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How to cook a quantum computer

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How to cook a quantum computer How to cook a quantum computer Presentation Transcript

  • How to cook a quantum computer A. Cabello, L. Danielsen, A. López Tarrida , P. Moreno, J. R. Portillo University of Seville, Spain University of Bergen, Norway ACCOTA Playa del Carmen, Mexico. November 2010 26/11/10 11:37 AM
  • How to cook a quantum graph state A. Cabello, L. Danielsen, A. López Tarrida , P. Moreno, J. R. Portillo University of Seville, Spain University of Bergen, Norway ACCOTA Playa del Carmen, Mexico. November 2010 26/11/10 11:37 AM
  • Optimal preparation of quantum graph states A. Cabello, L. Danielsen, A. López Tarrida , P. Moreno, J. R. Portillo University of Seville, Spain University of Bergen, Norway ACCOTA Playa del Carmen, Mexico. November 2010 26/11/10 11:37 AM
  • Some ideas Bit vs. qubit
  • Some ideas Bit vs. qubit Quantum states: superposition and entaglement Stabilizer states graph states
  • Some ideas Bit vs. qubit Quantum states: superposition and entaglement Stabilizer states graph states ← Oh! Graph Theory
  • Some ideas Bit vs. qubit Entaglement measures Representative graph state Quantum computers are made with graph states, but are unstable Quantum states: superposition and entaglement Stabilizer states graph states ← Oh! Graph Theory
  • Bit 0 and 1 (on/off, true/false, yes/no).
  • Qubit 2-dimensional quantum physic system, Hilbert space isomorphic to C 2 . Schumacher, 1995 E.g., BASIC STATE VECTORS ½ spin particle. Photon polarization. Two relevant states physic system. …
  • Qubit PHYSICS MATHEMATICS
  • Quantum Mechanics: superpositions If it is posible and then IRL Photons: Atoms: laser
  • Qubit qubit INFINITE PURE STATES: Lineal superposition (coherent) of basic states: BLOCH's sphere Or:
  • Comp. SYSTEMS ENTAGLEMENT STATES: PRODUCT STATES: PHYSCIS MATHEMATICS
  • Classification of states by entaglement Entagled states CANNOT be preparated with local dispositives . much stronger correlated than all possible classic systems. Quantum Mechanics => ENTAGLEMENTS Theory / Applications Pure state of a multipartite quantum system is ENTAGLED if it is NOT a product of states .
  • Classification of states by entaglement CRITERIA (pure states, multipartites) W. Dür, G. Vidal and J. I. Cirac, Phys. Rev. A 62 , 062314 (2000). F. Verstraete et al ., Phys. Rev. A 65 , 052112 (2002). Infinite classes, (bipartites too). Equivalent entaglement: Infinite classes, (three parts or more). Equivalent entaglement:
  • Classification of states by entaglement n>3 qubits: INFINITE amount of different, INEQUIVALENT classes of ENTAGLED STATES Subsets of states: Graph states
  • Stabilizer states n- qubits stabilizer state: Simultaneous by n independent operators of Pauli group of order n Stabilizer state by an operator A if :
  • Pauli group. Stabilizer state N -QUBITS STABILIZER STATE PAULI GROUP PAULI MATRICES
  • graph state An n- qubits graph state is a special kind of stabilizer state .
  • graph state An n- qubits graph state is a pure quantum state asociated to a simple connected graph G(V,E). Each vertex represents a qubit and each edge a qubits entaglement
  • graph state? Definition Only state satisfying: Generator operator stabilizer
  • graph state An n- qubits graph state is a pure quantum state asociated to a simple connected graph G(V,E). Each vertex represents a qubit and each edge a qubits entaglement
  • graph state An n- qubits graph state is a pure quantum state asociated to a simple connected graph G(V,E). Each vertex represents a qubit and each edge a qubits entaglement Applications: Quantum computation based on measures (cluster states) Quantum correction of errors Secret sharing protocols Proof of Bell's Theorem (e.g.; all-versus-nothing) Reduction of communication complexity Teletransportation... Theory of entaglement.
  • graph states in REAL LIFE (lab)? Now, we can: 6- qubits 4 -photons g raph states 8- qubits 4 -photons g raph states 10- qubits 5 -photons g raph states n- qubits n -photons g raph states up to n = 6 .
