstica compartida por los estados fundamentales de los Hamiltonianos con interacciones de corto alcance entre los componentes del sistema y con una brecha (“gap”) entre el estado fundamental y los niveles excitados, que consiste en una entropĂa de entrelazamiento subextensiva, que crece sĂłlo con la superficie de la biparticiĂłn. Esta propiedad se traduce en una menor complejidad de dichos sistemas, permitiendo simulaciones asequibles, con una reducciĂłn exponencial de los costes computacionales. Además del uso de los algoritmos ya existentes basados en TN, se desarrollará uno nuevo adecuado para sistemas en dimensiones altas. Si bien se dispone de muchos resultados Ăştiles para el entrelazamiento en muchos contextos diferentes, se sabe menos sobre el papel jugado por la no-localidad. Formalmente, un estado de un sistema compuesto de muchas partes, se define como no-local si sus correlaciones violan alguna “desigualdad de Bell” (“Bell inequality”, BI). La derivaciĂłn de dichas desigualdades para sistemas compuestos de muchas partes es un reto y sĂłlo recientemente se ha propuesto una clase de ellas, relevante para estados no triviales. En un capĂtulo importante de la tesis, aplicamos estas BIs para caracterizar completamente la transiciĂłn de fase de un modelo de Ising ferromagnĂ©tico con interacciones de largo alcance, haciendo una comparaciĂłn con los resultados basados en el entrelazamiento y luego haciendo uno de los primeros esfuerzos en el estudio de los SCMC desde una perspectiva de la no-localidad
Universality of Z3 parafermions via edge-mode interaction and quantum simulation of topological space evolution with Rydberg atoms
Parafermions are Zn generalizations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. Here we investigate the generation of quantum gates by allowing Z3 parafermions to interact in order to achieve universality. In particular, we study the form of the nontopological gate that arises through direct short-range interaction of the parafermion edge modes in a Z3 parafermion chain. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We illustrate how to access highly noncontextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3 topological states with Rydberg atom technology