CSE 599d - Quantum Computing Introduction to Quantum Error Correction

Transcription

1 CSE 599d - Quantum Computng Introducton to Quantum Error Correcton Dave Bacon Department of Computer Scence & Engneerng, Unversty of Washngton In the last lecture we saw that open quantum systems could nteract wth an envronment and that ths couplng could turn pure states nto mxed states. As we have argued before, ths s a bad process, because t can lessen or destroy the nterference effects whch are vtal to dstngushng a quantum from a classcal computer. Ths s called the decoherence problem. In ths lecture we wll begn to see how to deal wth ths problem. Note, however, rght off the bat, that there are other problems besdes the decoherence problem. For example we stll haven t address the fact that we don t have perfect control over our quantum system when attemptng to perform untary gates, preparaton of states, or measurement. We wll deal wth all of these n good tme, but for now we wll focus on the problem of decoherence. I. SIMPLE CLASSICAL ERROR CORRECTION Well when we are stumped about what to do n the quantum world, t s often useful to look to the classcal world to see f there s an equvalent problem, and f so, how that problem s dealt wth. Ths leads us from quantum nose to classcal nose. Suppose we have the followng classcal stuaton, we have a bt whch we send through a channel and wth probablty 1 p nothng happens to our bt, but wth probablty p the bt s flpped. Ths channel s called a bnary symmetrc channel. We can descrbe t by one of our doubly stochastc matrces f we feel lke t [ ] 1 p p M =. (1) p 1 p Now f we use ths channel once, then the probablty that we wll receve the wrong bt s p. Is there a way to use ths channel n such a way that we can decrease ths probablty? Yes, and t s rather smple. We just use the channel multple tmes and use redundancy. In other words, f we want to send a 0, we use the encodng and and send each of these bts through the channel. Now of course there wll stll be errors on the channel. Wth probablty (1 p) 3 no errors occur on the bts. Wth probablty 3(1 p) p one error occurs on the bts. Wth probablty 3(1 p)p two error occur on the bts. And wth probablty p 3 three errors occur on the bts. Now assume that p s small for ntutons sake (we wll calculate what small means n a second.) Notce that the three probabltes we have lsted above wll then be n decreasng order. In partcular the probablty of no or one error wll be greater than there beng two or three errors. But f a sngle error occurs on our bt, we can detect ths and correct t. In partcular f, on the other end of the channel we decode the states by {000, 001, 010, 100} 0 and {111, 110, 101, 011} 1, then n the frst two cases we wll have correctly transmtted the bt, even though a sngle error occurred on ths bt. We can thus calculate the probablty that ths procedure, encodng, sendng the bts ndvdually through the channel, and decodng fals. It s gven by 3(1 p)p + p 3 = 3p p 3. Now f ths s less than p we have decreased the probablty of falng to transmt our bt. Indeed ths occurs when 3p p 3 p or when p < 1. Thus f the probablty of our bt s less than 1, then we wll have decreased our falng (from p to 3p p 3.) Ths s great and s known as a redundancy error correctng code. The classcal theory of error correctng codes s devoted to expandng on ths basc observatons, that redundancy can be used to protect classcal nformaton. We won t delve too deeply nto classcal error correcton, although we wll fnd t useful to learn some thngs. Because we are nterested n understandng whether we can port ths dea of over to quantum theory! When we frst encounter classcal error correcton and thnk about portng t over to the quantum world, there are some nterestng reasons to beleve that t wll be mpossble to make ths transton. We lst these here, snce they are rather nterestng (although clams that these were major blockades to dscoverng quantum error correctng are probably a bt exaggerated, at least accordng to the people I ve talked to who were workng n quantum computng when ths happened.) No Clonng We ve seen that the no-clonng theorem states that there s no machne whch perform ψ ψ ψ. Thus a nave attempt to smply clone quantum nformaton n the same way that we copy nformaton n a redundancy code fals.