  • graph state? Constructive definition STEP 1 Asociate each vertex with a qubit in the state:
  • What is a graph state? CONSTRUCTIVE . STEP 2 Apply, for each edge, controlled-Z to the qbits:
  • What is a graph state? CONSTRUCTIVE . 1 2 3 4
  • Graph states equivalence LU (local unitary) equivalence: Graph states “are entaglement-equivalent” iff are LU-equivalent. LC (local Clifford) equivalence: conjecture LU LC:
  • Graph states equivalence FALSE Z. Ji, J. Chen, Z. Wei y M. Ying; arXiv: 0709.1266 Conjecture LU LC: But… True for small n. Small known counterexamples: 27 qubits. Probably inferior limit . Z. Ji, J. Chen, Z. Wei y M. Ying; arXiv: 0709.1266 True for some classes of graph states. M. Van den Nest et al ., Phys. Rev. A 71 , 062323 (2005) B. Zeng et al ., Phys. Rev. A 75 , 032325 (2007)
  • LC equivalence and local complementation Theorem ( M. Van den Nest et al ., Phys. Rev. A 69 022316 (2004) ): There exists a sequence of local complementation operator that maps graph G into graph G’ . LC LC LC LC
  • LC equivalence and local complementation j Theorem ( M. Van den Nest et al ., Phys. Rev. A 69 022316 (2004) ): There exists a sequence of local complementation operator that maps graph G into graph G’ .
  • LC equivalence and local complementation. Orbit. LC equivalence class. ORBIT: REPRESENTANTIVE? ÓRBIT (LC class)
  • LC equivalence and local complementation. Orbit. LC equivalence class = orbit: #Orbit: 802 non isomorph graphs ORBIT
  • Entaglements in Graph states. Classification n<8: M. Hein, J. Eisert y H. J. Briegel Phys. Rev. A 69 , 062311 (2004). n=8: A. Cabello, A. J. López-Tarrida, P. Moreno y J. R. Portillo Phys. Lett. A 373 , 2219 (2009). 45 classes for graphs up to 7 vertices 101 classes for 8 vertices graphs . # orbits for n vertices: 1 2 3 4 5 6 7 8 9 10 11 12 1 1 1 2 4 11 26 101 440 3,132 40,457 1,274,068 Classification and propierties: n<=12: A. Cabello, L.E. Danielsen, A. J. López-Tarrida, P. Moreno y J. R. Portillo Submitted (2010).
  • Entaglements in Graph states. Classification n<8:
  • Entaglements in Graph states. Classification. n<8 : CLASS ORDER CRITERIA LC INVARIANT Minimum number of controlled- Z gates for its preparation. Schmidt measure for the 8-partite split. Rank index (Schmidt ranks for all bipartite splits).
  • Sort criteria n<9 (our previous work) Minimum number of controlled- Z gates for preparation. For each class, a representative with minimum # edges AND minimum chromatic index Both minimums ever are in the same representative (n<9) EXPERIMENTAL corresponds to:
    • Minimum # controlled-Z gates .
    • Minimum preparation deepth (time units).
  • Sort criteria n<9 (our previous work) Schmidt measure of n -partite split. It shows the entaglement degree of a multipartite quantum system. r is the minimum # R of term in the SUM , in all the lineal decompositions in product states. SCHMIDT MEASURE
  • Sort criteria SCHMIDT RANKS Rank index (Schmidt rank of all bipartite splits). RANK INDEX
  • Schmidt measure bounds MAXIMUM SCHMIDT RANK PAULI PERSISTENCE MINIMAL VERTEX COVER
  • ATTENTION: PROBLEM!!!! Graph states entaglement. Classifition n<7: NO DISTINCTION NO EQUIVALENT CLASS!
  • Graph states entaglements. Clasification n=8:
  • ATTENTION: PROBLEM!!!! Solved (n<9) in Phys. Rev. 80 012102 (2009). 4 invariants are enough! Entrelazamiento en Graph states. Clasificación NO DISTINCTION NO EQUIVALENT CLASS!
  • n Download Size 8 entanglement8 101 graphs 9 entanglement9 440 graphs 10 entanglement10 3132 graphs (509 KB) 11 entanglement11.bz2 40,457 graphs (1.2 MB compressed) 12 entanglement12.bz2 1,274,068 graphs (45 MB compressed) Sort criteria n<13 Minimum number of controlled- Z gates for preparation. For each class, a representative with minimum # edges OR minimum chromatic index Both minimums NOT ever are in the same representative (n>9) EXPERIMENTAL corresponds to:
    • Minimum # controlled-Z gates .
    • Minimum preparation deepth (time units).
