Quantum Convolutional Error Correcting Codes H. F. Chau Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong (May 9, 998) I report two general methods to construct quantum convolutional codes for N-state quantum systems. Using these general methods, I construct a quantum convolutional code of rate /4, which can correct one quantum error for every eight consecutive quantum registers. PACS numbers: 03.67.-a, 03.67.Lx, 89.70.+c, 89.80.+h A quantum computer is more ecient than a classical computer in useful applications such as integer factorization [] and database search []. However, decoherence remains one of the major obstacles to building a quantum computer [3]. Nevertheless, the eect of decoherence can be compensated for if one introduces redundancy in the quantum state. By rst encoding a quantum state into a larger Hilbert space H. Then by projecting the wave function into a suitable subspace C of H. And nally by applying a unitary transformation to the orthogonal complement of Caccording to the measurement result; it is possible to correct quantum errors due to decoherence. This scheme is called the quantum error correction code (QECC) [4]. Many QECCs have been discovered (see, for example, Refs. [4{5]) and various theories on the QECC have also been developed (see, for example, Refs. [8{8]). In particular, the necessary and sucient condition for a QECC is [6{9] hi encode ja y Bjj encode i A;B ij ; () where ji encode i denotes the encoded quantum state jii using the QECC; A; B are the possible errors the QECC can handle; and A;B is a complex constant independent ofji encode i and jj encode i. Note that the above condition for a QECC is completely general, working for nite or innite number of N-state quantum registers. All QECCs discovered so far are block codes. That is, the original state ket is rst divided into nite blocks of the same length. Each block is then encoded separately using a code which is independent of the state of the other blocks (cf. Refs. [0,]). In addition to block codes, convolutional codes are well known in classical error correction. Unlike a block code, the encoding operation depends on current as well as a number of past information bits [0,]. For instance, given a (possibly innite) sequence of classical binary numbers (a ;a ;:::;a m ;:::), the encoding (b ;c ;b ;c ;:::;b m ;c m ;:::) with b i a i + a i mod ; c i a i + a i + a i mod () for all i, and a 0 a 0 is able to correct up to one error for every two consecutive bits []. In classical error correction, good convolutional codes often can encode with higher eciencies than their corresponding block codes in a noisy channel [0,]. It is, therefore, instructive to nd quantum convolutional codes (QCC) and to analyze their performance. In this letter, I rst report a way to construct a QCC from a known quantum block code (QBC). Then I discuss a way toconstruct a QCC from a known classical convolutional code. Finally, I report the construction of a QCC of rate /4 which can correct one quantum error for every eight consecutive quantum registers. Let me rst introduce some notations before I construct QCCs. Suppose each quantum register has N orthogonal eigenstates for N. Then, the basis of a general quantum state consisting of many quantum registers can be written as fjkigfjk P ;k ;:::;k m ;:::ig for all k m Z N. And I abuse the notation by dening k m 0 for all m 0. Suppose jki 7! i ;i ;:::;i m a (k) i ;i ;:::;i m ji ;i ;:::;i m i be a QBC mapping one quantum register to a code of length m. Hence, the rate of the code equals m. The eect of decoherence can be regarded as an error operator acting on certain quantum registers. I denote the set of all possible errors that can be corrected by the above quantum block code by E. Based on this QBC, one can construct a family of QCCs as follows: e-mail: hfchau@hkusua.hku.hk Perhaps the simplest way to see that Eq. () holds for innite number of N-state registers is to observe that Gottesman's proof in Ref. [9] does not depend on the niteness of the Hilbert space for encoded state.
