where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b

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1 CS /11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1} n {0,1} m, given s primitive finite olletion of gtes eh of whih implements oolen funtion on k its for some smll k. The piture of the iruit is s follows: i 1 i 2 i 3 i 4 i 5 i n 1 i n 0 1 f o 1 o 2 o 3 o m 1 o m where the ox ontins finite numer of gtes from the given olletion. Exmples of gtes tht re ommonly used re the following: (+) NOT AND NAND NOR There exist finite gte sets tht n e used to implement every multi-output oolen funtion. Suh sets re lled universl gte sets nd the following gte sets n esily e proved universl: {AND, NOT} {OR, NOT} {NAND} {NOR} Every oolen funtion n e implemented using gtes from universl gte set. This implies tht the iruit omplexity (numer of gtes in the minimum iruit) is the sme with respet to ny finite universl gte set up to onstnt ftor. A sutle point when we onsider iruit design is the fn out. In lssil iruits we n tke fn out s grnted, euse fn out is trivilly implemented in lssil relity. However, this is not trivil in quntum mehnis s we ll see shortly. CS 294-2, Fll 2004, Leture 4 1

2 1.2 Quntum Ciruits A quntum iruit implements unitry opertor in Hilert spe C 2n, given s primitive (usully finite) olletion of gtes eh of whih implements unitry opertor on k quits for some smll k. Unitrity implies tht quntum iruits hve the sme numer of inputs nd outputs. The piture of quntum iruit is s follows: i 1 i 2 i 3 i n 1 i n o 1 o 2 o 3 o n 1 o n where the ox ontins finite numer of quntum gtes. Clerly no finite set of gtes n generte ll unitry opertors. If suh set existed, then for ll vlues of θ R we should e le to uilt up quntum iruit for the opertor R θ using gtes from this set. With finite set of gtes tht s impossile. It seems though tht we need notion of pproximtion to define universlity. For tht purpose we ll use the opertor norm whih is defined s B = mx v =1 B v for every opertor B. Using tht mesure the distne etween opertors U nd U will e: U U = mx v =1 (U U ) v nd we ll sy tht opertor U simultes opertor U to within ε if U U ε After the ove definitions it s interesting to see how we define the universlity in quntum mehnis. We shll ll set G of quntum gtes universl if: U (unitry opertor on k quits), ε > 0, g 1,g 2,...,g l G : U U g1 U g2...u gl ε where y U gi ( usge of g i ) we represent the tensor produt of the gte g i with the identity opertor for n pproprite numer of quits so tht U gi is unitry opertor for k quits wheres g i might e unitry opertor for less thn k quits. 1.3 Known Universl Gte Fmilies for Quntum Mehnis The following fmilies of iruits re universl: CNOT, ll 1-quit gtes CNOT gte, Hdmrd gte, suitle phse flips Tofolli gte, Hdmrd gte CS 294-2, Fll 2004, Leture 4 2

3 where the Tofolli gte (or C-CNOT gte for ontrolled-ontrolled NOT gte ) is three-quit gte tht omplements the third it if the first two ontrol its re oth 1. 2 Solovy-Kitev Theorem The Solovy-Kitev theorem sttes the following: If G SU(d) is universl fmily of gtes (where SU(d) is the group of unitry opertors in d-dimensionl Hilert spe), G is losed under inverse (i.e. g G g 1 G) nd G genertes dense suset of SU(d), then U SU(d),ε > 0, g 1,g 2,...,g l G : U U g1 U g2...u gl ε nd l = O(log 2 1/ε) 3 Complexity Clsses - Clss BQP 3.1 Clss P - Polynomil Time A definition of the lss P in terms of iruits is the following: L P iff there is fmily F = { } n N of iruits suh tht: poly(n), n N there is polynomil time Turing Mhine tht on input 1 n outputs (Uniformity Condition) if x = n then (x) = ( L?) x 4 x 5 1 0/1 3.2 Clss BPP - Bounded Error Polynomil Time A definition of the lss BPP in terms of iruits is the following: L BPP iff there is fmily F = { } n N of iruits suh tht: every iruit hs n input x of x = n its nd rndom input r of r = O(poly(n)) its CS 294-2, Fll 2004, Leture 4 3

4 poly(n), n N there is polynomil time Turing Mhine tht on input 1 n outputs (Uniformity Condition) moreover: if x L nd x = n then Pr[ (x,r) = yes ] 2/3 if x / L nd x = n then Pr[ (x,r) = no ] 2/3 input x x = n 1 0/1 rndom string r r = m = O(poly(n)) r 1 r 2 r m 3.3 Clss BQP - Bounded Error Quntum Polynomil Time A definition of the lss BQP in terms of iruits is the following: L BQP iff there is fmily F = { SU(n)} n N of quntum iruits (unitry opertors) suh tht: every iruit hs n input x of x = n its nd m = O(poly(n)) dditionl inputs of vlue 0 > the output of the omputtion is onsidered to e the outome of the mesurement on the first output of the iruit poly(n), n N there is polynomil time Turing Mhine tht on input 1 n outputs (Uniformity Condition) moreover: if x L nd x = n then Pr[mesure = 1] 2/3 if x / L nd x = n then Pr[mesure = 0] 2/3 input x x = n 1 M 0/1 m 0 > 0 > 0 > CS 294-2, Fll 2004, Leture 4 4

5 3.4 Reversiility nd P BQP Quntum evolution is unitry nd every quntum iruit K orresponds to unitry opertor U K in some Hilert spe. U K eing unitry mens tht U K U K = U K U K = I, whih mens tht U K hs n inverse opertor. Thus every quntum iruit is reversile. This is not the se for lssil iruits, however. For exmple if we hve n AND gte, then going from two input its to one output it involves some loss of informtion, whih mkes reversiility infesile. However, strting from nonreversile iruit we n onstrut reversile iruit tht does the sme omputtion ut my require more inputs or outputs thn the initil nonreversile iruit. There re numer of wys to do this nd here we will show how to do it using the ontrolled swp gte (Fredkin gte). The ontrolled swp gte on input (,,) outputs (,,), if = 0, nd (,,), if = 1. It s ovious tht the ontrolled swp gte is the inverse of itself. We use the following nottion for the ontrolled swp gte. =0 =1 Sine the set of lssil gtes {AND,NOT } is universl, every lssil iruit n e uilt using these two gtes, s well s possily fnning out. Thus, in order to prove tht every lssil iruit n e extended to reversile one, we only hve to show tht we n simulte the AND nd NOT gtes, s well s fn out, using the ontrolled swp gte. The extr inputs tht we supposed will ome in hndy. In order to simulte the NOT gte vi the ontrolled swp gte we n give the ltter the triplet (, = 0, = 1) s input. Then t the third output we lwys get. In order to simulte the AND gte vi the ontrolled swp gte we n give the ltter the triplet (,, = 0) s input. Then t the third output we lwys get. Finlly, in order to fn out s permitted in lssil iruits vi the ontrolled swp gte, we n give the ltter the triplet (, = 0, = 1) s input. Then the first nd seond outputs lwys hve the vlue. It s ovious tht the reversile iruit tht orresponds to lssil iruit nd is onstruted in the ove wy will hve extly the sme output s the lssil one on lssil inputs. Thus, the ove onstrution shows s well tht P BQP. CS 294-2, Fll 2004, Leture 4 5

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