arXiv:1906.11137v4 [quant-ph] 12 Feb 2020 optn,MliaudLogic. Multivalued Computing, ro,(i u oecnraiycretbterr nasingl a in in previously, correcting errors require used not bit correction does subspaces. two-step correction correct error the readily phase to (iii) can opposed as code step our (ii) error, o noig i)orcd a orc n arbitrary any correct requi can qutrits code of number our the (ii) (i) outperfo encoding, in code code for This 9-qutrit codeword. the previous on the error single any correct rv htteerrmdlcniee nti ae pn the spans paper this in sing considered a qutrits. model of correct error number entire the to the in 5- that necessary optimal a prove is are stabil code propose its proposed qutrits our provide we error, 5 and paper, Since code quantum this correcting formulation. of In error sys number quantum mandated. quantum qutrit given is multi-valued correction a of operation error for reliable ones For states. binary than formation | neatwt h niomn n neg oeunwanted some undergo can system and quantum environment ( evolution The the [7]. with gates interact quantum called erators, rbe nQatmErrCreto 1] 1] [13]. also [12], has [11], [10] Correction Error code nearest-neighb Quantum the surface in solves as problem which termed extensively, code the studied of While been qubits. family [7] [9] code of new QECC concatenated number of a 5-qubit examples the qubits are The in codes nine [8]. optimal mentioned above error into be single to qubit a shown single was correct (QECC) a to code order of in correcting information error the quantum encoded first The evolution. 5,[] enr ytmi h ipethge dimensional qutr form higher or coins the state, simplest quantum dimensional has ternary the arbitrary higher is An with system. system quantum algo- Ternary deal walk [6]. often random [5], graph quantum on Furthermore, rithms resour [4]. less proto secure using cryptographic makes more space its space search search to the larger Hilbert Increasing due [3]. a in importance represent gaining vector to Multi is ability unit multi-valued. computing a inherently quantum computer, as are valued quantum systems represented of Quantum is unit Space. qubit, functional as The termed [2]. [1], terparts Q α | ne Terms Index Abstract vlto faqatmsaei oendb ntr op- unitary by governed is state quantum a of Evolution 2 + anpolm atrta hi nw lsia coun- classical known their cer- than solving faster of problems promise tain the hold computers UANTUM (3 | β × | 2 Mlivle unu ytm a tr oein- more store can systems quantum —Multi-valued E 3) + ,called ), prtrsae hrfr,orpooe oecan code proposed our Therefore, space. operator | QatmErrCreto,TrayQuantum Ternary Correction, Error —Quantum γ | ψ | 2 i 1 1 = dacdCmuig&Mcolcrnc nt ninStatis Indian Unit, Microelectronics & Computing Advanced = pia ro orcigCd For Code Correcting Error Optimal .I I. error . α NTRODUCTION | 0 ro orcinam oud this undo to aims correction Error . i + e nr unu Systems Quantum Ternary β | 1 i + iai Majumdar Ritajit γ | 2 * i mi:[email protected] Email: , ,β γ β, α, dividual (3 ∈ tems, × 1,* cols rms and izer our red We C ce 3) it, le e - n umt Sur-Kolay Susmita and , , o enr ytm,any systems, ternary For ro.Cnie a Consider error. n[0,adasnl ro a orce sn qutrits. 9 using corrected arbit correct was not systems could error also quantum (3 single model error ternary a this for Nevertheless, and considered [20], was in model error h opeiy n ec h ierqieet o error for requirement, time increases the procedure fo hence correction stabilizers correction. and two-step of complexity, set the the the and basis propose the code, not of did erro that subspaces authors phase pairwise and The correcting error, states. by bit corrected correcting for was steps cascading two eepoeteerrmdl n hwta tecmassall encompasses it that show and model, possible of error the number optimal. explore code we the in proposed on our errors makes bound phase which lower of five, and provable is set bit qutrits The proposed both steps. our correct single code, readily Laflamme 9-qutrit of can previous code stabilizers fo 5-qubit the the formulation Unlike a to stabilizer similar [9]. on is the error which provided QECC, single also this arbitrary have any We correct qutrit. can QECC proposed ceefrtepooe EC n ics t stabilizer 5. its Section in discuss conclude and We 4. 