Research Collection. Quantum error correction (QEC) Student Paper. ETH Library. Author(s): Baumann, Rainer. Publication Date: 2003

Transcription

1 Reserch Collection Student Pper Quntum error correction (QEC) Author(s): Bumnn, Riner Publiction Dte: 3 Permnent Link: Rights / License: In Copyright - Non-Commercil Use Permitted This pge ws generted utomticlly upon downlod from the ETH Zurich Reserch Collection. For more informtion plese consult the Terms of use. ETH Librry

2 Quntum Error Correction (QEC) Riner Bumnn, ETH Zurich 3,

3

4 Contents Contents... 3 Prefce Preliminry Topics..... Quntum computers nd qubits..... Quntum gtes Mesurement Shor s fctoring lgorithm..... Clssicl liner codes.... Bsics of quntum error correction..... Quntum chnnel..... Difficulties for quntum error correction Quntum error correction codes (QECC) Errors Properties of quntum codes Stbilizer formlism The bsics More formlly Unitry gtes nd the stbilizer formlism Further exmples Clderbnk-Shor-Stene (CSS) codes Stndrd form for stbilizer code Encoding, decoding, mesuring nd correcting Quntum error-correction without mesurement Fult-tolernt quntum computtion Bsics of fult-tolernt quntum computtion Fult-tolernt procedure Fult-tolernt gtes Fult-tolernt mesurement Fult-tolernt stte preprtion Conctented coding nd threshold theorem..... Conctented codes..... The threshold theorem Bounds Quntum Hmming bound Quntum Gilbert-Vrshmov bound Quntum singleton / Knill-Lflmme bound Simultions nd sttistics Trce distnce nd fidelity Simultions Comprison of bsic codes Summry of most importnt topics...8 A. Appendix...9 A.. Non-cloning theorem...9 B. Appendix - Quntum codes...9 C. Figure Tble...63 D. References...64 Riner Bumnn, ETH Zurich 3 Quntum Error Correction 3/66

5 Prefce Quntum error correction is young discipline rooting in the mid nineties. Due to lrge reserch efforts, tody s knowledge over this topic is lredy very profound. But it is still hot topic for scientists. This pper is not generl introduction in quntum computtion. It focuses on the spect of quntum error correction nd requires bsic understndings of quntum mechnics, clssicl computtion nd bit group theory. Anywy in the first prt very short, generl introduction to some preliminry topics is given. In this pper, I tried to use s much mthemtics s necessry for generl understnding without giving detiled mthemticl derivtion. Mny proofs nd mthemticl concepts would rther need n own pper for their discussion nd don t fit in here. This pper origintes from term project t the Swiss Federl Institute of Technology Zurich ETH ( Deprtment of Physics ( Institute for Theoreticl Physics, Prof. G. Bltter. I d like to thnk my sponsor Prof. G. Bltter for his support. I m grteful tht he hs given me the opportunity to write this work. Riner Bumnn,, 3 Riner Bumnn, ETH Zurich 3 Quntum Error Correction 4/66

6 . Preliminry Topics Computers becme very importnt tools in tody s world. They cn solve mny problems much fster thn ny humn being could do. But there re still few clsses of problems, which cnnot be solved efficiently on clssicl computers s for exmple the simultion of quntum mechnics or fctoring of lrge numbers. Moore s lw predicts tht computers get continuously fster nd smller. Alredy tody computer engineers fce the physicl limittions concerning the speed of light nd the size of toms. Tht s why new technologies for future computers re wnted. During his studies of quntum mechnics, Feynmn discovered tht quntum system of n spin-/ prticles is represented in n - dimensionl spce due to entnglement nd not in n-dimensionl spce s in the clssicl n bit problem. This fct led him to conjecture tht computer bsed on quntum mechnics might be much more power full thn clssicl one... Quntum computers nd qubits The memory of clssicl computer is n rry of bits ( or ). For exmple []. More formlly vector over the finite field : {,}. So n bits spn n-dimensionl vector spce over. While quntum bit, lso clled qubit, lies in, with the complex number spce (without normliztion). So n qubits re in n. The two bsis of re clled nd in nlogy to the vlues of clssicl bit. A single qubit is in the stte α + β with normliztion α + β. So normlized qubit lives in. But for simplicity we still write out α nd β. A qubit lso cn be represented in the stte vector form: α β A memory of three qubits would look like: α α α 3 β β β 3 Multiple qubits cn lso be entngled so tht they cnnot be decomposed into two seprte sttes. For exmple the EPR pir ( + ) cn not be written in terms of Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

7 ( α + β ) ( α + ) β α α + αβ + βα + ββ becuse α α nd β β must not be zero while α β nd β α hve to be zero. In this pper we denote clssicl bit with double line nd qubit with single line. Figure.. Bit representtion: clssicl bit (left), qubit (right) There is nother specilty in quntum sttes we hve to look t iθ briefly. Consider the stte e where is norml qubit stte iθ nd Θ rel number. We sy tht e re equl up to globl phse shift. We cn do this becuse the mesurements for them re bsolutely the sme. Assume generl mesurement opertor M m with probbility outcome m. The mesurement eqution would look like this: m M Tt m M m e iθ M t m M m e iθ Note tht especilly / i e iπ... Quntum gtes To perform clcultions on quntum computer, some gtes or opertions re needed s in clssicl computers. Such gtes re represented s unitry mtrices which cn be pplied on the vector spce, the qubits live in. For us the most importnt gtes re the Puli mtrices X,Y,Z, the Hdmrd gte H, the phse shift gte S, the π/8 gte T nd finlly the controlled-not CNOT. Lter on in this pper we will permnently mke use of this gtes nd there representtion so look t them very crefully! I ; X ; Y i ; Z ; i H S ; π /8 T i e iπ /4 ; CNOT Figure.. Quntum gtes Riner Bumnn, ETH Zurich 3 Quntum Error Correction 6/66

8 Riner Bumnn, ETH Zurich 3 Quntum Error Correction 7/66 In quntum circuit we represent CNOT nd quntum gte M in the following wy: Figure.3. Quntum gte representtion for circuits: CNOT (left) nd M gte (right) Note tht ll of these gtes operte only on one qubit except the CNOT. It is short notion of the controlled X gte. Similrly every rbitrry gte M cn be executed conditionlly (control qubits, blck point) on severl qubits (trgets, M-Box) on prllel. Figure.4. Multiple controlled M gtes (control qubit +, trget qubits 3+4) Lets hve look t the chrcteristics of some of these gtes to for getting fmilir with them. The identity I simply lets qubit unchnged b b I. The Puli X gte is lso clled the flip gte becuse it flips the bsis b b X. The Puli Z gte cuses phse flip b b Z. The Hdmrd gte brings bse stte in superposition + b b b H.