  • * No.: Number of the equivalence class. * |LC|: Number of nonisomorphic graphs in the class. * |V|: Number of vertices. * (|E|, χ', #): |E| is the minimum number of edges in the class. χ' is the minimum chromatic index of the graphs with |E| edges. # is the number of nonisomorphic graphs with |E| edges and chromatic index χ'. * (χ', |E|, #): χ' is the minimum chromatic index in the class. |E| is the minimum number of edges of the graphs with chromatic index χ'. # is the number of nonisomorphic graphs with chromatic index χ' and |E| edges. * ES: Schmidt measure. * RIi: (for n/2 ≥ i ≥ 2): Rank index for bipartite splits with i,n-i vertices. * C-M: (for 0 ≤ i ≤ x) Cardinality-multiplicities. Value i is the multiplicity of the cardinality i. Only the multiplicities of cardinalities 0 to x are listed, * 2-col: Does the class contain a two-colorable graph? * A representive graph from the class with minimum number of edges. * A representive graph from the class with minimum chromatic index. Graph states entaglements. Clasification n<13:
  • Cooking graph states A few invariants for 9<=n<=12 n #orbits #problems prob. p 9 440 2 0,0012218 10 3132 8 0,0006996 11 40457 78 0,0011929 12 1274068 472 0,0000949 4 invariants for n<=8 LC class #54 We calculate invariants and identify LC class:
  • Cooking graph states If we need prepare a GRAPH STATE LC-class 54 LC LC LC LC We calculate invariants and identify LC class: We preparate the BEST representative and we do LC transformation*
  • Cooking graph states If we need prepare a GRAPH STATE
  • CONCLUSIONS Extended up to 12 qubits g raph states entaglement classification. Best (in the sense of minimum time preparation and/or minimum work) representative of each new 1300000+ LC equivalence class. An (almost) complete sort criteria and new invariants for labeling class. Help to new proofs (AVN type) of Bell's theorem . Research of non-locality.
  • Procedure for the optimal preparation of 1.65 × 10 11 graph states with up to 12 qubits OPTIMAL: minimum number of entangling gates minimum number of time steps
    • Main goal: to provide in a single package all the tools needed to rapidly identify the entanglement class the target state belongs to, and then easily find the corresponding optimal circuit(s) of entangling gates, and finally the explicit additional one-qubit gates needed to prepare the target
    CONCLUSIONS
  • Arxiv: http://arxiv.org/abs/1011.5464 Nov 24, 2010 has been submitted to Physical Review A Nov 25, 2010 PUBLISHED
  • Thanks for your attention! ¡Gracias por escucharme!
  • Entrelazamiento en Graph states. Clasificación CONCLUSIÓN: NO podemos utilizar los invariantes propuestos por Hein et al . para decidir inequívocamente a qué clase de entrelazamiento pertenece un graph state dado. NUEVO PROBLEMA: debemos buscar un conjunto de invariantes que permita etiquetar de manera unívoca las clases de equivalencia de entrelazamiento, discriminando sin ambigüedad entre ellas.
  • Invariantes de Van den Nest-Dehaene-De Moor Teorema ( M. Van den Nest et al ., Phys. Rev. A 72 014307 (2005) ):
  • Invariantes de Van den Nest-Dehaene-De Moor Sobre los invariantes NDM (I): Conjuntos cuyos elementos son tuplas de operadores del estabilizador del estado, tales que sus soportes cumplen una serie de condiciones . Cardinales de los conjuntos. Son invariantes LC . Hay jerarquías o familias de invariantes NDM, caracterizadas por el valor del parámetro r . La familia con r = n es la que caracteriza y determina cada clase de equivalencia de cualquier estado de estabilizador .
  • Invariantes de Van den Nest-Dehaene-De Moor Sobre los invariantes NDM (II): El número de invariantes NDM crece muy rápidamente con r, n . ¡¡NO CALCULADOS EXPLÍCITAMENTE EN NINGÚN SITIO!!
  • Invariantes de Van den Nest-Dehaene-De Moor Sobre los invariantes NDM (III): NDM creen que su conjunto de invariantes puede ser mejorado u optimizado, si no para todos los estados de estabilizador, sí al menos para algunas clases interesantes de ellos. NDM afirman que probablemente con menos invariantes se pueda reconocer la equivalencia LC , y que quizá existan listas de invariantes que exhiban menos redundancias.
  • Formulación del problema Nuestro problema: ¿Qué ocurre si nos limitamos a Graph states de hasta n = 8 qubits , y utilizamos invariantes NDM partiendo de las familias con parámetro r más pequeño ? ¿ A partir de qué valor de r lograríamos la caracterización unívoca de las 146 clases de equivalencia LC correspondientes?
