I construct a quantum error correction code (QECC) in higher spin systems
using the idea of multiplicative group character. Each $N$ state quantum
particle is encoded as five $N$ state quantum registers. By doing so, this code
can correct any quantum error arising from any one of the five quantum
registers. This code generalizes the well-known five qubit perfect code in
spin-1/2 systems and is shown to be optimal for higher spin systems. I also
report a simple algorithm for encoding. The importance of multiplicative group
character in constructing QECCs will be addressed.Comment: Revised version, to appear in Phys.Rev.A (Rapid Communications). 4
  pages in Revtex 3.1, using amssymb.st

Chau, H. F.

English

arXiv

I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each N-state quantum particle is encoded as five N-state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.published_or_final_versio

Chau, HF

HKU Scholars Hub

Title Five quantum register error correction code for higher spinsystemsAuthor(s) Chau, HFCitation Physical Review A - Atomic, Molecular, And Optical Physics,1997, v. 56 n. 1, p. R1-R4Issued Date 1997URL http://hdl.handle.net/10722/43202Rights Creative Commons: Attribution 3.0 Hong Kong LicenseRAPID COMMUNICATIONSPHYSICAL REVIEW AATOMIC, MOLECULAR, AND OPTICAL PHYSICSTHIRD SERIES, VOLUME 56, NUMBER 1 JULY 1997Five quantum register error correction code for higher spin systemsH. F. Chau*Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong~Received 18 February 1997!I construct a quantum error correction code ~QECC! in higher spin systems using the idea of multiplicativegroup character. Each N-state quantum particle is encoded as five N-state quantum registers. By doing so, thiscode can correct any quantum error arising from any one of the five quantum registers. This code generalizesthe well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems.I also report a simple algorithm for encoding. The importance of multiplicative group character in constructingQECCs will be addressed. @S1050-2947~97!50707-4#PACS number~s!: 03.65.Bz, 02.10.Lh, 89.70.1c, 89.80.1hRAPID COMMUNICATIONSThe Rapid Communications section is intended for the accelerated publication of important new results. Since manuscripts submittedto this section are given priority treatment both in the editorial office and in production, authors should explain in their submittal letterwhy the work justifies this special handling. A Rapid Communication should be no longer than 4 printed pages and must be accompaniedby an abstract. Page proofs are sent to authors.The power of a quantum computer is perhaps best illus-trated by the powerful Shor quantum polynomial time fac-torization algorithm @1#. However, the real power of a quan-tum computer may be much more limited because it isextremely vulnerable to disturbance @2#. Nevertheless, Shorpointed out later that the effect of quantum decoherence canbe compensated for if we introduce redundancy in the quan-tum state in a suitable way. We first encode the quantumstate into a larger Hilbert space H . Then we measure thewave function in a suitable subspace C of H . And finally weapply a unitary transformation to the orthogonal complementof C according to our measurement result; it is possible tocorrect quantum errors due to decoherence with the environ-ment @3#. This kind of scheme is now called the quantumerror correction code ~QECC!. Since then, many QECCshave been discovered ~see, for example, Refs. @4–10#! andvarious theories on the QECC have also been developed~see, for example, Refs. @7–14#!. In particular, the necessaryand sufficient condition for a QECC is @12–14#^iencodeuA†Bu jencode&5lA ,Bd i j , ~1!where uiencode& denotes the encoded quantum state ui& using*Electronic address: hfchau@hkusua.hku.hk561050-2947/97/56~1!/1~4!