2 Measurement When we measure a quantum system, our descrpton of the quantum system changes. Another way ths s stated s that measurement dsturbs the state of a quantum system. In error correcton, we read out classcal nformaton n order to correctly recover our classcal nformaton. How do we perform measurements on quantum systems that don t destroy the quantum nformaton we are tryng to protect. Quantum Nose As we noted n the last lecture, quantum nose has a contnuous set of parameters to descrbe t. So we mght thnk that ths wll cause a problem, snce n classcal theory we could nterpret the nose as probabltes of determnstc evolutons occurrng, but n quantum theory we don t have such an nterpretaton (at least not yet.) Of course ths feels lke a lttle bt of a red herrng, snce, classcal nose also has contnuous parameters (say the probabltes of the errng procedures) to descrbe t. It s just that those parameters are probabltes whch makes us feel safer n the classcal case, although we have learned that dstnctons lke ths usually fade away when we examne them closer. For the reasons we mght expect that an equvalent to classcal error correcton n the quantum world does not exst. One of the surprsng dscoveres of the md-nnetes was that ths s not true! II. REVERSIBLE SIMPLE CLASSICAL ERROR CORRECTION So where to begn. Well one place to begn s to try to understand how to perform the classcal error correcton we descrbed above usng classcal reversble crcuts. So the frst part of our procedure s to encode our classcal bt. Suppose that we represent our three bts by three wres. Then t s easy to check than an encodng procedure for takng a bt n the frst wre to the encoded 000 and 111 confguratons s b b 0 b 0 b Next we send each of these bts through the bt flp channel ndependent of each other. We wll denote ths by the gate M on these bts () b M (3) 0 0 Now we need to descrbe the procedure for dagnosng and error and fxng ths error. Consder what the two controlled-nots do n our encodng crcut. They take , , , , , , , and Notce that except n the case where the last two bts are 11, after ths procedure, the frst bt has been restored to t s proper value, gven that only zero or one bt flp has occurred on our three bts. And when the last two bts are 11, then we need to flp the frst bt to perform the proper correcton. Ths mples that the decodng and fxng procedure can be done by the two controlled-nots, followed by a Toffol gate. In other words M M b M 0 M (4) 0 M It s easy to check that f only one or no error occur where the M gates are, then the output of the frst bt wll be b (and n the other cases the bt wll be flpped.) Ths s exactly the procedure we descrbed n the prevous secton and we ve done t usng completely reversble crcut elements (except for the M s, of course.) III. WHEN IN ROME DO AS THE CLASSICAL CODERS WOULD DO Now that we have a classcal reversble crcut for our smple error correctng procedure, we can see what happens when we use ths crcut on quantum nformaton nstead of classcal nformaton. One thng we need to do s to chose

3 the approprate nose channel for our code (we ll come back to more general nose eventually, don t stress about t, yet!) A natural choce s the bt flp channel we descrbed n the last lecture whch had the Kraus operator The crcut we want to evaluate s now A 0 = 1 pi A 1 = px (5) 3 ψ $ (6) 0 $ 0 $ where now ψ s now an arbtrary quantum state α 0 + β 1 whch we wsh to protect. So what does ths crcut do to our quantum data? Well after the frst two controlled-nots, we can see that the state s gven by α β (7) Notce that we have done somethng lke redundancy here: but we haven t coped the state, we ve just coped t n the computatonal bass. Next what happens? Well recall that we can nterpret $ as a bt flp error X happenng wth probablty p and nothng happenng wth probablty 1 p. It s useful to use ths nterpretaton to descrbe what wll happen next. In partcular t s useful to use ths nterpretaton to say that wth probablty (1 p) 3 no error occurred on all three qubts, wth probablty (1 p) p a sngle error occurred on the frst qubt, etc. So what happens f no error occur on our system (the Kraus operator I I I happens on our system.) Then we just need to run those two controlled-nots