Lecture 23: Quantum Computation

Transcription

1 princeton university cos 522: computtionl complexity Lecture 23: Quntum Computtion Lecturer: Snjeev Aror Scribe:Zhifeng Chen, Ji Xu This lecture concerns quntum computtion, n re tht hs become very populr recently becuse it promises to solve certin difficult problems fctoring nd discrete logrithm in polynomil time. 1 Quntum physics Quntum phenomen re counterintuitive. To see this, consider the bsic experiment of quntum mechnice tht proves the wve nture of electrons: the 2-slit experiment. (See Figure 1.) A source fires electrons one by one t wll. The wll contins two tiny slits. On the fr side re smll detectors tht light up whenever n electron hits them. We mesure the number of times ech detector lights up during the hour. The results re s follows. When we cover one of the slits, we observe the strongest flux of electrons right behind the open slit, s one would expect. owever, when both slits re open, we will see the interference phenomenon of electrons coming through two slits. In prticulr, t severl detectors the totl electron flux is lower when both slit re open s compred to when single slit is open. This defies explntion if electrons behve like little blls, s implied in some high school textbooks. A bll could either go through slit 1 or slit 2, nd hence the flux when both slits re open should be simple sum of the fluxes when one of them is open. The only explntion physics hs for this experiment is tht n electron does not behve s bll. It should be thought of s simultneously going through both slits t once, kind of like wve. It hs n mplitude for going through ech slit, nd this mplitude is complex number (in prticulr, it cn be negtive number). The chnce of n electron ppering t prticulr point p on the other side of the wll is relted to mplitude of reching p vi slit 1 + mplitude of reching p vi slit 2. Thus the points where the electron flux decreses when we open both slits re those where the two mplitudes hve opposite sign. Nonsense! you might sy. I need proof tht the electron ctully went through both slits. So you propose the following modifiction to the experiment. Position two detectors t the slits; these light up whenever n electron pssed through the slit. Now you cn test the hypothesis tht the electron went through both slits simultneously. Unfortuntely, when you put such detectors t the slits, the interference phenomenon disppers on the other side of the wll! The explntion is roughly s follows: the quntum nture of prticles disppers when they re under observtion. More specificlly, quntum system hs to evolve ccording to certin lws, nd nonreversible opertions such s observtion from nosy humns nd their detectors re not llowed. If these nonreversible opertions hppen, the quntum stte collpses. (One morl to drw from this is tht 1

2 # of electrons detected per hour 2 A A electron source B B A B Figure 1: 2-slit experiment quntum computers, if they re ever built, will hve to be crefully isolted from externl influences nd noise, since noise tends to be n irreversible opertion. Of course, we cn never completely isolte the system, which mens we hve to mke quntum computtion tolernt of little noise. This is topic of ongoing reserch.) 2 Quntum superpositions Now we describe quntum register, bsic component of the quntum computer. Recll the clssicl register, the building block of the memory in your desktop computer. An n-bit clssicl register with n bits consists of n prticles. Ech of them cn be in 2 sttes: up nd down, or nd 1. Thus there re 2 n possible configurtions, nd t ny time the register is in one of these configurtions. The n-bit quntum register is similr, except t ny time it cn exist in superposition of ll 2 n configurtions. (Drwing inspirtion from phenomen such s the 2-slit experiment, reserchers hve successfully implemented quntum registers using subtomic prticles.) Ech configurtion S {, 1} n hs n ssocited mplitude α S C where C is the set of complex numbers. α S = mplitude of being in configurtion S Physicists like to denote this system stte succinctly s S α S S. This is their nottion for describing generl vector in the vector spce C 2n, expressing the vector s liner combintion of bsis vectors. The bsis contins vector S for ech configurtion S. The choice of the bsis used to represent the configurtions is immteril so long s we fix bsis once nd for ll. At every step, ctions of the quntum computer physiclly, this my involve shining light of the pproprite frequency on the quntum register, etc. updte α S ccording to some physics lws. Ech computtion step is essentilly liner trnsformtion of the system stte. Let α denote the current configurtion (i.e., the system is in stte S α S S )