Theorem Given the above QBC and a quantum state jki jk ;k ;:::;k n ;:::i making up of possibly innitely many quantum registers, then the encoding jki jk ;k ;:::;k n ;:::i 7!jk encode i +O 4 i P ( p a ipkp) j i;j i;:::;j im 3 j i;j i;:::;j im jj i ;j i ;:::;j im i5 ; (3) forms a QCC of rate m provided that the matrix ip is invertible. This QCC can handle errors in the form E E. Proof: I consider the eects of errors EE E and E 0 E 0 E 0 EE on the encoded quantum registers by computing hk 0 encodeje 0y Ejk encode i Y 4 i j i;:::;j im;j 0 i ;:::;j0 im Substituting Eq. () into Eq. (4), we have a (P p 0 ip 0 k0 p 0 ) j 0 i ;:::;j0 im (P p a ipkp) j i;:::;j im hj 0 i ;:::;j0 imje 0y 3 i E ijj i ;:::;j im i5 : (4) hk 0 encodeje 0y Ejk encode i + Y hdp p ipkp 0 i + Y i h encode Pp ipkp;p p ipk0 p E i;e 0 i E 0y i E i P Ei p ipk p encode i (5) for some constants Ei;E 0 i independent of k and k0. Since the matrix is invertible, k i k 0 i solution of the systems of linear equations P p ipk p P p ipk 0 p. Consequently, for all i is the unique hk 0 encodeje 0y Ejk encode i k;k 0 E;E 0 (6) for some constant E;E 0independent ofkand k 0. Thus, the encoding in Eq. (3) is a QECC. At this point, readers should realize that the above scheme can be generalized to construct a QCC from a QBC which maps n quantum registers to m (> n) registers. It is also clear that the following two useful corollaries follow directly from Theorem : Corollary The encoding scheme given by Eq. (3) gives a QCC from a QBC provided that () the elements in the matrix are either zeros or ones; () ip is a function of i p only; and (3) ip (i p) consists of nitely many ones. Corollary The encoding scheme given by Eq. (3) gives a QCC from a QBC if () N is a prime power; () is not a zero matrix; and (3) ip is a function of i p only. Let me illustrate the above analysis by an example. Example Starting from the spin ve register code in Ref. [], one knows that the following QCC can correct up to one error in every ve consecutive quantum registers: 7! jk ;k ;:::;k m ;:::i " +O i N! (ki+ki )(pi+qi+ri)+piri N 3 p i;q i;r i0 N jp i ;q i ;p i +r i ;q i +r i ;p i +q i +k i +k i i (7) where k m Z N,! N is a primitive N th root of unity, and all additions in the state ket are modulo N. The rate of this code equals /5.
Although the QCC in Eq. (3) looks rather complicated, the actual encoding process can be performed readily. Because is invertible, one can reversibly map jk ;k ;:::;k n ;:::ito j P p pk p ; P p pk p ;:::; P p npk p ;:::i[3{5]. Then, one obtains the above ve register QCC by encoding each quantum register using the procedure in Ref. []. Now, I turn to the construction of QCCs from classical convolutional codes. Let me rst introduce two technical lemmas (which work for both QBCs and QCCs). Lemma Suppose the QECC jki 7! a (k) j jj ;j ;::: ;j ;:::i (8) j ;j ;::: corrects (independent) spin ip errors in certain quantum registers with j i Z N. Then, the following QECC, which is obtained by discrete Fourier transforming every quantum register in Eq. (8), jki 7! a (k) j ;j ;::: j ;j ;:::;p ;p ;::: + Y i p N! jipi N jp ;p ;:::i (9) corrects (independent) phase errors occurring in the same quantum registers. The converse is also true. Proof: Observe that one can freely choose a computational P basis for the encoded quantum state. In particular, if one N chooses the discrete Fourier transformed basis fj ~mig f j0!jm N jjig for each ofthe encoded quantum register, then the encoding in Eq. (9) is reduced to the encoding in Eq. (8). Thus, the code in Eq. (9) handles spin ip errors with respected to the discrete Fourier transformed basis fj ~mig. Consequently, the same code handles phase errors