/$10.00the QECC; A ,B are the possible errors that can be handledby the QECC; and lA ,B is a complex constant independent ofuiencode& and u jencode&.Early QECCs deal with decoherence of individual spin-12particles with the environment. Besides, the information lossto the environment is assumed to be unrecoverable. Morerecently, Duan and Guo considered the decoherence of spin-12 particles with the same environment. Based on a specificmodel of the environment in thermal equilibrium, they founda new coding scheme @15#. Another investigation concen-trates on the mutual decoherence between the quantum spinsinside the quantum computer. Chau pointed out that the abil-ity to correct quantum errors among various registers inside aquantum computer is equivalent to the ability to correct thequantum error of a single quantum higher spin particle @10#.Thus, it is interesting to construct QECCs for quantum reg-isters with higher spin.The QECC for particles with spin higher than 12 was foundby Chau using group-theoretical methods. He encodes eachquantum particle as nine quantum registers. And by doing so,his code can correct any quantum error involving exactly onequantum register @10#. Nonetheless, his code is not perfect.1So, it is natural to ask if it is possible to construct moreeconomical codes for higher spin systems.1See Ref. @4# for a precise definition of a perfect code.R1 © 1997 The American Physical SocietyRAPID COMMUNICATIONSR2 56H. F. CHAUAn affirmative answer is provided in this paper. I report away to encode each quantum particle as five quantum regis-ters, which can correct an error in at most one of the fiveregisters. I also show that this code is optimal in the sensethat no QECC with codeword length less than five can cor-rect a general one quantum register error. For spin-12 par-ticles, this code is equivalent to the perfect codes discoveredby Laflamme et al. @4# and Bennett et al. @14# up to unitarytransformations. As you will see in the derivation, the suc-cess of this five-register code relies heavily on the sum ruleof the multiplicative group character of the finite additivegroup ZN .The ~multiplicative! group character of the finite additivegroup ZN is a map x:ZN!C satisfying @16#x~a1b !5x~a !x~b ! ~2!for all a ,bPZN . Then x satisfies the sum rule @16#(mPZNx~m !5HN if x is the trivial character,0 otherwise. ~3!More concretely, the above sum rule can be written as(m50N21vNmk5HN if k50 modN ,0 for k51,2, . . . ,N21 modN , ~4!where vN is a primitive Nth root of unity.To see how we use Eq. ~4! to construct our five quantumregister code, let us begin by denoting the N mutuallyorthogonal eigenstates in each quantum register byu0&,u1&, . . . ,uN21&. Then, I claim that the following encod-ing scheme can correct any quantum error occurring in atmost one of the quantum registersuk&°1N3/2 (p ,q ,r50N21vNk~p1q1r !1prup1q1k& ^ up1r&^ uq1r& ^ up& ^ uq&[1N3/2 (p ,q ,r50N21vNk~p1q1r !1pr3up1q1k ,p1r ,q1r ,p ,q& ~5!for k50,1, . . . ,N21, where all the additions in the statekets and in the sum are modulo N .Let me denote the one-bit quantum error Ea occurring atthe ith quantum register by the symbol Ei ,a . To prove theabove claim, it suffices to show that Eq. ~1! holds for anyquantum errors A5Ei ,a and B5E j ,b for 1<i< j<5.First, I consider the case when (i , j)5(1,4) as a warm up.We have^kencodeuE1,a† E4,bukencode8 &51N3 (p ,q ,r ,p8,q8,r850N21vNk8~p81q81r8!1p8r82k~p1q1r !2pr3^p1q1kuEa† up81q81k8&^p1rup81r8&3^q1ruq81r8&^puEbup8&^quq8&51N3 (p ,q ,r ,p8,q8,r850N21vNk8~p81q81r8!1p8r82k~p1q1r !2pr3dq1r ,q81r8dq ,q8dp1r ,p81r83^p1q1kuEa† up81q81k8&^puEbup8&51N3 (p ,q ,r50N21vN~k82k !~p1q1r !3^p1q1kuEa† up1q1k8&^puEbup&5dk ,k81N2 (p ,q50N21^p1quEa† up1q&^puEbup&[dk ,k8L1,a;4,b , ~6!where L1,a;4,b is independent of k . Thus, Eq. ~1! holds when(i , j)5(1,4). Using the same trick, it is easy to verify thatEq. ~1! holds when (i , j)5(1,1), (2,2), (3,3), (4,4), (5,5),(1,5), and (3,5).Now, I proceed to the more difficult case when (i , j)5(1,2). We have^kencodeuE1,a† E2,bukencode8 &51N3 (p ,q ,r ,p8,q8,r850N21vNk8~p81q81r8!1p8r82k~p1q1r !2pr3dp ,p8dq ,q8dq1r ,q81r8^p1q1kuEa† up81q81k8&3^p1ruEbup81r8&51N3 (p ,q ,r50N21vN~k82k !~p1q1r !