3 3 nd U be the liner opertor. Then the next system stte is β = U α. Physics lws require U to be unitry, which mens UU = I. (ere U is the mtrix obtined by trnsposing U nd tking the complex conjugte of ech entry.) Note n interesting consequence of this fct: the effect of pplying U cn be reversed by pplying the opertor U : thus quntum systems re reversible. This imposes strict conditions on which kinds of computtions re permissible nd which re not. As lredy mentioned, during the computtion steps, the quntum register is isolted from the outside world. Suppose we open the system t some time nd observe the stte of the register. If the register ws in stte S α S S t tht moment, then Pr[we see configurtion S] = α S 2 (1) In prticulr, we hve S α S 2 = 1 t ll times. Note tht observtion is n irreversible opertor. We get to see one configurtion ccording to the probbility distribution described in (1) nd nd the rest of the configurtions re lost forever. Wht if we only observe few bits of the register so-clled prtil observtion? Then the remining bits still sty in quntum superposition. We show this by n exmple. Exmple 1 Suppose n n-bit quntum register is in the stte s {,1} n 1 α s s + β s 1 s (2) (sometimes this is lso represented s s {,1} n 1(α s + β s 1 ) s, nd we will use both representtions). Now suppose we observe just the first bit of the register nd find it to be. Then the new stte is 1 α s s (3) α s 2 s {,1} n 1 where the first term is rescling term tht ensures tht probbilities in future observtions sum to 1. 3 Clssicl computtion using reversible gtes Motivted by the 2nd Lw of Thermodynmics, reserchers hve tried to design computers tht expend t lest in principle zero energy. They hve invented reversible computtion, which, using reversible gtes, cn implement ll clssicl computtions. We will study reversible clssicl gtes s stepping stone to quntum gtes; in fct, they re simple exmples of quntum gtes. The Fredkin gte is populr reversible gte. It hs 3 Boolen inputs nd on input (, b, c) it outputs (, b, c) if = 1 nd (, c, b) if =. Itisreversible, in the sense tht F (F (, b, c)) = (, b, c). Simple induction shows tht if circuit is mde out of Fredkin gtes lone nd hs m inputs then it must hve m outputs s well. Furthermore, we cn recover the inputs from the outputs by just pplying the circuit in reverse. ence Fredkin gte circuit is reversible. The Fredkin gte is universl, mening tht every circuit of size S tht uses the fmilir AND, OR, NOT gtes (mximum fnin 2) hs n equivlent Fredkin gte circuit of size

4 4 b Fredkin gte b b Implementing AND Figure 2: Implementing AND with Fredkin Gte 1 Fredkin gte Implementing NOT (nd copy) Figure 3: Implementing NOT nd COPY with Fredkin Gte O(S). We prove this by showing tht we cn implement AND, OR, nd NOT using Fredkin gte some of whose inputs hve been fixed or 1 (these re control inputs ); see Figure 2 for AND nd Figure 3 for NOT; we leve OR s exercise. We lso need to show how to copy vlue with Fredkin gtes, since in norml circuit, gtes cn hve fnout more thn 1. To implement COPY gte using Fredkin gtes is esy nd is the sme s for the the NOT gte (see Figure 3). Thus to trnsform norml circuit into reversible circuit, we replce ech gte with its Fredkin implementtion, with some dditionl control inputs rriving t ech gte to mke it compute s AND/OR/NOT. These inputs hve to be initilized ppropritely. The trnsformtion ppers in Figure 4, where we see tht the output contins some junk bits. With little more work (see Exercises) we cn do the trnsformtion in such wy tht the output hs no junk bits, just the originl control bits. Thus the reversible circuit hs no effect prt from trnsforming the input bits into output bits. (We re ssuming tht the number of input bits equls the number of output bits.) 4 Quntum gtes A 1-input quntum gte (Figure 2) is represented by unitry 2 2 mtrix U =( U U 1 U 1 U 11 ). When its input bit is the output is the superposition U + U 1 1 nd when the input is 1 the output is the superposition U 1 + U When the input bit is in the superposition α + β 1 the output bit is superposition of the corresponding outputs (α U + β U 1 ) +(α U 1 + β U 11 ) 1. (4)