in the original fjmig basis. Conversely, suppose one chooses the original fjmig basis to encode a phase error correcting code. Then with respected to the fj ~mig basis, it is easy to check that the same code corrects spin ip errors. Lemma Suppose a QECC handles errors E and E satisfying (a) for all E i E i (i ;), there exists E 0 E such that E y E E y E0 ; and (b) for E i; E 0 i E i (i ;), E y i E0 i E i whenever errors E i and E 0 i occur at the same set of quantum registers; then the QECC actually handles errors in E E fe E :E E ;E E and errors E ; E occur at the same set of quantum registersg. Proof: One knows from Eq. () that hk 0 encode jey i E0 i jk encodei k;k 0 Ei;E i 0 for some E i;ei 0 independent ofk(i;). Also, Eq. () implies that the eect of an error E i is simply to rigidly rotate and to contract (or expand) the encoded ket space independent of the state jk encode i itself. Thus, one concludes that D E k 0 encode (E + E ) y (E + E ) k encode k;k 0 E ;E (0a) and D E k 0 encode (E + ie ) y (E + ie ) k encode k;k 0 0 E ;E (0b) for all E i E i (i ;), where E ;E and 0 E ;E are independent of k. By expanding Eqs. (0a) and (0b), one arrives at hk 0 encodeje y E jk encode i k;k 0 E;E () for some E;E independent ofk. Finally, I consider errors E i ; Ei 0 E i (i ;) occurring at the same set of quantum registers, then D E k 0 encode (EE 0 ) 0 y (E E ) k encode hk 0 encode je0y E0yE E jk encode i hk 0 encodeje 00 E 00 jk encode i () for some E 00 i E i (i ;). Hence from Eqs. () and (), I conclude that the QECC handles errors in the set E E. The next corollary follows directly from Lemma. 3
Corollary 3 A QECC handles general quantum error if and only if it handles both spin ip and phase errors in the corresponding quantum registers. Now, I am ready to prove the following theorem regarding the construction of quantum codes from classical codes. Theorem Suppose QECCs C and C handle phase shift and spin ip errors, respectively, for the same set of quantum registers. Then, pasting the two codes together by rst encodes the quantum state using C then further encodes the resultant quantum state using C, one obtains a QECC C which corrects general errors in the same set of quantum registers. Proof: From Corollary 3, it suces to show that the new QECC C corrects both spin ip and phase errors. By the construction of C, it clearly can correct spin ip errors. And using the same trick in the proof of Lemma, it is easy to check that C can correct phase shift errors as well. Readers should note that the order of pasting in Theorem is important. Reversing the order of encoding does not give a good quantum code. Also, proofs of Corollary 3 and Theorem for the case of N can also be found, for example, in Ref. [9]. Theorem 3 Suppose C is a classical (block or convolutional) code of rate r that can correct p (classical) errors for every q consecutive registers. Then, C can be extended to a QECC of rate r that can correct at least p quantum errors for every q consecutive quantum registers. Proof: Suppose C is a classical code. By mapping m to jmi for all m Z N, C can be converted to a quantum code for spin ip errors. Let C 0 be the QECC obtained by Fourier transforming each quantum register of C. Then Lemma implies that C 0 is a code for phase shift errors. From Theorem, pasting codes C and C 0 together will create a QECC C 00 of rate r. Finally, one can verify the error correcting capability ofc 00 readily [6]. Theorem 3 is useful to create high rate QCCs from high rate classical convolutional codes. Note that one of the simplest classical convolutional code with rate isgiven by Eq. (). Being a non-systematic and non-catastrophic 3 code [], it serves