on our state (C X ) 13 (C X ) 1 (α β) = α β 100 = (α 0 + β 1 ) 00 (8) The Toffol then does nothng to ths state and we see that our quantum nformaton has survved. But ths sn t too surprsng snce no error occurred. What about when a sngle bt flp error occurs? Let s say an error occurs on the second qubt. Then our state s α β 101. The effect of the controlled-nots s then (C X ) 13 (C X ) 1 (α β 101 ) = α β 110 = (α 0 + β 1 ) 10 (9) Agan, the Toffol wll do nothng to ths state. And we see that our quantum nformaton has survved ts encounter wth the bt flp error! One can go through the other cases of a sngle bt flp error. In the case where the bt flp error s on the frst qubt, the Toffol s essental n correctng the error, but n the case where t s on the thrd qubt, then the Toffol does nothng. But n all three cases the quantum nformaton s restored! One can go through and check what happens for the cases where two or three bt flp errors occur. What ones fnds out that s n these cases the resultng state n the frst qubt s β 0 + α 1. Thus f we look at the effect of ths full crcut, t wll performs the evoluton ρ (1 p) 3 ρ (1 p) pρ (1 p) pρ (1 p) pρ (1 p)p XρX (1 p)p XρX (1 p)p XρX (1 p)p XρX (10) Tracng over the second and thrd qubt, ths ammounts to the evoluton ρ [(1 p) 3 + 3p(1 p) ]ρ + [3p (1 p) + p 3 ]XρX (11) If we compare ths wth the evoluton whch would have occurred gven no encodng, ρ (1 p)ρ + pxρx (1) we see that f p < 1, then our encodng acts to preserve state better than f we had not encoded the state. Actually how do we know that we have preserved the state better? What measure should we use to deduce ths and why would ths be a good measure? In partcular we mght note that quantum error wll effect dfferent states n dfferent ways. Thus we should probably worry about the worst case for an nput state here. We ll take a lttle break here to dscuss brefly ths queston.

4 4 A. Fdelty The fdelty between two densty matrces ρ and σ s defned as [ ] F (ρ, σ) = Tr ρ 1 σρ 1 (13) We wll be mostly concerned wth the fdelty between a densty matrx ρ and the case where σ s a pure state σ = ψ ψ, [ ] [ ] F ( ψ, ρ) = Tr ρ 1 ψ ψ ρ 1 = Tr ψ ψ ρ ψ ψ = ψ ρ ψ (14) The fdelty s a measure of how close two states are. It s equal to 1 ff ρ = σ. More mportantly, the fdelty can serve as an upper and lower bound on the trace dstance, D(ρ, σ), 1 F (ρ, σ) D(ρ, σ) 1 F (ρ, σ) (15) Well there I ve gone agan, talkng about the trace dstance wthout defnng what t s. The trace dstance between two densty matrces s D(ρ, σ) = 1 Tr [ ρ σ ] (16) where M = M M. The trace dstance s a beautful metrc on densty matrces, because t s related drectly to how dfferent a POVM on these states can be. In partcular, assume that we perform a POVM wth element E. Then on the two state ρ and σ, ths results n measurement probabltes p = P r( ρ) = Tr[ρE ] and q = P r( σ) = Tr[σE ]. The Kolmogorov dstance between the two probablty dstrbutons produced by these measurements s D(p, q) = 1 p q (17) Let s show that D(ρ, σ) s equal to the maxmum over all possble POVMs of the Kolmogorov between the resultng probablty dstrbutons. Note that D(p, q) = 1 Tr[E (ρ σ)] (18) Now decompose ρ σ, whch s hermtan nto a postve hermtan part, P and a negatve hermtan part N: ρ σ = P N and the support of P and N s orthogonal. Then D(p, q) = 1 Tr[E (P N)] 1 Tr[E (P + N)] = 1 Tr[E ρ σ ] = D(ρ, σ) (19) Thus D(p, q) D(ρ, σ). But now choose a POVM wth elements whch project along the egenvectors of P and N. Then D(p, q) = 1 Tr[E (ρ σ)] = 1 Tr[E (P N)] = 1 Tr[(P N)] = D(ρ, σ) (0) Thus D(p, q) s always less than D(ρ, σ) and n fact D(p, q) can attan D(ρ, σ). Thus we see that D(ρ, σ) s equal to the maxmum over all POVMS of the Kolmogorov dstance of the probabltes resultng from ths POVM. Back the fdelty. We ve seen that the fdelty can be used to bound the trace dstance whch s related to how bad our probabltes wll be for the worst POVM resultng from measurng these states. Thus the fdelty s a nce quantty to use to measure how close two states are. Now one queston s why not use the trace dstance. One reason s that the trace dstance s often hard to calculate. The other reason s that the fdelty between a densty matrx and a pure state s rather easy to calculate.

5 5 B. Back To The Regularly Scheduled Program So what happens to the fdelty of the two cases we had above, one n whch no error correcton was performed and one n whch error correcton was performed. In the frst case the fdelty, assumng we start n a pure state s F 1 = [ ψ [(1 p) ψ ψ + px ψ ψ X] ψ ] 1 = [ (1 p) + ψ X ψ ] 1 (1) Ths s mnmzed (remember we want hgh fdelty) when the ψ X ψ = 0 and s equal to Smlarly f we perform error correcton we obtan F 1 1 p () F 3 = [ ψ [ ((1 p) 3 + 3p(1 p) ) ψ ψ + (3p (1 p) + p 3 )X ψ ψ X ] ψ ] 1 = [ (1 p) + ψ X ψ ] 1 (3) whch s agan bounded by F 3 (1 p) 3 + 3p(1 p) (4) The fdelty s greater usng error correcton, then, when p < 1. So our nave analyss was not so much the dunce after all. IV. LESSONS What lessons should we draw from our frst success n quantum error correcton? One observaton s that nstead of copyng the quantum nformaton we encoded the quantum nformaton nto a subspace. In partcular, we have encoded nto the subspace spanned by { 000, },.e. we have encoded our quantum nformaton as α β. Ths s our way of gettng around the no-clonng theorem. The second problem we brought up was measurement. Somehow we have made a measurement such that we could fx our quantum data (f needed) usng the Tofoll. Let s examne what happens to our subspace bass elements, 000 and under the errors whch we could correct. Notce that they enact the evoluton I I I {}}{ 000 X I I {}}{ I X I {}}{ I I X {}}{ (5) Now thnk about what ths s dong. These error processes are mappng the subspace where we encoded the nformaton nto dfferent orthogonal subspaces for each of the dfferent errors. Further when ths map s performed, the orthogonalty between the bass elements s not changed (.e. 000 and are orthogonal and after the error occurs, they are mapped to states whch reman orthogonal.) Now ths second fact s nce, because t means that the quantum nformaton hasn t been dstorted n an rreversble fashon. And the frst fact s nce because, f we can measure whch subspace our error has taken us to, then we wll be able to fx the error by applyng the approprate operaton to reverse the error operaton. In partcular consder the operators S 1 = Z Z I and S = Z I Z. These operators square to dentty and so have egenvalues +1 and 1. In fact, we can see that these egenvalues don t dstngush between states wthn a subspace, but do dstngush whch of the four subspaces our state s n. That s to say, for example that 000 and are have egnevalues +1 for both S 1 and S. Further, the subspace whch occurs f a sngle bt flp occurs on our frst qubt, 100 and 011 have egenvalues 1 for S 1 and 1 for S. We can smlarly calculate the other cases:

6 S 1 S Error { 000, } I I I { 100, 011 } 1 1 X I I { 010, 101 } 1 +1 I X I { 001, 110 } +1 1 I I X Thus we see that f we could perform a measurement whch projects onto the +1 and 1 egenstates of S 1 and S, then we could use the results of ths measurement to dagnose whch subspace the error has taken us to and apply the approprate X operator to recover the orgnal subspace. So s t possble to measure S 1 and S? Well we ve already done t but n a destructve way n our crcut. Consder the followng crcut What does ths crcut do? Well f the nput to ths crcut s α 00 + β 11, then the measurement outcome wll be 0 and f the crcut s α 01 + β 10, then the measurement outcome s 1. Assocatng 0 wth +1 and 1 wth 1, we thus see that ths s equvalent to measurng the egenvalue of the operator Z Z. Notce, however, that ths s a destructve measurement,.e. t doesn t leave the subspace ntact after the measurement. In the crcut we have constructed, then, the controlled-nots after the error have enacted thngs whch, f we had measured the qubts, would have been the egenvalues of S 1 and S. Ths s enough to dagnose whch error as occurred. Snce ths also does decodng of our encoded quantum nformaton, only n the case where the error occurred on the frst qubt do we need to do anythng: and ths s the case where the measurement outcomes are both 1 and so we use the Toffol to correct ths error. Ths suggest that a dfferent way to mplement ths error correctng crcut and that s to measure the second and thrd qubts. Snce measurements commute through control gates turnng them nto classcal control operatons, we could thus have performed