9 Riner Bumnn, ETH Zurich 3 Quntum Error Correction 8/66 It s nice to see tht X H nd Z S. When you look t the eqution below you ll understnd why the T gte is lso clled the π/8 gte. T e e e i i i 4 / 8 / 8 / 8 / π π π π Now we wnt to hve look t gtes cting on two qubits. The most populr is the controlled-not which flips the trget qubit if the control qubit is set. According to clssicl bits, we cll qubit set if it is in the bse stte nd not set if it is in the bse stte. b b b b b b b b b b CNOT CNOT Putting ll together we re now ble to construct circuit which prepres entngled EPR pirs. EPR Figure.. Quntum circuit to crete entngled EPR stte ( ) + + b b b b b b b b b b b b b I H CNOT For concrete EPR stte this would look like: ( ) + I H CNOT EPR Nice to know is lso tht X I nd XZY up to globl phse shift. Y i XZ

10 Often it is good to write out the bsis to understnd wht hppens. Below you see the bsis of, nd 3 qubit system. bse ; bse ; bse 3 In multiple qubit systems, we often simply write the index of the qubits lowered right to the gte insted of lwys writing on which qubit we pply certin gte. Assume we hve three qubit system. For pplying X on the second qubit, we simply write X s short form for I X I.... Reversible quntum gtes nd irreversible clssicl gtes Mthemticlly quntum gte is described s unitry mtrix. Since every unitry mtrix hs n inverse, every clcultion on quntum computer must be reversible. Note tht this is different to clssicl computer where most gtes re irreversible. For exmple logicl XOR ( XOR b c) is not reversible, becuse it is impossible to determine the vlues of nd b from c. So the question comes up how to implement this irreversible gtes in reversible mnner? This problem cn be solved very esily by just keeping one of the originl vlues. For exmple to clculte XOR we cn use the following circuit: A A B A B Figure.6. Reversible XOR circuit Riner Bumnn, ETH Zurich 3 Quntum Error Correction 9/66

11 ... Universl quntum gtes In clssicl computers ll logic cn be build up from smll set of so clled universl gtes or even from one single (e.g. NAND). This is not surprising, s the truth tble of clssicl gte hs finite combintions of outputs. This is very importnt for hrdwre construction. If we re ble to implement such set of universl gtes we cn compute wht ever we wnt. But in quntum computtion, infinitely mny different gtes exist. Implementing ll of them is impossible. But fortuntely, it turns out tht lso in quntum computtion sets of universl quntum gtes exist which cn pproximte every other quntum gte with n rbitrry smll error. This first set consists of the CNOT, Hdmrd nd T gtes. The second set consists of the CNOT, Hdmrd, phse nd Toffoli gtes.(the Toffoli gte is CNOT with two control qubits nd one trget qubit.) We will see in section four tht ll of this gtes cn be implemented fult-tolerntly. The Hdmrd nd T gte re used to pproximte ny single qubit unitry opertion. The CNOT llows us to pproximte multiple qubit unitry opertions. A detiled proof of this would relly go beyond the scope of this pper nd cn be found in the book of Nielsen nd Chung. The min ide is tht we hve to construct rottion by n irrtionl multiple of π. Combining T nd HTH let us construct rottion on the Bloch sphere long the xis π n r cos, 8 sin π, 8 π cos through the ngle 8 π Θ cos : 8 T * HTH e π π i Z i X 8 8 π * e cos π π π I i cos ( ) sin sin 8 X + Z + Y Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

12 .3. Mesurement The vlue of clssicl bit is lwys deterministiclly or. When mesuring single qubit in n rbitrry stte, the probbility of the outcome is α nd for β. Such mesurement is done by Hermitin projection on the bsis sttes,. So mesurement is represented by Hermitin mtrices. So we get clssicl bit out of mesurement. But the xioms of quntum mechnics tells us tht fter the mesurement n rbitrry qubit is collpsed in the mesured bses stte. So we destroy qubit with simple mesurement. In quntum circuit we use the following symbol: Figure.7. Symbol for mesurement We often omit the two lines representing the clssicl bit t the right of mesurement becuse they normlly do not interct gin with the rest of circuit. Concerning generl opertor M, which is Hermitin nd unitry with eigenvlues ±, the following circuit cn be used for mesuring its eigenvlues without completely destroying the mesured qubit. After mesurement, the qubit will be in the corresponding eigenvector. This is very importnt trick, but note, tht we do not gin ny informtion bout α or β. Φ Φ Φ3 Φ 4 Figure.8. Mesurement of generl opertor M Lets verify this for the Puli mtrix Z. It is obvious tht Z is hermitin, unitry mtrix with eigenvlues ±. Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

13 Φ Φ Φ Φ b ( + ) ( + + b + b ) ( + + b b ) ( ( + ) + ( ) + b( + ) b( ) ) ( ( + + b b ) + ( + b + b ) ( + b ) Mesuring the first qubit, let us now determine the eigenvlue of Z leving in the corresponding eigenstte..4. Shor s fctoring lgorithm For long time it ws not cler how to use the power of quntum computers. The mjor problem ws the collpse of coherent sttes during mesurements of generl qubit. It seemed impossible to get out detiled result of quntum computtion. Peter Shor published in 994 surprising lgorithm for the fctoriztion of lrge numbers, which hndled this problem in very clever wy. Most of modern public cryptogrphy is bsed on the ssumption tht the fctoriztion of lrge numbers is extremely difficult. But the new lgorithm would llow solving this problem in polynomil time. This ws gret improvement. For the interested reder, there re mny good introduction to this lgorithm so we won t disuse it in here... Clssicl liner codes Liner codes hve been developed for mking clssicl communiction fult tolernt. With them it is possible to communicte over noisy nd erroneous dt chnnels. Encoding is used very often, becuse in relity there is no errorless communiction. Fmous exmples for such communictions re telecommuniction (fix net or mobile), modem connections to the Internet, Ethernet networks. The mjor ide is to encode k dt bits in n bits. The n-k bits re used for introducing dditionl informtion, the so clled redundncy. This redundncy cn be used to detect bit flips occurred during trnsmission. Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

14 Assume we wnt to send the dt bit x. We encode this with the genertor mtrix G by multipliction. This results in the code word c. () x ; G ; x G c A code which encodes k dt bits in n bits is clled [n,k] code. So the code we introduced bove is [3,] code. It simply repets the dt bit three times. We cll it the three-bit repetition code. This encoded bit c cn now be trnsmitted over communiction chnnel. Assume tht during delivery the second bit hs been flipped by the error e. So we ll receive r. e ; c + e r For the correction of the received word we need two more things. The so clled prity check mtrix H nd the syndrome decoder tble. The prity check mtrix is defined in the following wy. When you multiply ny vlid codeword c with the trnsposed of the prity check mtrix H you get zero. c H T There is stright forwrd wy getting the prity check mtrix from the genertor mtrix, which cn be found in every clssicl coding book. When you solve the sme eqution for received word r you ll get the syndrome s, which llows mking drwbcks on the resulted error. s (c+e) H T (r) H T c H T +e H T +e H T e H T Given the prity check mtrix for our exmple we cn clculte the syndrome. T H ; r H s ; Syndrome Error Pttern Riner Bumnn, ETH Zurich 3 Quntum Error Correction 3/66