as an ideal starting point to construct good QCCs. First, let me write down this code in quantum mechanical form: Lemma 3 The QCC jk ;k ;:::i7! +O i jk i + k i ;k i +k i +k i i (3) for all k i Z N, where all additions in the state ket are modulo N,can correct up to one spin ip error for every four consecutive quantum registers. Proof: Using notations as in the proof of Theorem, I consider hk 0 encode je0y Ejk encode i. Clearly, the worst case happens when errors E and E 0 occur at dierent quantum registers. And in this case, Eq. (3) implies that exactly two ofthe following four equations hold: 8 >< >: k i + k i k 0 i + k0 i k i + k i + k i k 0 i + k 0 i + k 0 i k i+ + k i k 0 i+ + k0 i k i+ + k i + k i k 0 i+ + k 0 i + k 0 i for all i. One may regard k i s as unknowns and ki 0 s as arbitrary but xed constants. Then, by straight forward computation, one can show that picking any two equations out of Eq. (4) for each i will form an invertible system with the unique solution k i ki 0 for all i. Thus, hk0 encode je0y Ejk encode i k;k 0 E;E 0 and hence this lemma is proved. (4) That is, both b i and c i are not equal to a i. 3 That is, a nite number of channel errors does not create an innite number of decoding errors. 4
Example Theorem 3 and Lemma 3 imply that the following QCC of rate /4: " +O jk ;k ;:::i7!jk encode i i N p ;q ;:::!(ki+ki )pi+(ki+ki +ki )qi N jp i + p i ;p i +p i +q i ;q i +q i ;q i +q i +p i i (5) for all k i Z N, where all additions in the state ket are modulo N can correct at least one error for every 6 consecutive quantum registers. But, in fact, this code is powerful enough to correct one error for every eight consecutive quantum registers (see also Ref. [6]). Proof: Let E and E 0 be two quantum errors aecting at most one quantum register per every eight consecutive ones. By considering hk 0 encode je0y Ejk encode i, I know that at least six of the following eight equations hold: 8 >< >: p i + p i p 0 i + p 0 i p i + p i + q i p 0 i + p0 i + q0 i q i + q i q 0 i + q0 i q i + q i + p i q 0 i + q0 i + p0 i p i + p i p 0 i + p0 i p i + p i + q i p 0 i + p0 i + q0 i q i + q i q 0 i + q0 i q i + q i + p i q 0 i + q 0 i + p 0 i for all i Z +. Let me regard p i and q i as unknowns; and p 0 i and q0 i (6) as arbitrary but xed constants. Then, it is straight forward to show that choosing any six equations in Eq. (6) for each i Z + would result in a consistent system having a unique solution of p i p 0 i and q i q 0 i for all i Z +. Consequently, hk 0 encode je0y Ejk encode i p ;q ;p ;q ;:::( +Y i P i ji pj (kj +kj k0 j kj 0 )+qj (kj +kj +kj k0 j kj 0 kj 0 )! N ) hf i je 0y i jf iihg i jejg i i (7) for some linearly independent functions f i (p ;q ;p ;q ;:::) and g i (p ;q ;p ;q ;:::). Now, I consider a basis fh i (p ;q ;p ;q ;:::)gfor the orthogonal complement of the span of ff i ;g i g iz +. By summing over all h i s while keeping f i s and g i s constant in Eq. (7), one ends up with the constraints that k i k 0 i for all i Z+. Thus, hk 0 encode je0y Ejk encode i k;k 0 p ;q ;p ;q ;:::" +Y i hf i (p ;q ;:::)je 0y jf i (p ;q ;:::)ihg i (p ;q ;:::)jejg i (p ;q ;:::)i : (8) Hence, Eq. (5) corrects up to one quantum error per every eight consecutive quantum registers. The above rate /4 QCC is constructed from a classical convolutional code of rate /. One may further boost up the code performance by converting other ecient classical convolutional codes (such as various k(k + )-rate codes in Ref. [7]) into QCCs. On the other hand, it is impossible to construct a four quantum register QBC that can correct one quantum error [,6]. With modication, the same argument can be used to show that no QCC can correct one error for every