Last time, we talked about how Equation (1) can simulate Equation (2). We asserted that Equation (2) can also simulate Equation (1).

Transcription

1 6896 Quatum Complexity Theory Sept 23, 2008 Lecturer: Scott Aaroso Lecture 6 Last Time: Quatum Error-Correctio Quatum Query Model Deutsch-Jozsa Algorithm (Computes x y i oe query) Today: Berstei-Vazirii Algorithm Simo s Algorithm Shor s Algorithm idde Subgroup Framework 2 Recap of Last Lecture Error Propagatio Recall from the previous lecture that error propagates differetly i quatum computig compared with i classical computig I particular, the (990 s) Threshold Theorem states that quatum computig is robust to small errors sice it is a liear theory This implies that small errors are ot magified over the course of the computatio ad ca be corrected usig sophisticated hierarchical error-correctig codes This is ulike classical computig where oise amplificatio was a major cosideratio i desigig digital systems 2 The Query Model I quatum complexity theory, we are iterested i determiig which problems are efficietly solvable by quatum algorithms ad which oes are ot This is a difficult problem to aswer sice we do t kow to quatify the umber of required computatioal steps We do t eve kow how to prove that the problems we care about do ot require a expoetial umber of steps We oly itroduced this 2 We did t get to this 6-

2 The Query Model allows us to make some aalysis of the computatioal complexity of quatum algorithms I this model, the complexity of a algorithm is measured as the umber of queries we make to some query box So, what do we mea by a query? I classical computig, the idea is fairly self-explaatory For example, to fid the majority of three iput bits x, x 2, ad x 3, we may query for the values for x ad x 3 If we fid that both bits are oe, the we will eed a total of two queries I quatum computig, the queries (like all computatioal steps) must be reversible ad, therefore, uitary I the previous lecture, we gave two equivalet models of quatum queries I the first model, the fuctio f( ) that we wat to query becomes xor-ed to some aswer bit b I the secod model, f( ) gets writte to a phase; we flip the amplitude if the aswer is oe ad, otherwise, we do t Writte out, the two defiitios of queries are: x b x b f(x) () x ( ) f(x) b x (2) Last time, we talked about how Equatio () ca simulate Equatio (2) We asserted that Equatio (2) ca also simulate Equatio () 3 Deutsch-Jozsa Algorithm The Deutsch-Jozsa algorithm is our first example where quatum computig is more efficiet tha classical computig The algorithm computes the exclusive-or of two bits i oe query rather tha two (I classical computig, we eed to query both bits to fid the exclusive-or) We ca exted this algorithm to exclusive-or -bits usig oly /2 queries So, this is a factor of two speed-up The algorithm is the followig We query i superpositio ad the apply a adamard + ( ) f(0) + ( ) f() 2 2 If f(0) is equal to f(), the the adamard is goig to map above state to ± Otherwise, if f(0) = f(), the the adamard is goig to map to ± ±, if f(0) = f() ±, if f(0) = f() 2 Berstei-Vazarai Algorithm [93] The Deutsch-Jozsa algorithm ca be geeralized to the problem of fidig a hidde liear structure Suppose that there is a boolea fuctio, f : {0, } {0, }, which maps -bit strigs to a sigle bit I additio, we have query access to the fuctio Like i the Deusch-Jozsa algorithm, we ca query ay -bit strig x for f(x); ad we ca also query i superpositio Let us cosider the case where f(x) is of the form s x mod 2, where s is a fixed ad secret strig (ere deotes the ier product, ie s x = s x + + s x mod 2) I other words, we are promised that there exists a secret strig s such that f(x) = s x mod 2 for all x The problem is to fid s 6-2