15 Riner Bumnn, ETH Zurich 3 Quntum Error Correction 4/66 With look t the syndrome tble we find the occurred error e. We now hve to subtrct this error from r getting c. Finlly we hve to decode c by the genertor mtrix getting bck our originl dt bit. c e r ; () G c For the interested reder the clcultion of nother exmple is given below, the [,] code. x ; G ; c G x ; r ; H ; s H r T ; e ; c e r ; G c Syndrome Error Pttern

16 . Bsics of quntum error correction.. Quntum chnnel In the lst prgrph we herd bout clssicl communiction chnnel. Lets hve look t the quntum nlogue. A quntum chnnel is connection over which qubits cn be sent nd coherence is preserved. In rel quntum chnnel lwys some error occurs. An input stte cn be chnged or even worse cn be entngled with the environment. For correcting such errors we need to introduce some dditionl informtion which let us figure out wht hppened, the so clled redundncy. In trnsmission, often redundncy is encoded for severl qubits together. For this purpose we need quntum error correction codes (QECC). Also for stored qubits we need certin redundncy becuse quntum memory is much more vulnerble to errors thn clssicl one. So it is self-evident to protect qubits by encoding them s in communiction... Difficulties for quntum error correction Now, one would think tht we just hve to tke the clssicl liner codes nd every thing gets fine. But there re three rther big difficulties to del with: No cloning theorem: Most clssicl codes require the possibility to copy bits s the simple repetition code, discussed bove, which just sends the sme bit severl times. Due to non-cloning we re not ble to do such copies of qubits. (See Appendix A. for the No cloning theorem) Errors re continuous: Clssicl errors re discrete, only bit flip cn occur. But quntum errors re continuous. A qubit cn be in ny superposition of the two llowed bses sttes. Determining which error occurred in order to correct would pper to require infinite precision, nd therefore infinite resources. Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

17 Mesurement destroys quntum informtion: In clssicl error correction we simply observe the vlue of the output dt bits to decide how to correct nd decode it. But in quntum mechnics such n observtion destroys the coherence of stte nd lets it collpse in bsis stte of the mesurement opertor. So we hve to find other wys for error detection nd correction. Becuse of these problems it ws believed for long time tht quntum computtion will never scle. Fortuntely none of these problems is very serious s we will see in this chpter..3. Quntum error correction codes (QECC) A quntum error correction code consists of four mjor steps. Assume you wnt to send sequence of dt qubits over quntum chnnel. Then you split this sequence in fix sized sets of qubits. In the figure below the size is two. These sets re encoded independently resulting in codeword. There re more qubits in codeword thn dt bits in the originl set. This codewords cn then be trnsmitted over the quntum chnnel. We ssume tht potentil errors cn only occur during this trnsmission. When you receive the messge you do some error detection nd potentilly lso some error recovery. These two phses together re simply clled error correction. The correct codeword cn then be send through decoder resulting in the originl set of dt bits. potentil errors Figure.. The mjor elements of quntum error correction codes The best thing for getting first understnding of the mjor elements of quntum error correction code is to look t some simple codes. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 6/66

18 .3.. Three-qubit flip code Assume we wnt to send single qubit + b through quntum chnnel. During trnsmission the qubit cn be flipped to X b + with probbility p. Such quntum chnnel is clled bit flip chnnel. Note tht qubit flip is the sme s you pply the Puli X gte to the qubit. + b + b X b + Figure.. Quntum bit flip chnnel We now cn try to use the sme ide s in the three bit repetition code from the lst chpter. We just send the qubit three times over the quntum chnnel. L denotes the originl (logicl) qubit to be encoded. The encoding for the two bse (pure) sttes re the following: L L For n rbitrry qubit b we get the encoding + b. L + But how to do this encoding without the possibility of qubit copying. The following circuit does this encoding. The two sttes re so clled ncills prepred in stndrd stte. Ancills re dditionl qubits generlly introducing the redundncy. Figure.3. Three-qubit flip code - encoding circuit We write out gin the bsis of three qubit system. The esiest wy to understnd the clcultions is in term of bit flipping. Note tht the first bit is the control bit. If it is set it simply interchnges the trget sttes. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 7/66

19 Riner Bumnn, ETH Zurich 3 Quntum Error Correction 8/66 3 bse ; 3 b b,3 3,,3 b CNOT CNOT CNOT b b Or just in non-vector form: ( ) ( ) 3 3 b b CNOT b b CNOT b After encoding, we now cn pss ech qubit independently over the described bit flip chnnel. With the used encoding, we re cpble to correct single or non qubit flip. This occurs with probbility ( ) ( ) p p p p p + +. So the probbility of n error remining uncorrected or wrong corrected with this code is 3 3 p p.

20 Assume now, tht during trnsmission bit flip occurred on the first qubit resulting in r + b. The error correction consists of the detection nd recovery phse. Detection: We perform mesurement with projection opertor P to determine the error syndrome. P + no error P + bit flip on qubit one P + bit flip on qubit two P 3 + bit flip on qubit three Note tht mesurements in these bsis do not chnge the stte becuse they re orthogonl. For our received codeword syndrome mesurements: r we get the following r r r r P P P P 3 r r r r Note tht the syndrome only contins informtion bout wht error occurred nd does not llow us to infer nything bout or b. Recovery: The syndrome tells us if nd how to correct the received stte. In our exmple the syndrome tells us, tht the first qubit ws flipped. We simply flip it bck by pplying X to it. This ll sounds very nice. But in the section bout mesurement, we lerned tht in relity we re not ble to implement this mesurement without destroying the mesured qubit. So we somehow hve to find other bsis which provides us with the sme informtion nd but which cn esily be implemented. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 9/66

21 Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66 Insted of mesuring ech single qubit with the four projectors P, P, P, P 3 we could perform two mesurements on the observble Z Z ( Z Z I ) nd Z Z 3 ( I Z Z ). Ech of them hve eigenvlue ±. The corresponding eigenvectors re the two vlid codewords nd. So we cn use the mesurement circuit introduced in the section bout mesurements. Z Z Z 3 Z We cn understnd the observble Z Z s comprison of the first two qubits. The eigenvlue + indictes they re the sme nd tht they re different. Like this we cn construct the following syndrome (right) tble nd mesurement bsis (left). Z Z + ( ) I + ( ) I Z Z 3 I + ( ) I + ( ) For our exmple we get: 3 + r r r r Z Z Z Z When we look up this error in the syndrome tble we find: (-,+) -> X Z Z Z Z 3 Error + + I + - X X - - X

22 As described in the lst section, mesurements usully re done with ncills. For the observble Z Z, Z Z 3 the mesurement nd correction circuit is shown below. The three upper lines denote the encoded qubit block. In our figures, we will not mrk them specilly. Figure.4. Three-qubit flip code - error detection nd correction circuit with ZZ We could even simplify this gte. Insted of the Z Z, Z Z 3 mesurements, we could perform two controlled-nots on the sme trget ncill but with different control qubit. If the two control qubits re the sme, the ncill either stys in the stte or gets flipped twice resulting lso in the stte. Like this we re ble to compre two qubits s in the circuit bove. Figure.. Three-qubit flip code - error correction circuit with CNOT The syndrome tble stys the sme: M M Error I X X X 3 Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