the followng crcut 6 (6) ψ $ X (7) 0 $ 0 $ What do we learn from the above analyss? We learn that quantum error correcton avods the fact that measurng dsturbs a quantum system by performng measurements whch project onto subspaces. These measurements do not dsturb the nformaton encoded nto the subspaces snce the measurement yeld degenerate values outcomes for any state n ths subspace. Ths technque, of performng measurements whch do not fully project onto a bass, s essental to beng able to perform quantum error correcton. V. DEALING WITH PHASE FLIPS Now the thrd problem we brought up was the fact that quantum errors for a contnuous set. For now, however, lets just move on to a dfferent error model. In partcular lets consder nstead of a bt flp model, a phase flp model. In ths model, the Kraus operators are gven by A 0 = 1 pi A 1 = pz (8) Now the effect of Z on a quantum state s to change the phase between 0 and 1. So how are we gong to correct ths error? It changes a phase, not an ampltude! Well we use the fact that phase changes are ampltude changes n a dfferent bass. In fact, we already know the bass change to perform. The Hadamard bass. Recall that HZH = X. Ths suggest that just pror to sendng our nformaton through the quantum channel and just after recevng the quantum nformaton we should apply Hadamard gates. Ths suggests the crcut ψ H $ H X (9) 0 H $ H 0 H $ H

7 Now wll ths work? Certanly t wll work, ths s just a bass change and the fdeltes wll be dentcal to the bt flp analyss. What does ths crcut do? Well nstead of encodng nto the subspace spanned by 000 and ths code encodes nto a subspace spanned by + ++ and, where ± = 1 ( 0 ± 1 ). Thus we see that by expandng our noton of encodng beyond encodng nto somethng smple lke the repeated computatonal bass states, we can deal wth a totally dfferent type of error, one that doesn t really have a classcal analogy n the computatonal bass. But we are just puttng off the queston of what happens when we have arbtrary errors. But notce somethng here. We sad n a prevous lecture that the bt flp error model was equvalent to the phase dampng model, and n that case our Kraus operators were [ 1 ] [ q ] [ ] q B 0 = B 1 = B = 0 1 q (30) q 7 When p = q, these were the same superoperator. We can express the above Kraus operators as B 0 = 1 qi, B 1 = q (I + Z), B = q (I Z) (31) Now surely, snce ths s the same superoperator, our analyss of the error correctng procedure wll be dentcal and we wll be able to ncrease the probablty of successfully correctng ths model gven q 1 4. But the B operators are sums of I and Z errors. Thus a code desgned to correct sngle Z errors seems to be workng on superoperators whch have Kraus operator s whch are sums of dentty and Z errors. Why s ths so? Well we have some encoded nformaton ψ. Then a Kraus operator whch s the sum of terms whch we can correct occurs (plus terms we can t correct.) Then we perform the measurement to dstngush whch subspace we our error has taken us to. But at ths pont, the Kraus operators whch are sum of error gets projected onto one of the error subspaces. Thus n effect the fact that the error s a sum of errors gets erased when we do ths projecton. We wll return to ths fact later, but ths s an mportant pont. Whle quantum errors may be a contnuous set, the error correctng procedure can, n effect, dgtze the errors and deal wth them as f they formed a dscrete set. VI. THE SHOR CODE So far we have dealt wth two models of errors, bt flp errors and phase flp errors. We have also seen that f a code corrects an error then t wll be able to correct Kraus operator errors whch have a sum of these errors. Thus we expect that f we can desgn a quantum error correctng code whch can correct a sngle X, Y, or Z error then ths can correct an arbtrary error on a sngle qubt. Indeed Peter Shor desgned just such a code (shortly after dscoverng hs factorng algorthm, genus!) How does ths code work? Well we ve already seen that f we encode nto the subspace spanned by 000 and, then we can correct a sngle bt flp error. What do sngle phase flp errors do to ths code? Well notce that Z I I 000 = I Z I 000 = I I Z 000 = 000 but Z I I = I Z I = I I Z =. Thus we see that, unlke bt flp errors, sngle phase flp errors on ths code act to