5 5 Norml circuit Fredkin version of c x 1 y x 1 y x c n y x n c... y m m g 1 n+k m g... junk outputs k Figure 4: Converting norml circuit c into n equivlent circuit c of Fredkin gtes. Note tht we need dditionl control inputs More succinctly, if the input stte vector is (α,β ) then the output stte vector is ( ) ( ) α α = U β If α 2 + β 2 = 1 then unitrity of U implies tht α 2 + β 2 =1. Similrly, 2-input quntum gte is represented by unitry 4 4 mtrix R. When the input is the superposition α + α α α 11 11, the output is β + β β β where β α β 1 β 1 β 11 = R In generl, quntum gte with k inputs is specified by unitry 2 k 2 k mtrix. Exmple 2 A Fredkin gte is lso vlid 3-input quntum gte. We represent it by n 8 8 mtrix tht gives its output on ll 2 3 possible inputs. This mtrix is permuttion mtrix (i.e., obtinble from the identity mtrix by pplying permuttion on ll the rows) since the output F (, b, c) is just permuttion of the input (, b, c). Exercise 2 sks you to verify tht this permuttion mtrix is unitry. A quntum circuit on n inputs consists of () n n-bit quntum register (b) sequence of gtes (g j ) j=1,2,....ifg j is k-input gte, then the circuit specifiction hs to lso give sequence of bit positions (j, 1), (j, 2),...,(j, k) [1,n] in the quntum register to which this gte is pplied. The circuit computes by pplying these gte opertions to the quntum register one by one in the specified order. The register holds the stte of the computtion, nd only one gte is pplied t ny given time. Exmple 3 Suppose we hve n n-bit quntum register in the stte S,1 n α S S. If we pply 1-input quntum gte U to the first wire, the new system stte is computed s follows. First fctor the initil stte by expressing ech n-bit configurtion s conctention of the first bit with the remining n 1 bits: S + α 1,S 1S. (5) S {,1} n 1 α,s β α 1 α 1 α 11

6 6 (Formlly we could express everything we re doing in terms of tensor product of vector spces but we will not do tht.) To obtin the finl stte pply U on the first bit in ech configurtion s explined in eqution (4). This yields (α,s U + α 1,S U 1 ) S +(α,s U 1 + α 1,S U 11 ) 1S (6) S {,1} n 1 We cn similrly nlyze the effect of pplying k-input quntum gte on ny given set of k bits of the quntum register, by first fctoring the stte vector s bove. 4.1 Universl quntum gtes You my now be little troubled by the fct tht the number of possible quntum gtes with single input is the set of ll unitry 2 2 mtrices, n uncountble set. Tht seems like bd news for the Rdio Shcks of the future, who my feel obliged to keep ll possible quntum gtes in their inventory, to llow their customers to build ll possible quntum circuits. Luckily, Rdio Shck need not fer. Reserchers hve shown the existence of smll set of universl 2-input quntum gtes such tht every circuit composed of S rbitrry k-input quntum gtes cn be simulted using circuit of size poly(k) S log S composed only of our universl gtes. The simultion is not exct nd the distribution on outputs is only pproximtely the sme s the one of the originl circuit. (All of this ssumes the bsence of ny outside noise; simultion in presence of noise is topic of reserch nd currently seems possible under some conditions.) In ny cse, we will not need ny fncy quntum gtes below; just the Fredkin gte nd the following 1-input gte clled the dmrd gte: 5 BQP = 1 ( ) Definition 1 A lnguge L {, 1},isinBQP iff there is fmily of quntum circuits (C n ) of size n c s.t. x {, 1} : x L Pr[C(x) 1 =1] 2 3 x/ L Pr[C(x) 1 =1] 1 3 ere C(x) 1 is the first output bit of circuit C, nd the probbility refers to the probbility of observing tht this bit is 1 when we observe the outputs t the end of the computtion. The circuit hs to be uniform, tht is, deterministic polynomil time (clssicl) Turing mchine must be ble to write down its description.