23 For our exmple we get: CNOT4 CNOT CNOT CNOT 4 3 r r + When we look up this error in the syndrome tble we find (-,+) - > X, wht we expected. Now the decoding circuit is esy. We just reverse the encoding circuit: Figure.6. Three-qubit flip code - decoding circuit.3.. Three-qubit phse flip code Let s hve look t different kind of error, which is not known from the clssicl theory, the phse flip error. Z Z ( + b ) b A quntum chnnel on which phse flip occurs with probbility p is clled phse flip chnnel. + b + b X b Figure.7. Quntum phse flip chnnel There is very esy wy to protect ginst phse flips. Suppose we work in the qubit bsis: ( ) + + ( ; ) In this bsis the phse flip opertor Z cts s n ordinry bit flip opertor nd we cn use the ordinry bit flip error correction code. The lterntive syndrome mesurement for this code cn be described s following H 3 Z Z H 3 X X nd H 3 Z Z 3 H 3 X X 3. Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

24 The encoding circuit is described below. The three Hdmrds t the right re responsible for the trnsformtion of the bsis. Figure.8. Three-qubit phse flip code - encoding circuit The code qubits cn now be pssed independently over the quntum phse flip chnnel. For the error correction circuit we just hve to convert our stte bck to the stndrd bsis using the Hdmrds gin. There we cn pply the sme circuit s in the three-qubit flip code. Also the syndrome tble stys the sme. Figure.9. Three-qubit phse flip code - error correction circuit with CNOT The decoding circuit lso stys the sme s in the three-qubit flip code becuse we hve lredy switched bck the bses by pplying the Hdmrds in the error correction circuit. Lets hve look t the clcultion of concrete exmple. We first encode our qubit: L L 3/ 4 / Riner Bumnn, ETH Zurich 3 Quntum Error Correction 3/66

25 Riner Bumnn, ETH Zurich 3 Quntum Error Correction 4/66 / 7 / 7 / / 7 / / / 7 / 4 / 3/ H H H CNOT CNOT H H H L c Assume now, the second qubit is ffected by phse flip error getting: / 7 / 7 / / 7 / / / 7 / / 7 / 7 / / 7 / / / 7 / Z c r Now we pply the Hdmrds to convert our received word bck in the norml bsis nd mesure it s in the three-qubit flip code. 4 / 3/ / 7 / 7 / / 7 / / / 7 / 3 3 H H H H H H r Hr Hr Hr CNOT CNOT CNOT CNOT The syndrome tble tells us tht qubit two hs been phse flipped. Now we hve to phse flip bck the qubit two. This would normlly be done by Z but we lredy chnged the bsis nd in this bsis

26 Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66 phse flip corresponds to n ordinry flip. So we simply pply X nd we get the originl code word. 4 / 3/ 4 / 3/ Hr X.3.3. Shor s nine-qubit code We now know codes which protects ginst n ordinry qubit flip or phse flip. Let us try to mke simple code out of them, which protects ginst both of these errors nd even the combintion of them. The min ide is to encode first the qubit using the phse flip lgorithm nd then to encode ech of the three resulted qubits gin with the common bit flip code. This is Shor s nine-qubit code. The two bse code words re the following: ( )( )( ) ( ) ( )( )( ) ( ) ) ( ) ( L L As lredy mentioned, this code even protects mong the combintion of the errors X Z Y.

27 The encoding circuit visulizes the used ide very well. At the left you see the three-qubit phse flip encoding circuit nd t the right three times the three-qubit bit flip encoding circuits. Figure.. Nine-qubit code encoding circuit The technique we used to build Shor s nine-qubit code out of the two previous codes is clled conctention. In conctention codes re cscded to build better codes. We will hve deeper look t this technique in lter section. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 6/66

28 The error correction nd decoding circuit for the nine-qubit code lso conctentes the lredy known circuits from the three-qubit codes. From left to right the three-qubit phse flip error correction circuit, the three-qubit phse flip decoding, the three-qubit bit flip error correction circuit nd finlly the three-qubit bit flip decoding circuit. Figure.. Nine-qubit code error correction nd decoding circuit As for previous codes we cn find n lterntive syndrome mesurement bsis. The pirwise comprison for flip detection cn be done in the following bsis: Z Z ; Z Z 3 ; Z 4 Z ; Z Z 6 ; Z 7 Z 8 ; Z 8 Z 9. For the phse flip we hve to compre two tripples: X X X 3 X 4 X X 6 ; X 4 X X 6 X 7 X 8 X 9. These bses will be importnt in further generliztion, in the stbilizer mechnism. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 7/66

29 .4. Errors Let s summrize ll the errors we lredy know: I no error X - bit flip Z phse flip Y bit nd phse flip But there re infinitely mny other errors, which could occur. For exmple qubit cn be chnged by n error E like: E 3 + b 3 Surprisingly these continuum errors cn be reduced to the discrete set of errors (I,X,Z,Y). Every generl error E cn be described s superposition of this discrete set: E e I + e X + e3z + e4y This is chrcteristic of the Puli mtrices. Lets decompose very nice error: E I + Z This lone won t help much. Now we mesure the error syndrome, in the bsis of the discrete error set (I,X,Z,Y). With the mesurement the stte collpses the superposition of errors into one of the four sttes. Like this we hve reduced n rbitrry error to one of our four bsic ones, from which we cn esily correct the error. Lets look t very simple exmple for getting the feeling wht this mens. We use the three-qubit flip code. The encoded qubit would look like: + Assume very nsty error E dultertes the encoded qubit getting: E ; E X + X 3 4 Riner Bumnn, ETH Zurich 3 Quntum Error Correction 8/66

30 Let s clculte wht hppens step by step Figure.. Three-qubit code error correction circuit ( + ) + ( + ) 4 The two lower qubits re in the superposition: + Mesuring these two qubits collpses the superposition in either or. Becuse of this the uncorrectble stte lso collpses in correctble one. Assume we mesure M nd M collpses into the following stte: 6 + Riner Bumnn, ETH Zurich 3 Quntum Error Correction 9/66

31 This stte cn then esily be corrected. A look t the syndrome tble tells us tht we just hve to flip the third qubit with X 3 bck resulting in: Properties of quntum codes Codes re generlly chrcterized by the triple [n,k,d], n>k n length of resulting code word k number of qubits to be encoded d minimum distnce An error on system of qubits cn be decomposed in errors on single qubits. In simple terms, the weight of n error is the number of such single errors it consists out. A code with miniml distnce d is ble to detect ll errors of weight (d-). After n error is detected, correction is not difficult thing ny more. The three-qubit flip nd phse flip code re both [3,,] codes. Shor s nine-qubit code is [9,,3] code. A quntum error-correcting code cn be viewed s mpping of k qubits ( k dimensionl Hilbert spce) into n qubits ( n dimensionl Hilbert spce). The dditionl (n-k) qubits provide the redundncy. In order for code to correct two errors E nd E b, we must lwys be ble to distinguish them cting on every combintion of two different bsis codewords. So for E i nd E b j to be distinguishble, they must be orthogonl. i E T E b j δ b δ ij Riner Bumnn, ETH Zurich 3 Quntum Error Correction 3/66