^p1q1kuEa† up1q1k8&3^p1ruEbup1r&. ~7!By relabeling x5p1q , y5p1r , and z5r , Eq. ~7! can berewritten as^kencodeuE1,a† E2,bukencode8 &51N3 (x ,y ,z50N21vN~k82k !~x1z !3^x1kuEa† ux1k8&^y uEbuy&5dk ,k81N2 (x ,y50N21^xuEa† ux&^y uEbuy&[dk ,k8L1,a;2,b , ~8!where L1,a;2,b is independent of k . Thus, Eq. ~1! holds when(i , j)5(1,2). In a similar way, one can show that Eq. ~1! isalso true for (i , j)5(1,3).Now, I move on to the case when (i , j)5(2,3). By directcomputation, we obtain^kencodeuE2,a† E3,bukencode8 &51N3 (p ,q ,r ,p8,q8,r850N21vNk8~p81q81r8!1p8r82k~p1q1r !2prRAPID COMMUNICATIONS56 R3FIVE QUANTUM REGISTER ERROR CORRECTION CODE . . .3dp ,p8dq ,q8dp1q1k ,p81q81k8^p1ruEa† up81r8&3^q1ruEbuq81r8&5dk ,k81N3 (p ,q ,r ,r850N21vN~r82r !~k1p !^p1ruEa† up1r8&3^q1ruEbuq1r8&. ~9!By relabeling x5r82r , y5p1r , z5q1r , and u5p , Eq.~9! can be written as^kencodeuE2,a† E3,bukencode8 &5dk ,k81N3 (u ,x ,y ,z50N21vNx~u1k !^y uEa† uy1x&^zuEbuz1x&5dk ,k81N2 (x ,y ,z50N21^y uEa† uy1x&^zuEbuz1x&[dk ,k8L2,a;3,b , ~10!where L2,a;3,b is independent of k . Hence, Eq. ~1! is alsosatisfied when (i , j)5(2,3). Using similar methods, it can beshown that Eq. ~1! holds if (i , j)5(2,4), (2,5), and (3,4).Finally, I consider the case when (i , j)5(4,5). By directcomputation, we find that^kencodeuE4,a† E5,bukencode8 &51N3 (p ,q ,r ,p8,q8,r850N21vNk8~p81q81r8!1p8r82k~p1q1r !2pr3dp1r ,p81r8dq1r ,q81r8dp1q1k ,p81q81k8^puEa† up8&3^quEbuq8&51N3 (p ,q ,r ,p8,q8,r850N21vNk8~p81q81r8!1p8r82k~p1q1r !2pr3d2p1k ,2p81k8d2q1k ,2q81k8d2r2k ,2r82k8^puEa† up8&3^quEbuq8&. ~11!Let us analyze the situation by considering the following twosubcases:Subcase (a): If k2k8 is odd and N is even, then it isimpossible to find p ,p8PZN such that2p1k52p81k8modN . Hence, the existence of thed2p1k ,2p81k8 term in Eq. ~11! implies that^kencodeuE4,a† E5,bukencode8 &50.Subcase (b): if either k2k8 is even or N is odd, then it ispossible to find p ,p8PZN such that 2p1k52p81k8modN . That is to say, it make sense to regard(k82k)/2 as an integer in ZN . Then Eq. ~11! becomes^kencodeuE4,a† E5,bukencode8 &51N3 (p ,q ,r50N21vN[~k82k !/2][3p12q1r2~3k82k !/2]3 K puEa† up2 k82k2 L K quEbuq2 k82k2 L5dk ,k81N2 (p ,q50N21^puEa† up&^quEbuq&[dk ,k8L4,a;5,b , ~12!where L4,a;5,b is independent of k . Therefore, the encodingscheme in Eq. ~5! satisfies Eq. ~1! for any (i , j) with1<i , j<5; and, hence, this scheme is able to correct anyquantum error arising at any one of the quantum registers aspromised.The key idea used in this five-register code is ~i! the mul-tiplicative group character sum rule in Eq. ~3!, ~ii! the rela-beling of some variables in the summation, and ~iii! thestrong correlation between the five quantum registers.2 Sincethe sum rule in Eq. ~3! plays a very important role in both thefive and the nine quantum register codes @10#, it will beinteresting to rewrite other existing QECCs for spin-12 par-ticles in a form similar to that of Eq. ~5!. This may provide away to generalize these codes to higher spin systems.Back to the case when N52. The above encoding schemeabove can be explicitly written asu0&°1A8@ u00000&1u01100&1u10101&1u11001&1u11010&2u10110&1u01111&2u00011&] ~13a!andu1&°1A8@ u10000&2u11100&2u00101&1u01001&2u01010&2u00110&1u11111&1u10011&]. ~13b!This scheme can be transformed to the perfect code obtainedby Laflamme et al. @4# ~and hence also Bennett et al.’s @14#!by a simple unitary transformation: first permute the fivequantum registers by P(13524), then add an extra phase ofp to the encoding state whenever p1r1k is even. That is tosay, Laflamme et al.’s perfect code can be written asuk&°1A8 (p ,q ,r ~21 !~p11 !