7 7 At first the uniformity condition seems problemtic becuse clssicl Turing mchine cnnot write complex numbers needed to describe quntum gtes. owever, the mchine cn just express the circuit pproximtely using universl quntum gtes, which is finite fmily. The pproximte circuit computes the sme lnguge becuse of the gp between the probbilities 2/3 nd 1/3 used in the bove definition. Theorem 1 P BQP Proof: Every lnguge in P hs uniform circuit fmily of polynomil size. We trnsform these circuits into reversible circuits using Fredkin gtes, thus obtining quntum circuit fmily for the lnguge. Theorem 2 BPP BQP Proof: Every lnguge in BPP hs uniform circuit fmily (C n ) of polynomil size, where circuit C n hs n norml input bits nd n dditionl n k input wires tht hve to be initilized with rndom bits. We trnsform the circuit into reversible circuit C n using Fredkin gtes. To produce rndom bits, we feed O(n c ) zeros into n rry of dmrd gtes nd plug their outputs into C n; see Figure 5. A uniform ckt of BPP Fredkin version of c x1... x c Control bits n x 1 Complex representtion rndom y x c of ll possible outputs n bits y of ckt c n c 1 2 i... dmnel Gte Figure 5: C n is Fredkin circuits by replcing ech norml gte of C n with respective quntum gte. An rry of dmrd gtes produce rndom bits. To see tht this works, imgine fixing the first n bits to string x in both circuits. A simple induction shows tht if we strt with n N-bit quntum register in the stte nd pply the dmrd gte one by one on ll the bits, then we obtin the superposition S,1 N n c 1 S (7) 2N/2 Thus the rndom inputs of circuit C n get uniform quntum superposition of ll possible bit strings. Thus the output bit of C n is uniform superposition of the result on ech bit string nd observing it t the end gives the probbility tht C n ccepts when fed rndom input.

8 8 6 Fctoring integers using quntum computer This section proves the following fmous result. Theorem 3 (Shor 1994) There is polynomil size quntum circuit tht fctors integers. We describe Kitev s proof of this result since it uses eigenvlues, more fmilir nd intuitive concept thn the Fourier trnsforms used in Shor s lgorithm. Definition 2 (Eigenvlue) λ is n eigenvlue of mtrix M if there is vector e (clled the eigenvector), s.t.: M e = λe Fct: If M is unitry, then λ = 1. In other words there is θ [, 1) such tht λ = e 2πiθ = cos(2πθ)+i sin(2πθ). Fct: If M e = λe then M k e = λ k e. ence e is still n eigenvector of M k nd λ k is the corresponding eigenvlue. Now we re redy to describe Kitev s lgorithm nd prove Theorem 3. Proof: Let N be the number to be fctored. As usul, ZN is the set of numbers mod N tht re co-prime to N. Simple number theory shows tht for every ZN there is smllest integer r such tht r 1 (mod N); this r is clled the order of. The lgorithm uses the well-known fct tht if we cn compute the order of rndom elements of ZN then we cn fctor N with high probbility. The reson is tht if (r 1) (mod N), then ( r 2 1)( r 2 +1) (mod N). If is rndom, with probbility 1 2, r 2 1 (mod N), r 2 1 (mod N) (this is simple exercise using the Chinese reminder theorm) nd hence gcd(n, r 2 1) N,1. Thus, knowing r we cn compute r/2 nd compute gcd(n, r 2 1). With probbility t lest 1/2 (over the choice of ) this method yields fctor of N. The fctoring lgorithm is mixture of clssicl nd quntum lgorithm. Using clssicl rndom bits it genertes rndom ZN nd then constructs quntum circuit. Observing the output of this quntum circuit few times followed by some more clssicl computtion llows it to obtin r, the order of, with resonble probbility. (Of course, we could in principle describe the entire lgorithm s quntum lgorithm insted of s mixture of clssicl nd quntum lgorithm, but our description isoltes exctly where quntum mechnics is crucil.) Consider clssicl reversible circuit tht cts on numbers in ZN, nd is described by U(x) =x (mod N). Then we cn view this circuit s quntum circuit operting on quntum register. If the quntum register is in the superposition 1 α x x, x ZN 1 Aside: This isn t quite correct; the crdinlity of ZN is not power of 2 so we cnnot construct quntum register whose only possible configurtions correspond to elements of ZN. owever, if we use the nerest power of 2, everything we re bout to do will still be pproximtely correct.