32 3. Stbilizer formlism The stbilizer formlism is very elegnt nd efficient wy to describe most quntum error correction codes. It s prtilly bsed on group theory. Best for fmilirizing ourselves with this topic is n exmple. 3.. The bsics Actully we lredy met the bsics of the stbilizer formlism in the lst section. Remember the lterntive mesurement bsis for error detection of the nine-qubit code: Z Z ; Z Z 3 ; Z 4 Z ; Z Z 6 ; Z 7 Z 8 ; Z 8 Z 9 ; X X X 3 X 4 X X 6 ; X 4 X X 6 X 7 X 8 X 9. Lets write them in tble where every column represents qubit nd every row bsis clled M i M Z Z I I I I I I I Z Z M I Z Z I I I I I I Z Z 3 M 3 I I I Z Z I I I I Z 4 Z M 4 I I I I Z Z I I I Z Z 6 M I I I I I I Z Z I Z 7 Z 8 M 6 I I I I I I I Z Z Z 8 Z 9 M 7 X X X X X X I I I X X X 3 X 4 X X 6 M 8 I I I X X X X X X X 4 X X 6 X 7 X 8 X 9 Figure 3.. Extended stbilizer tble for the nine-qubit code This is lredy the stbilizer tble for the nine-qubit code. We now wnt to understnd the centrl insight of the stbilizer formlism. It is esily illustrted by n exmple. Consider the EPR stte: + It is obvious tht this stte stisfies the identities X X nd Z Z. We sy tht is stbilized by the opertors Z Z nd X X. Or more formlly, n opertor fixes stte if the stte is n eigenvector with eigenvlue + of this opertor. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 3/66

33 X X / / / / Z Z / / / / A little less obvious is tht is the unique stte which is stbilized by the two opertors. This is n effect from liner lgebr. Two different 4-dimensionl mtrices with eigenvlues ± hve only one common eigenvector with eigenvlue +. The bsic ide of the stbilizer formlism is now tht quntum sttes cn be more esily described by working with the opertors tht stbilizes them thn by working explicitly with the stte itself. Coming bck to the nine-qubit code, we find tht lso the two vlid bse codewords L ; L of the code re eigenvectors of ll eight opertors M to M 8 with eigenvlue +. L ( ) L ( + + ) All the opertors tht fix both bse codewords L ; L cn be written s the product of the eight opertors M i. The set of opertors tht fix L ; L form group S, clled stbilizer of the code. The M i s re the genertors g i of this group. Depending if we focus on the genertor or the opertor spect of stbilizer we will denote it by M i or g i interchngeble. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 3/66

34 As we herd in the previous section, the eigenvlue mesurement of the opertor M, let us determine if bit flip error hs occurred on either qubit or. We sw tht this qubit flip errors cn be represented by the opertors X nd X. Note tht both of these errors X nd X nticommute with M while ll other errors, which cnnot be detected by M, commute with it. { M X } -> M X X M Or more generl:, [ X ], 3 X M -> M X + X M + { i, E j } [, ] X M -> M i E j E j M i E j M -> M E + E M + E i E j i j j i j 3.. More formlly The Puli group G n on n qubits is defined to consist of ll the Puli mtrices, together with the multiplictive fctors ±, ±i. We write out here just the simplest G Puli group over one qubit: G { ±I, ±ii, ±X, ±ix, ±Y, ±iy, ±Z, ±iz } This set forms group under mtrix multipliction. The stbilizer S is n belin subgroup of G n. S defines the set of sttes V s to be the set of n qubit sttes which re fixed by every element of S. V s is the vector spce stbilized by S. Normlly S is described by its genertors personlized by the stbilizer tble seen bove. Note tht not ny subgroup G n cn be used s stbilizer. There re two importnt conditions: The elements of S commute with ech other -I is not n element of S (Becuse ( I) hs only the trivil solution ) It mkes sense tht the genertors of S re independent so tht S is s smll s possible. The check mtrix formlism could be used to chieve this. So [n,k] stbilizer code is defined to be the vector spce V s stbilized by S<g,, g n-k >. In order to perform error-detection opertions for stbilizer code, ll we need to do is mesuring the eigenvlue of ech genertor of the stbilizer. This mesurement thn uniquely identifies the syndrome from which cn conclude on the error occurred. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 33/66

35 3.3. Unitry gtes nd the stbilizer formlism Let s nlyze wht hppens when unitry gte U is pplied to vector spce V s stbilized by S. Let be ny element of V s. Then for ny element g of the stbilizer group S, U Ug UgU + U nd thus the stte U is U gte CNOT g IN X X Z g OUT X X X Z stbilized by UgU +. Group theory resons tht for computing the chnges of the stbilizer, we only need to compute how it ffects the genertors of the stbilizer. In the tble t the right you see list of such trnsformtions. Any unitry opertion tking n element from G n to n elements of G n under conjugtion cn be decomposed in H, S nd CNOT gtes. We sy the set of U such tht UG n U + G n is the normlizer of G n denoted by N(G n ). With this formlism we cn define the distnce of stbilizer code to be the minimum weight of n element of N(S)-S. H S X Y Z Z X Z X Z X Z X Z X Z Z Z Z X Y Z X -Z -X -Z -X Z Figure 3.. Gte trnsformtion tble under unitry opertions (right) Lets check this for two concrete exmples: H T XH Z H T ZH X Riner Bumnn, ETH Zurich 3 Quntum Error Correction 34/66

36 3.4. Further exmples The three-qubit flip code [3,,] As developed bove, the three-qubit code consists of the genertors t the right. X nd Z re the X nd Z opertors trnsformed to the code spce. They re usully lso included in the stbilizer tble becuse they lso belong to N(S) mong with g nd g. Error recovery for this code is done s lredy explined in previous section. g Z Z I g I Z Z X X X X Z Z Z Z Figure 3.3. Stbilizer tble for the three-qubit code (right) The nine-qubit Code [9,,3] Here gin the complete stbilizer tble for Shor s nine-qubit code: g Z Z I I I I I I I g I Z Z I I I I I I g 3 I I I Z Z I I I I g 4 I I I I Z Z I I I g I I I I I I Z Z I g 6 I I I I I I I Z Z g 7 X X X X X X I I I g 8 I I I X X X X X X X X X X X X X X X X Z Z Z Z Z Z Z Z Z Z Figure 3.4. Stbilizer tble for the nine-qubit code The five-qubit code [,,3] But wht is the minimum size of quntum code, which protects single qubit mong ny single errors. As we will see in lter section the nswer is the five-qubit code. You might think now, tht this is the most importnt code, but due to its complexity it is not used very often. Let s hve look t its the stbilizer tble: g X Z Z X I g I X Z Z X g 3 X I X Z Z g 4 Z X I X Z X X X X X X Z Z Z Z Z Z Figure 3.. Stbilizer tble for the five-qubit code Riner Bumnn, ETH Zurich 3 Quntum Error Correction 3/66

