Nesta se¸c˜ao mostraremos que podemos construir um conjunto de portas quˆanticas para um e dois q-bits que permitem que se fa¸ca qualquer rota¸c˜ao de um q-bit na esfera de Bloch e se possa manipular dois q-bits que interagem entre si. O conceito de univer- salidade na computa¸c˜ao quˆantica est´a relacionado com o uma opera¸c˜ao unit´aria qualquer (uma rota¸c˜ao) que pode ser constru´ıda com precis˜ao arbitr´aria por um circuito quˆantico evolvendo somente aquelas portas [120]. O conjunto de portas quˆanticas universais mais conhecido na Literatura ´e formado pelas portas Hadamard, porta de fase, porta T e CNOT.
Vamos ent˜ao mostrar que as portas Hadamard e T constituem um conjunto de portas quˆanticas universal para um q-bit. Para isso, considere-se a porta T e a porta constru´ıda pela combina¸c˜ao HT H. A combina¸c˜ao HT H resulta em
HT H = 1 2 1 + eiπ/4 1− eiπ/4 1− eiπ/4 1 + eiπ/4 = e iπ/8 2 cosπ 8 −i sin π 8
−i sin π8 cos π 8
que nada mais ´e que uma rota¸c˜ao de π/4 radianos em torno do eixo x da esfera de Bloch, ou seja, HT H = Rx(π/4). Sendo a porta quˆantica T uma rota¸c˜ao de π/4 radianos em
torno do eixo z da esfera de Bloch, pode-se gerar uma rota¸c˜ao na dire¸c˜ao de um vetor arbitr´ario ˆν com a combina¸c˜ao
Rν(θ) = Rz(π/4)Rx(π/4) = e−i π 8σ z e−iπ8σ x = hcosπ 8I − i Z sin π 8 i h cosπ 8I − i X sin π 8 i (B.17) = cos2 π 8I − i h cosπ 8X + sin π 8 Y + cos π 8 Z i sinπ 8,
que corresponde a uma rota¸c˜ao em torno de um eixo ~ν = cosπ8, sinπ8, cosπ8 da esfera de Bloch, sendo o ˆangulo θ definido por cosθ
2 = cos 2 π
8. Portanto, usando a portas
quˆanticas Hadamard e π/8 pode-se obter uma rota¸c˜aoRν(θ). Aplicando-se, ent˜ao,Rν(θ)
diversas vezes, tem-se uma boa aproxima¸c˜ao para uma rota¸c˜ao qualquer Rν(α) com
precis¸c˜ao arbitr´aria. N˜ao iremos mostrar a prova desta ´ultima afirma¸c˜ao aqui para n˜ao nos estendermos muito. Maiores detalhes sobre a aplica¸c˜ao repetida deRν(θ) para obter
uma rota¸c˜ao qualquer Rν(α) com precis˜ao arbitr´aria, consultar a referˆencia [120].
Dessa forma, como a porta CNOT atua sobre dois q-bits interagentes e as portas Hadamard, π/8 e a porta de fase sendo aplicadas diversas vezes produzem rota¸c˜oes com precis˜ao arbitr´aria, ent˜ao, temos que esse conjunto de portas quˆantica servem para realizar qualquer opera¸c˜ao para a implementa¸c˜ao da computa¸c˜ao quˆantica.
Dimens˜oes Extras: Teoria de
Kaluza-Klein
A id´eia de utilizar dimens˜oes extras na relatividade geral foi proposta por Th. Kaluza [155] em 1921 com a finalidade de unificar a gravita¸c˜ao e o eletromagnetismo. Kaluza adotou a hip´otese que o Universo poderia possuir outras dimens˜oes espaciais al´em das dimes˜oes x, y, z que s˜ao compactadas em um espa¸co m´ınimo, isto ´e, em um espa¸co t˜ao pequeno que facilmente escapa `a detec¸c˜ao. Desse modo, Kaluza considerou um espa¸co- tempo pentadimensional ou um espa¸co-tempo em (4 + 1) dimens˜oes, onde era imposta `a dimens˜ao espacial extra que todas derivadas em rela¸c˜ao a quinta dimens˜ao fossem nulas. Assim, nenhuma grandeza f´ısica dependeria da coordenada extra.
A contribui¸c˜ao dada por O. Klein [156] em 1926 foi em compactificar essa dimens˜ao extra proposta por Kaluza. A compactifica¸c˜ao significa que ao inv´es de termos mais uma teoria com a dimens˜ao sendo infinita, pode-se mudar a teoria de modo que esta dimens˜ao extra tenha um comprimento finito e que deva ser peri´odica. A compactifica¸c˜ao ´e feita assumindo que a dimens˜ao extra tem a topologia de um c´ırculo de raio muito pequeno. Assim, a dimens˜ao extra tem a propriedade de ser peri´odica, ou seja, sendo y a dimens˜ao extra e T o per´ıodo, ent˜ao y′ = y + T . Assim, as trˆes dimens˜oes espaciais
x1, x2, x3 s˜ao consideradas como dimens˜oes estendidas e qualquer dimens˜ao espacial extra
´e considerada pequena e circular, ou seja, compacta. Nessa dimens˜ao extra, os campos f´ısicos dependem da periodicidade da dimens˜ao extra e podem ser expandidos em modos de Fourier1 [157, 158].