9 9 b b x Cond-U gte x if b= Ux if b=1 Figure 6: Conditionl-U circuit then pplying U gives the superposition α x x (mod N). x Z N Interpret this quntum circuit s n N N mtrix lso denoted U nd consider its eigenvlues. Since U r = I, we cn esily check tht ech eigenvlue hs the form e 2πiθ where θ = j r for some j r. The lgorithm will try to obtin rndom eigenvlue. It thus obtins in binry expnsion number of form j r where j is rndom. It turns out tht the chnce is pretty good tht this j is coprime to r, which mens tht j r is n irreducible frction. Even knowing only the first 2 log N bits in the binry expnsion of j r, the lgorithm cn round off to the nerest frction whose denomintor is t most N (this cn be done; esy number theory) nd then it reds off r from the denomintor. Now we describe how to compute the first 2 log N bits of rndom eigenvlue of U. Assume for now tht the lgorithm hs quntum register whose stte is superposition, denoted e, corresponding to rndom eigenvector of U. (See below for detils on how to put register in such stte.) Then pplying U gives the finl stte λ e, where λ is the eigenvlue ssocited with e. Thus the register s stte hs undergone phse shift i.e., multipliction by sclr lthough there is yet no direct wy to mesure λ. Now define conditionl-u circuit (Figure 6), whose input is (b, x) where b is bit, nd cond-u(b, x) =(b, x) ifb = nd (b, x (mod N)) if b = 1. Agin, notice tht this cn be implemented using clssicl Fredkin gtes. Using cond-u circuit nd two dmrd gtes, we cn build quntum circuit shown in Figure 7. When this is pplied to quntum register whose first bit is nd the remining bits re in stte e, then we cn mesure the corresponding eigenvlue λ by repeted mesurement of the first output bit. 2 1 e 1 1 e e 2 2 cond-u 1 2 e + λ 2 1 e ((1 + λ) e +(1 λ) 1 e ) (8) 2