dstort the nformaton encoded nto the subspace. But also notce that these sngle phase flp errors act lke phase flp errors on the encoded bass 000,. But we ve seen how to deal wth phase flp errors. Suppose we defne the states p = 1 ( ) m = 1 ( 000 ) (3) Then a sngle phase flp error on these two states sends p to m and vce versa,.e. t looks lke a bt flp on these qubts. We can thus deal wth these sngle phase flp errors by usng a bt flp code. In partcular defne the two nne qubt states 0 L = p p p = ppp = 1 ( ) ( ) ( ) 1 L = m m m = mmm = 1 ( 000 ) ( 000 ) ( 000 ) (33) Suppose we encode a qubt of quantum nformaton nto these the subspace spanned by these two states. Now sngle phase flp errors can be fxed by dagnosng what subspace the ppp and mmm subspace has been sent to. Further

8 sngle bt flp errors can be dealt wth from wthn each p and m state: these states are encoded states n our orgnal bt flp code. Puttng ths together, we see that we should be able to use ths error correctng code to correct bt flps and phase flps. Actually t can do more and ndeed can handle a sngle Y error as well To see ths lets construct the crcut for encodng nto ths code and then performng the decodng and correcton. 8 ψ H $ X H X (34) 0 $ 0 $ 0 H $ X H 0 $ 0 $ 0 H $ X H 0 $ 0 $ Now, frst of all, sn t ths beautful! Now notce the three blocks of three n ths code. Also notce how these blocks, when there s ether no error or a sngle X error n the superoperators (of course n general our superoperators wll each contan error terms, but we wll smply talk about the case where there s one such error snce ths s, to frst order, the most mportant error, assumng that the errors are weak enough. Yeah, all rather vague rght now, but we need to make progress wthout gettng bogged down n the detals...yet!) produces a channel on the outer three wres (just after the H and before the H) whch s dentty (because these are desgned to correct just that error. But what about f a sngle Z error occurs. As we noted above, ths means that on the encoded quantum nformaton, ths acts lke a Z error on the p, m states. Snce only a Z error occurs, we are not taken out of the p and m subspace by ths error, and so the effect of the error correcton n an nner block wll be to produce a state whch has an sngle qubt Z error on the outer wre. Now we see how the code corrects ths sngle Z error: ths s just the phase flp error correctng code! But what happens f, say a sngle Y = XZ error occurs? Frst notce that the global phase doesn t really matter. So we can assume that the error s XZ. Now the Z error actng on the encoded state, acts wthn the encoded space as a Z error. Thus magne performng ths error, and then runnng the bt flp code. The bt flp code wll correct the X error, but the end result wll be that a Z error has occurred on the encoded nformaton. But then the Z error wll be corrected by the outer code. Thus we see that a sngle Y error wll be corrected by ths code. So what have we n Shor s code? We have a code whch can correct any sngle qubt error from the set {X, Y, Z}. But as we have argued above, ths means that any sngle qubt error whch s a sum of these errors wll also be corrected (we wll make ths argument rgorous later.) So f a sngle arbtrary error occurs on our qubt, Shor s code wll correct t. We call ths a quantum error correctng code whch can correct a sngle qubt error. So what have we learned? We ve learned that s possble to crcumvent the objectons we rased early: clonng, measurement, and contnuous sets of errors, and to enact error correcton on quantum nformaton. What does ths mean for practcal mplementatons of a quantum computer? Well t means that more sophstcated versons of quantum error correcton mght be able to make a very robust computer out of many nosy components. There are many ssues whch we need to dscuss n ths theory, and the full story of how to buld such a quantum computer s the subject of fault-tolerant quantum computaton. A man am n the next lectures wll be to obtan an understandng of the theory of quantum error correcton, buldng up to the theory of fault-tolerant quantum computaton. Acknowledgments The dagrams n these notes were typeset usng Q-crcut a latex utlty wrtten by graduate student extraordnares Steve Flamma and Bryan Eastn.