  • Invariantes NDM, familia r = 1 Si r = 1: un soporte dado clase de equivalencia asociada al soporte (engloba a los operadores de estabilización que tienen el mismo soporte) cardinal de la clase de equivalencia asociada al soporte
  • Clases de equivalencia de soportes y cardinalidad 3 2 1
  • Primera respuesta al problema Nuestra primera respuesta problema: ¿Qué ocurre si nos limitamos a Graph states de hasta n = 8 qubits , y utilizamos invariantes NDM partiendo de las familias con parámetro r más pequeño ? ¿ A partir de qué valor de r lograríamos la caracterización unívoca de las 146 clases de equivalencia LC correspondientes? A. Cabello, A. J. López-Tarrida, P. Moreno y J. R. Portillo Phys. Rev. A 80 , 012102 (2009). SUFICIENTE CON r = 1
  • Conjunto compacto de invariantes LC Número de invariantes NDM con r = 1: Coincide con el número de posibles soportes para un graph state de n qubits. A partir de n = 3 el número de invariantes NDN ( r = 1) de una clase LC es demasiado grande para ser práctico . Una tabla con los invariantes citados, hasta n = 8 , tendría 30060 valores. COMPRIMIR LA INFORMACIÓN EN MENOS INVARIANTES LC
  • Conjunto compacto de invariantes LC Criterios para construir nuevos invariantes LC: (por orden de prioridad) CANDIDATA: LA DISTRIBUCIÓN DE PESOS . (comentario de NDM) 1 2 3 Que sean invariantes LC (obviamente). Que discriminen inequívocamente entre clases de equivalencia LC no equivalentes. Fácilmente legibles.
  • Distribución de pesos como invariante LC 2 DEFINICIÓN Número de operadores del estabilizador con peso d DISTRIBUCIÓN DE PESOS
  • Conjunto compacto de invariantes LC Criterios para construir nuevos invariantes LC: (por orden de prioridad) NUEVO ENFOQUE: LA EQUIPOTENCIA. 1 2 3 Que sean invariantes LC (obviamente). Que discriminen inequívocamente entre clases de equivalencia LC no equivalentes. Fácilmente legibles.
  • Clases de equivalencia equipotentes Definición de equipotencia: El número de clases [  ] equipotentes para un cardinal A  es un invariante LC. MULTIPLICIDAD O POTENCIA DE A 
  • Conjunto compacto de invariantes LC. Solución Nuevos invariantes: Notación compacta basada en dos índices, un valor cardinal A  y su multiplicidad     1 2 3
  • Conjunto compacto de invariantes LC. Ejemplo 3 2 1
  • Conjunto compacto de invariantes LC. Resultados
  • Conjunto compacto de invariantes LC. Resultados Hemos comprobado que basta con cuatro invariantes cardinal-multiplicidad para caracterizar y distinguir cualquier clase de equivalencia LC hasta n = 8. A. Cabello, A. J. López-Tarrida, P. Moreno y J. R. Portillo Phys. Rev. A 80 , 012102 (2009). Por tanto, podemos decidir a qué clase de equivalencia LC pertenece cualquier graph state de hasta 8 qubits calculando de manera intrínseca (es decir, sin generar la clase LC completa) esos cuatro invariantes cardinal-multiplicidad.
  • Conclusiones: importancia práctica Necesitamos preparar un graph state concreto: Clase LC-54 LC LC LC LC Calculamos los invariantes del grafo e identificamos la clase LC: Preparamos el representante óptimo (y aplicamos sucesivas LCs) :
  • Conclusiones: caracterización de Graph states En lo referente a los fundamentos: Para decidir a qué clase de entrelazamiento pertenece un graph state de hasta 8 qubits basta con calcular cuatro cantidades invariantes LC (las multiplicidades de ciertos cardinales asociados a los soportes del grafo correspondiente). Con este resultado resolvemos el problema de discriminación entre clases de equivalencia LC surgido en la clasificación de los Graph states. Se puede aplicar la misma estrategia para generar un conjunto compacto de invariantes LC que discriminen entre clases, para valores de n más altos . Numéricamente factible hasta n = 12 , más allá de la capacidad actual de preparación. El resultado responde a la conjetura de V. den Nest – Dehaene - De Moor sobre la posibilidad de caracterizar subclases especiales de estados de estabilizador con subfamilias de invariantes NDM.