10 1 e Cond-U gte Figure 7: Bsic block: Conditionl-U gte nd two dmrd gtes. Thus the probbility of mesuring inthefirst bit is proportionl to 1 + λ. We will refer to this bit s the phse bit, since repetedly mesuring it llows us to compute better nd better estimtes to λ. Actully, insted of repeted mesuring we cn just design quntum circuit to do the repetitions by noticing tht the output is just sclr multiple of e gin, so we cn just feed it into nother bsic block with fresh phse bit, nd so on (see Figure 8). We mesure phse bits ll t once t the end. Are we done? Unfortuntely, no. Obtining n estimte to the first m bits of λ would involve circuit with 2 m phse bits (this is simple exercise bout probbilistic estimtion), nd when m = 2 log N, this number is bout N, wheres we re hoping for circuit size of poly(log N). Thus simple repetition is very inefficient wy to obtin ccurte informtion bout λ. A more efficient technique involves the observtion tht U hs specil property: powers of U lso hve smll circuits. Specificlly, U 2k (x) = 2k x (mod N), nd 2k is computble by circuits of size poly(log N + log k) using fst exponentition. Thus we cn implement conditionl-u 2k gte using quntum circuit of size poly(log N+ k). The eigenvlues of U 2k re λ 2k. If λ = e 2πiθ where θ [, 1) (see Figure 9) then λ 2k = e 2πiθ2k. Of course, e 2πiθ2k is the sme complex number s e 2πiα where α =2 k θ (mod 1). Thus mesuring λ 2k gives us 2 k θ (mod 1) nd in prticulr the most significnt bit of 2 k θ (mod 1) is nothing but the kth bit of θ. Using k =, 1, 2,...2log N we cn obtin the first 2 log N bits of θ. As in Figure 8, we cn bundle these steps into single cscding circuit where the output of the conditionl-u 2k 1 circuit feeds into the conditionl-u 2k circuit. Ech circuit hs its own set of O(log N) phse bits; mesuring the phse bits of the kth circuit gives n estimte of the kth bit of θ tht is correct with probbility t lest 1 1/N. All phse bits re mesured in single stroke t the end. To finish, we show how to put quntum register into stte corresponding to n eigenvector.

11 11 e Cond-U Cond-U Cond-U Figure 8: Repeting the bsic experiment to get better estimte of λ. λ = e 2πi θ 2π θ Figure 9: Eigenvlue λ = e 2πiθ in the complex plne.

12 Uniform superpositions of eigenvectors of U Actully, we show how to put the quntum register into uniform superposition of eigenvectors of U. This suffices for our cscding circuit, s we will rgue shortly. First we need to understnd wht the eigenvectors look like. Recll tht { 1,, 2,..., r 1} is subgroup of ZN. Let B be set of representtives of ll cosets of this subgroup. In other words, for ech x ZN there is unique b B nd l {, 1,...,r 1} such tht x = b l (mod N). Then the following is the complete set of eigenvectors, where ω = e 2πi r : r 1 j {, 1,...,r 1}, b B e j,b = ω jl b l (mod N) (9) The eigenvlue ssocited with this eigenvector is ω j = e 2πij r. Fix b nd consider the uniform superposition: 1 r 1 e j,b = 1 r 1 r 1 ω jl b l (mod N) (1) r r j= j= l= l= j= l= = 1 r 1 r 1 ω jl b l (mod N). (11) r Seprting out the terms for l = nd using the formul for sums of geometric series: = 1 r 1 r ( r 1 b + j= l=1 (ω l ) r 1 ω l b l (mod N) ) (12) since ω r = 1 we obtin = b (13) Thus if we pick n rbitrry b nd feed the stte b into the quntum register, then tht cn lso be viewed s uniform superposition 1 r j e j,b. 6.2 Uniform superposition suffices Now we rgue tht in the bove lgorithm, uniform superposition of eigenvectors is just s good s single eigenvector. Fixing b, the initil stte of the quntum register is 1 r e j,b, j where denotes the vector of phse bits tht is initilized to. After pplying the quntum circuit, the finl stte is 1 c j e j,b, r j

13 13 where c j is stte vector for the phse bits tht, when observed, gives the first 2 log N bits of j/r with probbility t lest 1 1/N. Thus observing the phse bits gives us whp rndom eigenvlue. Exercises 1 Implement n OR gte using the Fredkin gte. 2 Given ny clssicl circuit computing function f from n bits to n bits, describe how to compute the sme function with reversible circuit tht is constnt fctor bigger nd hs no junk output bits. (int: Use our trnsformtion, then copy the output to sfe plce, nd then run the circuit in reverse to erse the junk outputs.) 3 Verify tht the Fredkin gte is vlid quntum gte. 4 Verify ll the number theoretic fcts mentioned in the description of the fctoring lgorithm.