~r11 !1k~p1q1r11 !up1q11&^ up& ^ up1r& ^ uq& ^ uq1r&, ~14!for k50,1.Now, I give a simple encoding algorithm for this code.Using a series of quantum binary conditional-NOT gates, wemay ‘‘copy’’ the state uk ,0,0,0,0& to uk ,0,k ,k ,k& efficiently.Next, we apply quantum discrete Fourier transforms similarto that used in Shor’s algorithm @1,17,18# separately to thethird, fourth, and fifth quantum registers. Then, we add anadditional phase of vNpr to the system using a Toffoli-likegate @19,20#. We then use a series of quantum binaryconditional-NOT gates to ‘‘copy’’ the fourth register to thesecond one. Finally, by suitably adding the quantum registers2In other words, the high entanglement entropy in this code.RAPID COMMUNICATIONSR4 56H. F. CHAUtogether using reversible quantum logic gates, we obtain thequantum code as required. The entire encoding procedurecan be summarized below:uk ,0,0,0,0&°uk ,0,k ,k ,k&°1N3/2 (p ,q ,r50N21vNk~p1q1r !uk ,0,r ,p ,q&°1N3/2 (p ,q ,r50N21vNk~p1q1r !1pruk ,0,p ,q ,r&°1N3/2 (p ,q ,r50N21vNk~p1q1r !1pruk ,p ,r ,p ,q&°1N3/2 (p ,q ,r50N21vNk~p1q1r !1pr3up1q1k ,p1r ,q1r ,p ,q&. ~15!Finally, I present a proof of the optimality of the aboveQECC. More precisely, I will show that it is not possible tocorrect a general quantum error involving exactly one quan-tum register by encoding each quantum particle by four ~orless! quantum registers. Following Sec. V B in Ref. @12# ~seealso Ref. @21#!, I suppose that a single error correcting quan-tum code with codeword length four exists. Then one canalways writeuiencode&5 (p ,q ,r ,s50N21apqrs~ i ! up ,q ,r ,s& ~16!for i50,1, . . . ,N21. Define the reduced density matricesrp8q8;pq~ i !5 (r ,s50N21~ap8q8rs~ i ! !*apqrs~ i ! ~17!for all i , and~E ~ i0 , j0!! i j5H 1 if i5i0 and j5 j0 ,0 otherwise. ~18!Now we consider the error operators E3,(i0 , j0) and E4,(i0 , j0),which act on the third and fourth register, respectively. Sup-pose uiencode&Þu jencode&; then from Eqs. ~1!, ~16!, and ~17!,one arrives atr~ i !r~ j !50 ~19!for all iÞ j . Similarly, we consider the actions of E1,(i0 , j0)and E2,(i0 , j0) on the encoded registers. Putting i5 j in Eq.~1!, one arrives atr~ i !5r~ j ! ~20!for all i , j . From Eqs. ~19! and ~20!, one concludes that allthe ~Hermitian! reduced density matrices r (i) are nilpotent.However, this is possible only if r (i)50 and henceapqrs(i) 50 for all i ,p ,q ,r ,s50,1, . . . ,N21. This contradictsthe assumption that (p ,q ,r ,s50N21 apqrs(i) up ,q ,r ,s& encodes thequantum state ui&. Thus, the codeword length must be at leastfive. Consequently, the five quantum register code reportedhere is optimal.I would like to thank E. Knill and R. Laflamme for point-ing out a mistake in the earlier version of this manuscript,and M. Grassl for pointing out the use of Ref. @12# to provethe optimality of this code. In addition, valuable discussionswith M. Ben-Or are gratefully acknowledged.@1# P. W. Shor, in Proceedings of the 35th Annual Symposium onthe Foundation of Computer Science, edited by S. Goldwasser~IEEE Computer Society, Los Alamitos, CA, 1994!, p. 124.@2# R. Landauer, in Proceedings of PHYSCOMP94, edited by D.Matzke ~IEEE Computer Society, Los Alamitos, CA, 1994!, p.54.@3# P. W. Shor, Phys. Rev. A 52, 2493 ~1995!.@4# R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Phys.Rev. Lett. 77, 198 ~1996!.@5# A. R. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098~1996!.@6# A. M. Steane, Phys. Rev. A 54, 4741 ~1996!.@7# A. M. Steane, Phys. Rev. Lett. 77, 793 ~1996!.@8# A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A.Sloane, Phys. Rev. Lett. 78, 405 ~1997!.@9# D. Gottesman, Phys. Rev. A 54, 1862 ~1996!.@10# H. F. Chau, Phys. Rev. A 55, 839 ~1997!.@11# A. Steane, Proc. R. Soc. London, Ser. A 452, 2551 ~1996!.@12# E. Knill and R. Laflamme, Phys. Rev. A 55, 900 ~1997!.@13# E. Knill ~unpublished!.@14# C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K.Wootters, Phys. Rev. A 54, 3824 ~1996!.@15# L.-M. Duan and G.-C. Guo ~unpublished!.@16# K. Ireland and M. Rosen, A Classical Introduction To ModernNumber Theory, 2nd ed. ~Springer, New York, 1990!, Chap. 8.@17# A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733 ~1996!.@18# P. Hoyer ~unpublished!.@19# H. F. Chau and F. Wilczek, Phys. Rev. Lett. 75, 748 ~1995!.@20# A. Barenco et al., Phys. Rev. A 52, 3457 ~1995!.@21# M. Grassl ~private communication!.

Five Quantum Register Error Correction Code For Higher Spin Systems

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