3 First, let us examie the query complexity i the classical world ow may queries do we eed to solve this problem classically? queries are sufficiet, because we ca query the basis strigs For example, for = 5, we ca query the basis strigs to fid s,, s 5 as follows: f(0000) = s f(0000) = s 2 f(0000) = s 3 f(0000) = s 4 f(0000) = s 5 I additio, is also the lower boud We ca prove that we eed at least queries usig a basic iformatio theoretic argumet There are possibilities for s, ad each query ca oly cut the space i half I cotrast, the Berstei-Vazarai algorithm solves the problem usig oly oe query The quatum circuit diagram of the Berstei-Vazarai algorithm is give below: s f s s s 2 3 Mathematically, this is equivalet to: = = x ( ) f(x) x ( ) s x x ( ) s x ( ) s x x After queryig the box i superpositio, the the secret strig s writte i the adamard basis So i order to read s, we covert back to stadard basis This is what is accomplished by applyig a secod adamard i the ed Notice that although we eeded a liear umber of gates, we oly have to make oe query This is a -to- speed-up 6-3

4 3 Simo s Algorithm [93] Usig the Berstei-Vazarai algorithm, we have demostrated that we eed oly oe query i the quatum world where we eeded at i the classical world While this is a iterestig result, what we really wish for is a expoetial-to-polyomial speed-up What we wat to ask is, Ca we use the quatum model to attack the Beast that is Expoetially? This questios leads us to Simo s algorithm 3 As i the Berstei-Vazarai algorithm, we are agai give a query box that computes some fuctio f(x) I this case, the fuctio f : {0, } {0, } maps -bits to -bits As before, there is a promise associated with the fuctio f(x) This time, the promise is that there exists a secret strig s, where s is ot the zero-vector, such that f(x) = f(y) if ad oly if x = y s The problem is to fid s To illustrate what we mea, the secret strig s is 0 i the followig example: f(000) = 5, f(00) = 7 f(00) = 4, f(0) = 42 f(00) = 7, f(0) = 5 f(0) = 42, f() = 4 First, let us examie the query complexity i the classical world ow may queries do we eed to solve this problem classically? I classical world, we eed o the order of /2 bits (To prove that we eed O(/2 ) queries, pick s radomly ad use Yao s Mimax Priciple) By the Birthday Paradox, eve with radomess, we still eed /2 bits I quatum computig, we ca use Simo s algorithm do this with liear umber of circuits, where each circuit is give by: f We apply a adamard to the first register, ad the query both registers The, we measure the secod register After we make the measuremet, what is left i the first register is: 3 x f(x) x +, where x y = s 2 2 y Simo s origial motivatio was to try to prove that the quatum model does ot provide ay expoetial speedup e failed i his attempt, ad what he came up with i the ed was a couter-example, which became Simo s algorithm Shor read this paper ad used the geeral idea to factor itegers So, as we will see later o, Shor s algorithm is Simo s algorithm plus some umber theory 6-4

5 To extract iformatio about s, we apply the adamard agai ad see what s left over from the mess x + y ( ) x z z + ( ) y z z = + z {0,} z {0,} [( ) x z + ( ) y z ] z z {0,} We ca t directly extract s from here, but we ca extract some useful iformatio about s ( ) x z = ( ) y z or else the amplitudes cacel out This leads us to a liear equatio for s x z = y z x z = (x s) z x z = x z s z s z = 0 We did t lear s, but we did lear that s satisfies the radom liear equatio s z = 0 We ca solve for s usig Gaussia Elimiatio o liearly idepedet equatios We oly eed to ru Simo s circuit O() times to get liearly idepedet equatios, so Simo s algorithm rus i O() queries This is a expoetial-to-polyomial speed-up! 4 BPP A = BQP A At the ed of the day, we did t prove that BPP = BQP owever, we did make some progress Namely, that via some relativizatio method, we ca prove that there exists some A, such that BPP A = BQP A 5 Shor s Algorithm [94] Where Simo s algorithm worked i Z 2, Shor s algorithm works i Z N So, f : [N] [N] The promise is that there exists some r, such that f(x) = f(x + r) = f(x + 2r) = We ll see more o Shor s algorithm ad o idde Subgroup ext lecture! 6-5

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