37 From our knowledge bout the genertors we cn build the unnormlized bse codewords very esily. For L we just mke summtion over ll elements of the stbilizer groups S multiplied with. For L we just trnsfer L by X which flips every qubit. The bse code words of the five-qubit looks thn like: L M M S + M + M + M 3 + M 4 +M M + M M 3 + M M 4 + M M 3 +M M 4 + M 3 M 4 + M M M 3 +M M M 4 + M M 3 M 4 + M M 3 M 4 +M M M 3 M L X L The very complex encoding circuit of the five-qubit code is presented below. Understnding this circuit would requires dozens of pges of clcultions. So just enjoy hving look t it: Figure 3.6. Five-qubit code encoding circuit Riner Bumnn, ETH Zurich 3 Quntum Error Correction 36/66

38 3.. Clderbnk-Shor-Stene (CSS) codes The Clderbnk-Shor-Sten codes or simply CSS codes re subclss of the stbilizer codes. They llow to construct quntum error correction codes from clssicl liner codes. Tking clssicl prity check mtrix P, we cn construct quntum code, correcting X errors, just by replcing every by Z nd putting I s elsewhere. The sme cn be done with second prity check mtrix Q. To detect Z errors, we just hve to plce X s insted of Z s in the genertors. We cn combine these codes if the derived genertors commute. This is true if the rows of P nd Q re orthogonl. This mens tht the dul code of ech code must be subset of the other code. Such combined code of P [n,k,d ] nd Q [n,k,d ] is CSS(P,Q) [n, k -k,d 3 ] code with d 3 min{d,d } The Stene seven-qubit CSS code [7,,3] The most populr exmple of CSS code is the seven-qubit Stene code. It is creted from the clssicl Hmming code [7,4,3] which is self dul. An explntion of clssicl Hmming codes nd self dulity is behind the scope of this pper but cn be find in every good book bout clssicl liner codes. For our exmple we tke for P the prity check mtrix H: H And for Q the trnsposed of its genertor G: G T Riner Bumnn, ETH Zurich 3 Quntum Error Correction 37/66

39 After reducing ll dependent genertors nd bringing them up in stndrd form we end up with the following stbilizer tble nd bse codewords: L L Figure 3.7. Stbilizer tble for the seven-qubit code (right) The encoding circuits looks like this: g I I I X X X X g I X X I I X X g 3 X I X I X I X g 4 I I I Z Z Z Z g I Z Z I I Z Z g 6 Z I Z I Z I Z X X X X X X X X Z Z Z Z Z Z Z Z Figure 3.8. Sten seven-qubit code ( CSS code) encoding circuit Riner Bumnn, ETH Zurich 3 Quntum Error Correction 38/66

40 Error correction for CSS codes is very esy becuse the genertors g i does not mix up X s nd Z s. So we cn correct the flip nd phse error seprtely with the sme correction circuit in different bsis. Figure 3.9. CSS code error correction circuit Figure 3.. Sten seven-qubit code error correction circuit Riner Bumnn, ETH Zurich 3 Quntum Error Correction 39/66

41 3.6. Stndrd form for stbilizer code Probbly you lredy noticed tht the choice of genertors is not t ll unique. We cn lwys replce genertor g i with g i g j for some other genertor g j. The gol of this prgrph is to develop unique representtion of the stbilizer. But first we hve to lern how to write stbilizer in binry vector form. Till now, we wrote ll the X s nd Z s in the sme mtrix. Now we seprte them in two seprte ones, writing them next to ech other delminting them by line in the middle. Then we replce ll the I s nd empty plces with s nd the X s nd Z s with s. For the five-qubit code this would look like this: To bring such binry vector in stndrd form, we first hve to bring it in the following shpe with Gussin elimintion: I A B D C E For our exmple this would look like: Then we perform nother Gussin elimintion on E to get: I A A B C C D I E D Our exmple does not need ny further elimintion becuse D nd E do not exist. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 4/66

42 The genertors in the first row will not commute with them in the third unless D, which relly implies tht S. So we cn omit the lst line finlly getting the stndrd form. I A A B C C D I E 3.7. Encoding, decoding, mesuring nd correcting One of the gret dvntges of the stndrd form of the stbilizer formlism is the bility for systemtic construction for encoding nd decoding Encoding nd decoding For this systemtic pproch we need the so clled Z nd X opertions. We hve to pick k opertors independently of the genertors commuting with them nd ll other opertors. But they cn esily be determined from the stndrd form: ( ) ( ) Z A T I X E T IC T We cn now prepre the system in n n stte, mesure ll observbles g,,g n-k,z,,z k nd fix the system with the Puli opertors. Once this is prepred we cn chnge it to ny rbitrry encoded computtionl bsis stte. There is lso much more complex wy how such n encoding cn be done directly, which we will not discus. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 4/66

43 3.7.. Mesuring nd correcting As in section. explined, there exists clever gte, mesuring single qubit opertors M with eigenvlues ±, using controlled-m opertions, two Hdmrd gtes nd n ncill. Hving this eigenvlues it is not gret thing nymore to mke the corrections with the corresponding Puli mtrices. So to mesure the seven-qubit Sten code you cn use the following circuit: Figure 3.. Sten seven-qubit code utomtic constructed error detection circuit Riner Bumnn, ETH Zurich 3 Quntum Error Correction 4/66

44 3.8. Quntum error-correction without mesurement Till now, for error-correction, we expected to mesure n opertor M i nd then executing conditionlly the unitry opertor U i. But it is lso possible, just to do this correction by unitry opertions nd ncills without ny mesurement. This cn be very useful in some systems when mesurements re very undesirble. Prepre n ncill system with bsis stte i corresponding to the possible error syndromes. The ncill strts in stndrd pure stte before error-correction. Define n unitry opertor U on the principl system plus ncill: U U M i i ( ) The effect of U is to effect the trnsformtion R ( σ ) i U M σm on the system being corrected, exctly the sme quntum opertion s performed while common error correction. For the nine-qubit Shor code, error-correction without mesurement is very intuitive. The decoding nd correcting circuit would look like: i i i i + i U + i Figure 3.. Shor nine-qubit code error correction circuit without mesurement Riner Bumnn, ETH Zurich 3 Quntum Error Correction 43/66