four consecutive quantum registers [6]. It is instructive to compare the performances of QBCs and QCCs in other situations. In addition, in order use QCCs in quantum computation, one must investigate the possibility of fault tolerant computation on them. Moreover, it would be ideal if the fault tolerant implementation of single and two-quantum register operations must involve only a nite number of quantum registers in the QCC. While a general QCC may not admit a nite fault tolerant implementation, many QCCs with nite memories 4 can be manipulated fault tolerantly. 4 That is, codes with encoding schemes which depend on a nite number of quantum registers in jki. 5
Example 3 By subtracting those quantum registers containing p i, p i+, q i, q i+ and q i+ by one in Eq. (5), one ends up with changing jk ;k ;:::;k i ;::: encode i to jk ;k ;:::;k i ;k i +;k i+ ;::: encode i. Clearly, the above operation is fault tolerant and involves only a nite number of quantum registers. Fault tolerant implementation of single register phase shift can be obtained in a similar way. Further results on fault tolerant implementation on QCCs will be reported elsewhere [9]. Finally, decoding a classical convolutional code can be quite involved [8]. So, it is worthwhile to investigate the eciency of decoding a QCC. I shall report them in future works [9]. ACKNOWLEDGMENTS Iwould like to thank T. M. Ko for introducing me the subject of convolutional codes. Iwould also like to thank Debbie Leung, H.-K. Lo and Eric Rains for their useful discussions. This work is supported by the Hong Kong Government RGC grant HKU 7095/97P. [] P. W. Shor, in Proceedings of the 35th Annual Symposium on the Foundation of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 994), p. 4. [] L. K. Grover, in Proceedings of the 8th Annual ACM Symposium on the Theory of Computing (ACM Press, New York, 996), p.. [3] R. Landauer, in Proceedings of PHYSCOMP94, edited by D. Matzke (IEEE Computer Society, Los Alamitos, CA, 994), p. 54. [4] P. W. Shor, Phys. Rev. A 5, 493 (995). [5] R. Laamme, C. Miquel, J. P. Paz and W. H. Zurek, Phys. Rev. Lett. 77, 98 (996). [6] A. R. Calderbank and P. W. Shor, Phys. Rev. A 54, 098 (996). [7] A. M. Steane, Phys. Rev. A 54, 474 (996). [8] A. M. Steane, Phys. Rev. Lett. 77, 793 (996). [9] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Phys. Rev. Lett. 78, 405 (997). [0] D. Gottesman, Phys. Rev. A 54, 86 (996). [] H. F. Chau, Phys. Rev. A 55, 839 (997). [] H. F. Chau, Phys. Rev. A 56, (997). [3] E. M. Rains, R. H. Hardin, P. W. Shor and N. J. A. Sloane, Phys. Rev. Lett. 79, 953 (997). [4] S. Llyod, and J.-J. E. Slotine, Los Alamos preprint archive quant-ph/970 (997). [5] S. L. Braunstein, Los Alamos preprint archive quant-ph/97049 (997). [6] E. Knill and R. Laamme, Phys. Rev. A 55, 900 (997). [7] E. Knill, Los Alamos preprint archive quant-ph/9608048 (996). [8] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys. Rev. A 54, 384 (996). [9] D. Gottesman, Ph.D. Thesis, Caltech (997), chap.. [0] Ph. Piret, Convolutional Codes: An Algebraic Approach (MIT Press, Cambridge, MA, 988). [] A. Dholakia, Introduction to Convolutional Codes with Applications (Kluwer, Dordrecht, 994). [] See, for example, chap. 4 in Ref. []. [3] C. H. Bennett, IBM J. Res. Dev. 7, 55 (973). [4] C. H. Bennett, SIAM J. Comp. 8, 766 (989). [5] H. F. Chau and H.-K. Lo, Cryptologia, 39 (997). [6] H. F. Chau, to appear in the Proc. of the st NASA International Conf. on Quant. Comp. and Quant. Comm. (998). [7] A list of good classical convolutional codes can be found, for example, in appendix A of Ref. []. [8] See, for example, chaps. 5{8 in Ref. []. [9] H. F. Chau, in preparation (998). 6