A teoria de Kaluza-Klein ´e uma forma de escrever a relatividade geral em dimens˜oes maiores que as 4 ou (3 + 1) estabelecidas por Einstein. A dimens˜ao ou dimens˜oes extras que sejam consideradas pela teoria de Kaluza-Klein deve sempre ter a topologia de um c´ırculo, cujo raio deve se considerando sendo muito pequeno. O elemento de linha para o
1
Nota: Com a expans˜ao do campo eletromagn´etico em modos de Fourier tornou-se poss´ıvel ter uma explica¸c˜ao te´orica para a quantiza¸c˜ao da carga el´etrica. Para maiores detalhes ver referˆencia [157].
espa¸co-tempo em (4 + 1) dimens˜oes ´e escrito na forma [157, 158, 159, 160] ds2 = gABdxAdxB
(C.1) = gµνdxµdxν + (dy + κ Aµ(x) dxµ)2,
onde os ´ındices gregos µ, ν indicam as coordenadas do espa¸co-tempo, a coordenada y indica a dimens˜ao extra onde a periodicidade da coordenada extra ´e dada aqui no intervalo 0≤ y≤ l, onde y′ = y + 2πl. κ ´e uma constante conhecida como constante de Kaluza que tem
dimes˜ao de (massa)−1 ou comprimento de modo que o termo κ Aµ(x) seja adimensional.
O termo Aµ(x) ´e o potencial de gauge do eletromagnetismo e tem unidade de massa
ou de (comprimento)−1 e satisfaz as transforma¸c˜oes de gauge abelianas discutidas na relatividade geral em (3 + 1) dimens˜oes, ou seja
Aµ → A′µ= Aµ+ ∂µǫ. (C.2)
Sem perda de generalidade, o campo de gauge Aµ(x) pode ser considerado como
n˜ao-abeliano, ou seja, ele ir´a satisfazer `a transforma¸c˜oes de gauge n˜ao-abelinas ou trans- forma¸c˜oes de gauge de Yang-Mills [157]
Aa
µ→ Aaµ′ = Aaµ+ ∂µǫa+ CabcǫbAcµ, (C.3)
onde os ´ındices a, b, c indicam que os campos de gauge s˜ao n˜ao-abelianos. ´
E interessante notar que os spinores s˜ao definidos na relatividade geral localmente atrav´es das componentes de uma base n˜ao-coordenada chamadas tetradas ou vierbein, isto ´e, ˆθa = ea
µ(x) dxµ. Da mesma forma, quando estamos trabalhando com a teoria de
Kaluza-Klein, os spinores devem continuar sendo definidos localmente onde a base n˜ao- coordenada fica senda dada por ˆθA¯ = eA¯
A(x) ˆθAe as componentes da base n˜ao-coordenada
(f¨unfbein) satisfazem a rela¸c˜ao
gAB(x) = e ¯ A A(x) e ¯ B B(x) ηA ¯¯B, (C.4)
com os ´ındices A, B = µ, y indicando os ´ındices do espa¸co-tempo µ = x0, x1, x2, x3 e
y sendo a dimens˜ao extra. Os ´ındices ¯A, ¯B = a, 5 indicam os referenciais locais dos observadores, onde a = 0, 1, 2, 3. Dessa forma, a express˜ao para a derivada covariante de um spinor na teoria de Kaluza-Klein deve ser escrita como
∇A= ∂A+ ΓA= (∂µ+ Γµ) + (∂y+ Γy) , (C.5)
onde a conex˜ao spinorial tem a mesma forma que discutimos no cap´ıtulo 1, express˜ao (1.97) Γµ= i 4ωµ ¯A ¯BΣ ¯ A ¯B; Γ y = i 4ωy ¯A ¯BΣ ¯ A ¯B, (C.6) onde ΣB ¯¯C = i 2
γB¯, γC¯ e com as matrizes de Dirac γA¯ satisfazendo a rela¸c˜ao γA¯, γB¯ =
−2ηA ¯¯B, ηA ¯¯B = diag (−1, 1, 1, 1, 1). O termo ω
A ¯B ¯C(x) satisfaz as equa¸c˜oes de estrutura de
Maurer-Cartan [159, 160]
Devido a periodicidade da coordenada extra, pode-se escrever o spinor de Dirac em termos dos modos de Fourier, ou seja,
Ψ (xµ, y) = ∞ X l=−∞ e2πilLy Ψl(xµ) = Ψ0(xµ) + e±2πi y LΨ ±1(xµ) + . . . , (C.8)
onde L ´e uma escala da dimens˜ao extra e Ψl(xµ) s˜ao os modos massivos. Em especial, o
modo Ψ0(xµ) corresponde a f´ermions sem massa que utilizamos na descri¸c˜ao do grafeno
feita na se¸c˜ao 7.1. Em outros estudo, o modo zero corresponde a f´ermions sem carga el´etrica [160].
Portanto, com esta breve introdu¸c˜ao sobre a teoria de Kaluza-Klein somos capazes de estudar a aplica¸c˜ao do teoria de Kaluza-Klein no grafeno. O grafeno tem uma propriedade vinda de sua rela¸c˜ao de dispers˜ao que nos permite estudar efeitos quˆanticos atrav´es da equa¸c˜ao de Dirac para um f´ermion sem massa em (2 + 1) dimens˜oes. Ao considerarmos uma dimen˜ao extra para no grafeno, poderemos dar uma descri¸c˜ao geom´etrica para o K-spin e estudar a equa¸c˜ao de Dirac para um f´ermion sem massa em (3 + 1) dimens˜oes.
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