45 4. Fult-tolernt quntum computtion One of the most powerful pplictions of quntum error-correction is not the protection of stored or trnsmitted quntum informtion, but the protection of quntum informtion s it dynmiclly undergoes computtion. This is very importnt becuse quntum gtes re not t ll s relible s clssicl ones nd the possible errors re much more complex. Assume we wnt to execute Shor s fctoring lgorithm. Although it tkes polynomil time in the size of the input, for resonble numbers, it would still tke millions of dependent clcultions. Combined with the error rte of the gtes, we would never get ny useful result. Becuse of this it ws believed for long time tht quntum computtion is just useless drem. But with Quntum error correction we re ble to construct fult-tolernt quntum gtes, due to which we cn solve this problem. 4.. Bsics of fult-tolernt quntum computtion Our gol is to construct quntum computer, which still produces useful results despite of error influences. Unfortuntely noise nd errors occur in every unit of quntum computer beginning t stte preprtions over gtes nd mesurements up to simple trnsmissions. To fight this problem, we replce every qubit with the encoded block of it nd the other units with their fult-tolernt nlogs Error model Assume single qubit hs been flipped nd we use this qubit s the control qubit for controlled-not. Like this the error will propgte on the trget qubit s well. Even worse, in conctented sttes phse error cn propgte bckwrds on the other qubits in the sme block. So we must correct such errors s fst s possible before they ffect the whole system. By performing error-correction periodiclly fter every step we prevent errors to propgte. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 44/66

46 4.. Fult-tolernt procedure Let us define the fult-tolernce of procedure to be the property tht if only one component fils then the filure cuses t most one error on ech encoded qubit block. Furthermore, we require tht if only one component fils then the mesurement result reported must hve error probbility O(p ), with p the mximum filure probbility over ll components. Like this we ensure tht, in such n component, single error is isolted nd does not ffect ny other qubits. A component is n independent building unit fter which error correction cn be done like quntum gte, mesurement or stte preprtion Fult-tolernt gtes Assume you wnt to execute n X gte on qubit which is encoded in seven qubits. Then we denote this by line over it like X. Note tht such gtes ct on prticulr code nd usully cn not be pplied on other codes. Below we will focus on the populr seven-qubit Sten code. We need to construct universl set of fult-tolernt quntum gtes to be ble to perform ll desired opertions Fult-tolernt Puli gtes In the stbilizer section, we hve lredy described the implementtion of the X nd Z gte for the Stene code: X X X X 3 X 4 X X 6 X 7 ; Z Z Z Z 3 Z 4 Z Z 6 Z 7. Out of them it is very esy to build Y X Z gte. We clled this kind of implementtion bitwise, becuse we just cn pply the desired opertion on every single qubit of code block. It is obvious tht single error, occurring on one qubit just before the opertion or in one of the single gtes like X, does not propgte ny further, becuse there is no informtion exchnge between the qubits. So every encoded gte implemented in bitwise fshion is utomticlly fult-tolernt. This property is clled trnsverslity. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 4/66

47 Figure 4.. Fult tolernt gtes: X (left), Z (middle), Y (right) Fult-tolernt Hdmrd An encoded Hdmrd gte H should interchnge X nd Z under conjugtion, just s the Hdmrd gte interchnges Z nd X. So the encoded Hdmrd cn be implemented s H H H H 3 H 4 H H 6 H 7. It is obvious tht single error, occurring on one qubit just before the opertion or in one of the H i s, does not propgte ny further, becuse there is no informtion exchnge between the qubits. Figure 4.. Fult tolernt Hdmrd gtes (right) Fult-tolernt phse gte Implementing the phse gte S is bit trickier. When we look t the tble in section 3.3, we see tht under conjugtion S should tke Z to Z nd X to Y ix Z. But bitwise implementtion of S would tke Z to Z nd X to Y. The minus sign cn be fixed with Z opertor in front. Like this we cn implement S trnsversly by pplying ZS to every single qubit. Figure 4.3. Fult tolernt phse gtes (right) Z Z Z Z Z Z Z S S S S S S S Riner Bumnn, ETH Zurich 3 Quntum Error Correction 46/66

48 Fult-tolernt CNOT Let s hve look t the CNOT gte next. The importnt question is, would bitwise implementtion violte the fult-tolernce. No. A CNOT involves two different blocks nd one error in one block only results in one error in the sme block. The error is propgted to one qubit in the other block but this does not violte our definition of fult-tolernce. Figure 4.4. Fult tolernt CNOT gtes Fult tolernt π/8 gte For the complete set of universl gtes we lck of the implementtion of non trnsversl gte s the π/8 or T gte, which is much more complex. It consists of three mjor steps: ) Prepre n ncill, in bse stte, pply Hdmrd nd then non fult-tolernt T gte on it resulting in the stte: Θ TH T + + e iπ / 4 b) Apply CNOT on the prepred stte Θ s control qubit nd on the stte we wnt to trnsform s trget qubit. The system would result in the following stte: CNOT CNOT Θ CNOT i + e π / 4 ( + b ) iπ / 4 [ ( + b ) + e ( + b )] iπ / 4 iπ / 4 [( + be ) + ( b + e ) ] Riner Bumnn, ETH Zurich 3 Quntum Error Correction 47/66

49 c) Now we hve to mesure the originl qubit. Resulting, we re done, with we hve to pply SX on the originlly ncill qubit to bring it in the stte T. For the whole thing to be fult-tolernt, we hve to show tht the three steps re fult-tolernt s well. Above we showed tht fulttolernt CNOT exists nd for the mesurement, we will show it in the next prgrph. But, wht bout the preprtion? HT is n eigenstte of Z nd while mesuring SX we collpse the stte in the desired stte. Like this we don t hve to cre bout fult tolernce here. THZHT + TXT + e iπ / 4 SX So we end up with the following circuit. Remember, the double line denotes clssicl bit. T Figure 4.. Fult tolernt π/8 gte 4.4. Fult-tolernt mesurement Remember from section.3, how generl hermitin, unitry opertor M is mesured. (All tensors in G n re hermitin nd unitry.) One first could think, tht we just hve to bring M in trnsversl form M nd then mesure bitwise with the sme control ncill. But such mesurement is not fult tolernt, becuse n error in the ncill block would cuse severl dt bits to be corrupted. A nice wy how to void this problem is illustrted bove. We prepre for every dt qubit n ncill in bse stte. Then we trnsform them to the so clled ct stte with Hdmrd pplied on the first of them nd CNOT with the first qubit s the control qubit nd the other s the trget. We then hve to verify this stte by mesuring them in pir like in the three-qubit flip code form section. If the ct is fine we continue, otherwise we just strt gin from the beginning. Then we perform the controlled M opertions bitwise. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 48/66

50 Finlly we combine the so gined results with CNOT controlled from the first ncill qubit on the other ncills. Finlly we hve to pply Hdmrd on the first ncill to get the eigenvlue of M. The lst prt is fult-tolernt nywy, becuse no error could ffect the dt qubits ny more. But the finl mesurement nd the verifiction mesurement of the ct stte cn be fulty itself with certin probbility. To reduce this error probbility we just hve to perform the whole mesurement procedure severl times nd tke the mjority vote. Like this fult-tolernt mesurement with rbitrry smll error cn be constructed. Figure 4.6. Fult tolernt mesurement circuit 4.. Fult-tolernt stte preprtion Now fult tolernt stte preprtion is very esy. You just prepre the desired stte nd execute verifiction s described in the section fult-tolernt mesurement before. If the stte is fine, continue, otherwise strt gin. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 49/66

51 . Conctented coding nd threshold theorem Encoding dt using quntum error-correcting codes nd pplying fult-tolernt opertions my lso introduce new errors. So if the rbitrry error rte p of component is to high we could do things even worse. It could be shown tht the probbility for fulttolernt circuit to introduce two or more errors into single block is t most cp for some constnt c 4. So the encoded procedure will succeed with probbility -cp. Assume p is smll enough, e.g. p< -4, there is rel benefit using this technology. But we would still hve filure probbility of cp. For n rbitrry long clcultion s Shor s fctoring lgorithm with millions of clcultion this is not useble. The success probbility for n step clcultion is (-cp n ) n. Assume million clcultions with filure probbility p< -4 thn the success probbility is lredy inexistent with < -. Even repeting the sme clcultion severl times nd tke the mjority result wnt help nymore... Conctented codes Knill nd Lflmme solved this problem by introducing the conctented codes we met lredy in Shor s nine-qubit code. A conctented code encodes set of k qubits with [n,k,d] code. It thn tkes every obtined qubit nd encodes it gin using [n,,d ], these qubits could be encoded gin with [n,,d ] code nd so on indefinitely. For k conctention, this results in [nn n..n k,k,dd d..d k ] code. A so conctented code hs filure probbility of (cp) k / c. This increse of relibility is remrkble. We now cn construct quntum error-correction codes with rbitrry filure probbility. Like this, it will be possible to execute rbitrry long clcultions on quntum computer. If such codes re trnsversl the effort for mesurements nd corrections re miniml, s we sw in the previous section. Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

52 The circuit below visulizes the ide of conctented codes with depth 3: k Figure.. [nn n,k,dd d ] conctented code Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

53 .. The threshold theorem Suppose we wnt to clculte n rbitrry long clcultion on quntum computer with filure probbility cp per opertion. For exmple the fctoring lgorithm. The execution of the lgorithm tkes P(n) gtes, where P(n) is polynomil function of the problem size n, chieving finl ccurcy of ε. For this we hve to conctente our code k times such tht: ( cp) k c ε p(n) This gives us the sufficient condition p<p th /c tht k cn be found. With respect to the growth of circuit we get the following theorem. Threshold theorem for quntum computtion: A quntum circuit contining p(n) gtes my be simulted with error probbility of t most ε using log p O poly ε ( n) p ( n) gtes on hrdwre whose components fil with probbility t most p, provided p is below constnt threshold, p<p th, nd given resonble ssumptions bout the noise in the underlying hrdwre. [Nielsen & Chung] p th depends minly on the ssumptions mde bout the computtionl cpbility. A conservtive estimte gives p th -4. This result ws very importnt becuse it dispelled the doubts tht no quntum computer could ever do rbitrry long clcultion due to error ccumultion. Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

54 6. Bounds Theoreticins lwys try to mke generl sttements over generl bounds especilly for proofs. An importnt chrcteristic of clssicl codes for such bounds does not held for quntum codes. In clssicl code, n error is lwys distinguishble from n other. Lets tke the nine-qubit Shor code nd consider the effects of Z nd Z error. They re bsolutely the sme nd just shift the phse of the first three qubits. This degenertion is bd for bound proofing but one of the resons for the power of quntum codes. 6.. Quntum Hmming bound An interesting question is, which is the best possible quntum code. The quntum Hmming bound gives simple formul for this. Assume code encoding k qubits into n qubits with the potentil to correct t errors. Suppose j t errors occur. So there re n possible j sets of error loctions. A single error could be X, Y or Z error resulting in 3 j possibilities. In order to encode k qubits in nondegenerte wy ech of these errors must correspond to n orthogonl k -dimensionl spce. The code spce is n -dimensionl wht leds to the quntum Hmming bound inequlity: n 3 j k n j E.g. for [n,] codes we get the following speciliztion: t j + ( 3n) n ; n As with clssicl code, we cn nerly rech this bound with rndom codes. But they re very hrd to implement. A better clss of perfect quntum codes reching this bound re the Hmming codes over GF(4). Although the quntum hmming bound is only for non-degenertive codes, it works very well for degenertive ones s well. No degenertive code is yet known, which brkes this bound. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 3/66

55 6.. Quntum Gilbert-Vrshmov bound For quntum non-degenertive distnce codes, it is generlly not possible to rech the quntum Hmming bound but nother. Assume [n,k,d] code with j error occurring. The Gilbert- Vrshmov Bound sys, tht we cn lwys find distnce d quntum code encoding k qubits in n qubits stisfying: d j n 3 j k j n 6.3. Quntum singleton / Knill-Lflmme bound For degenerted [n,k,d] quntum codes which re ble to correct errors on ny t qubits must follow the following bound: n 4t + k Especilly this limits the miniml size of quntum code resolving ny rbitrry single error to n 4 +. We lredy sw code which ccomplishes this bound, the five-qubit code. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 4/66

56 7. Simultions nd sttistics 7.. Trce distnce nd fidelity Distnces describe how fr two code words re wy from ech other. This mens how mny single errors re required t minimum to trnsfer code word into nother. So fr we met the trce distnce d between to quntum sttes: D ( ρ, σ ) tr ρ σ The fidelity is n other distnce metric which is not so obvious but it hs some very nice chrcteristics in which we cnnot go in here. It is defined s: / ( ρ, σ ) tr ρ / σρ F 7.. Simultions In this section we summrize the most importnt results from J. Niw s nd K. Mtsumoto s pper Simulting the Effects of Quntum Error Correction Schemes. For more detiled informtion you ll find link to the online version in the references. Simultions nd clcultions of hrdwre error behvior correlted with quntum error correcting codes is very importnt to build useble quntum computer. In the chrt below QECC/X mens tht error-correction is done every X gtes. The y-xis in the figure shows the fidelity nd the x- xis the number of Hdmrd trnsformtions. It s interesting tht not lwys higher frequency of error-correction is effectively better. Riner Bumnn, ETH Zurich 3 Quntum Error Correction /66

57 Figure 7.. Simultion results error correction frequency Simultion shows tht the seven-qubit code, with decoherence probbility p< -4 nd error-correction every ~ gtes, is very effective. But for p< -3 there is no error-correction rte which is nerly useble. We lso see tht for p< - the differences between the different rtios re not very lrge. Below the effects of opertionl errors with stndrd devition of σ< -3 nd σ< - is shown. It cn be seen tht for rbitrry long computtion stndrd devition of σ< -3 is clerly required. Figure 7.. Simultion results opertionl error frequency We lern tht quntum computing hrdwre should hve decoherence rtio of p< -4 nd n opertionl error rtio of σ< -3 for rel world pplictions. Riner Bumnn, ETH Zurich 3 Quntum Error Correction 6/66