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[15], [14], arxi o ntr o l ausof values all for unitary not is matrix I E II. × hr,i eea,ec ftevalues the of each general, in where, nerr en unu prtr sauiaymti [7]. matrix unitary a is operator, quantum a being error, An nti ae,w rps -urtqatmerrcorrectin error quantum 5-qutrit a propose we paper, this In h eto h ae sognzda olw nScin2 Section in - follows as organized is paper the of rest The ncnrs,teeaefwrsuiso ut-audQECC multi-valued on studies fewer are there contrast, In 3) unu ros utemr,te9qti oerequired code 9-qutrit the Furthermore, errors. quantum RROR (3 × M 3) DLFOR ODEL ros nscin3w rvd h encoding the provide we 3 section In errors. (3 (3 × M × ia nttt,India Institute, tical 1 ternary 3) 3) = (3 arxo h form the of matrix T × ntr prtr n ec our hence and operator, unitary ERNARY c b a j h f g e d 3) unu ytm.Ageneralized A systems. quantum ntr prtri probable a is operator unitary Q UANTUM ,b j , . . . b, a, ,b j , . . . b, a, S YSTEMS n hence and ∈ C This . rary (1) rly ng g r r r . does not represent a quantum error. Nevertheless, we stick Extending the similar argument to the other two sets dictates with this matrix for the time being. Now, we consider the set 9 that if Λiσi =0, then Λi =0 ∀i. of matrices σ ,i =1, 2,... 9. =1 i iP 1 0 0 1 0 0 Lemma II.2. For any matrix M, λ ,i , ,..., 2 (3 × 3) ∃ i =1 2 9 σ1 = 0 0 1 σ2 = 0 0 ω 9 0 1 0 0 ω 0 such that λiσi = M. i=1 1 0 0 0 0 1 P Proof. Consider the matrix M as in Eq. 1. Putting M = σ3 = 0 0 ω σ4 = 0 1 0 9 2 0 ω 0 1 0 0 λiσi yields the following 9 equations 2 =1 0 0 ω 0 0 ω iP σ5 = 0 1 0 σ6 = 0 1 0 λ1 + λ2 + λ3 = a 2 ω 0 0 ω 0 0 2 2 λ1 + ω λ2 + ωλ3 = f 0 1 0 0 ω 0 2 λ1 + ωλ2 + ω λ3 = h σ7 = 1 0 0 σ8 = ω 0 0 0 0 1 0 0 1 λ4 + λ5 + λ6 = e 2 0 ω 0 λ4 + ω λ5 + ωλ6 = c 2 σ9 = ω 0 0 2 λ4 + ωλ5 + ω λ6 = g 0 0 1 λ7 + λ8 + λ9 = i where ω is the cube-root of unity. 2 λ7 + ω λ8 + ωλ9 = b 2 Lemma II.1. σi, i =1, 2,..., 9 are linearly independent. λ7 + ωλ8 + ω λ9 = d
Proof. Let us assume there exists Λi,i =1, 2,..., 9 such that 9 The determinant of the L.H.S. of the first three equations is Λiσi =0, and Λi 6=0 ∀i. We have the set of equations: always non-zero, which implies that it is always possible to i=1 P find λ1, λ2, λ3 which satisfy the set of three equations. Since Λ1 +Λ2 +Λ3 = 0 each set of three equations has disjoint set of coefficients, 2 Λ1 + ω Λ2 + ωΛ3 = 0 similar arguments hold for the other two sets also. 2 Λ1 + ωΛ2 + ω Λ3 = 0 Therefore, for any such matrix M, it is always possible to find linearly independent parameters λi,i = 1, 2,..., 9 such Λ4 +Λ5 +Λ6 = 0 9 2 that λ σ = M. Λ4 + ω Λ5 + ωΛ6 = 0 i i i=1 2 P Λ4 + ωΛ5 + ω Λ6 = 0 Theorem II.3. A QECC which can correct the matrices 7 8 9 Λ +Λ +Λ = 0 σi,i =1, 2,..., 9 can correct any error on a qutrit. 2 Λ7 + ω Λ8 + ωΛ9 = 0 2 Proof. The proof follows directly from Lemma II.1 and II.2. Λ7 + ωΛ8 + ω Λ9 = 0 If {M} is the set of all (3 × 3) matrices, then the set of all Note that these nine equations can be grouped into three possible quantum errors {E}, such that every E ∈{E} is a sets, each set containing three equations. No two sets of unitary matrix, is {E}⊂{M}. Therefore, any E∈{E} can equations involve the same coefficients. In the above nine also be written as a linear combination of σi,i =1, 2,..., 9. equations, the first three, second three, and the last three If a QECC can correct each of σi, it can also correct any error equations form such sets. We show the proof for the set of first E on the quantum system. three equations involving coefficients Λ1, Λ2, Λ3. The proof If we consider the matrices σ ,i =1, 2,..., 9 as errors, then for the other two sets will be identical. i it is easy to see that each of those matrices keep one of the If the set of matrices σ , i = 1, 2,..., 9 are not linearly i basis states unchanged, and affects the other two bases. For independent, at least two of the three coefficients Λ1, Λ2, Λ3 example, if the quantum state is |ψi = α |0i + β |1i + γ |2i, must be non-zero. If only one of them is non-zero, then it then σ1 |ψi = α |0i+β |2i+γ |1i. The basis state |0i remains does not satisfy the first equation, and hence such a scenario unaffected while the other two bases are flipped. Such errors is ruled out. Without loss of generality, let us assume that prove to be difficult to correct in our framework. Hence, we Λ1 = −(Λ2 +Λ3) 6= 0. Substituting Λ1 in the second and aim for some further tuning. We consider the matrices X1, X2, third equations respectively yields Z1 and Z2 as errors and show that all the σi,i = 1, 2,..., 9 2 Λ2 1 − ω matrices can be written as the linear combination of these (ω − 1)Λ2 = (1 − ω)Λ3 ⇒ = 2 Λ3 ω − 1 matrices and their products. Therefore, any (3 × 3) quantum 2 2 Λ2 1 − ω error E can be written as a linear combination of X1, X2, Z1 (ω − 1)Λ2 = (1 − ω )Λ3 ⇒ = Λ3 ω − 1 and Z2 and their products. 2 Equating the ratios of Λ2 and Λ3 gives ω = ω , which is not 0 0 1 0 1 0 possible. Therefore, to satisfy the first set of three equations, X1 = 1 0 0 X2 = 0 0 1 each of the coefficients must be zero. 0 1 0 1 0 0 1 0 0 1 0 0 2 Z1 = 0 ω 0 Z2 = 0 ω 0 0 0 ω2 0 0 ω |0Li = |00000i + |01001i + |02002i + |10100i ω ω2 ω2 We explicitly show the formulation (upto a scalar coeffi- + |11101i + |12102i + |20200i + |22211i 2 cient) of σ1 and σ2 matrices using X1, X2, Z1 and Z2. The + ω |22202i + |01010i + ω |02011i + ω |00012i other matrices can also be formulated similarly. + ω |11110i + |12111i + ω |10112i + ω2 |21210i 2 2 2 1 0 0 1 0 0 1 0 0 + ω |22211i + ω |20212i + |02020i + ω |00021i 2 2 2 0 0 1 = 0 1 0 + 0 ω 0 + ω |01022i + ω |12120i + ω |10121i + ω |11122i 0 1 0 0 0 1 0 0 ω2 2 + ω |22220i + ω |20221i + |21222i + |00101i 1 0 0 0 1 0 + ω |01102i + ω2 |02100i + ω |10201i + |11202i 2 + 0 ω 0 + 0 0 1 2 2 2 2 + ω |12200i + ω |20001i + ω |21002i + ω |22000i 0 0 ω 1 0 0 2 2 2 + |00202i + ω |01200i + ω |02201i + ω |10002i 0 ω 0 0 ω 0 2 2 2 + 0 0 1 + 0 0 1 + ω |11000i + ω |12001i + ω |20102i + ω |21100i 2 ω2 0 0 ω 0 0 + |22101i + ω |01111i + |02112i + ω |00110i 2 0 0 1 0 0 ω2 + |11211i + |12212i + |10210i + ω |21011i 2 2 + 1 0 0 + ω 0 0 + |22012i + ω |20010i + ω |01212i + ω |02210i 0 1 0 0 1 0 2 2 + ω |00211i + ω |11012i + |12010i + ω |10011i 0 0 ω 2 2 2 + ω |21112i + ω |22110i + |20111i + ω |02121i + ω 0 0 2 2 2 0 1 0 + ω |00122i + ω |01120i + ω |12221i + |10222i 2 2 ω ω ω = I + Z1 + Z2 + X2 + ωZ2X2 + ω Z1X2 + |11220i + |22021i + |20022i + |21020i 2 2 2 ω ω ω +X1 + ω Z2X1 + ωZ1X1 + |02222i + |00220i + |01221i + |12022i + ω |10020i + |11021i + |22122i + |20120i 1 0 0 1 0 0 1 0 0 + |21121i 2 2 0 0 ω = 0 ω 0 0 0 1 The other two logical states are reported in Appendix A. 0 ω 0 0 0 ω 0 1 0 For error correction, the logical states should conform to the 2 = I + Z1 + Z2 + Z2X2 + ωZ1X2 + ω X2 Knill-Laflamme condition [21] which states that for any error 2 +Z2X1 + ω Z1X1 + ωX1 E, h0L| E |0Li = h1L| E |1Li = h2L| E |2Li. One can check easily that the condition is satisfied for this encoding. In accordance with [20], we call the errors X1 and X2 as “bit shift errors”, and the errors Z1 and Z2 as “phase errors”. B. Optimality Of 5-Qutrit Code The action of these errors on a general qutrit |ψi = α |0i + There are two bit errors (Xi), two phase errors (Zj), and β |1i + γ |2i are shown below: therefore four possible Yij = XiZj errors in the error model.
i 2i Therefore, there are eight possible error states for each of the Zi |ψi = α |0i + ω β |1i + ω γ |2i physical qutrits and one error free state. In an n-qutrit code,
Xi |ψi = α |0+ ii + β |1+ ii + γ |2+ ii (8n + 1) error states are possible for each of the |0iL, |1iL and |2iL states, leading to 3(8n + 1) possible error states. For where i ∈ {1, 2} and the addition is modulo 3. Therefore, successful error correction, it is necessary that these possible the error model to correct any (3 × 3) quantum error can be error states reside in orthogonal subspaces of the Hilbert Space summarized as in Eq. 2. associated with the logical qutrit. Since an n-qutrit state is 2 2 associated with a 3n-dimensional Hilbert Space, the necessary E = δI3 + ηiZi + (µj Xj + ξij Yij ) (2) condition is =1 =1 Xi Xj Xi,j 3(8n + 1) ≤ 3n I where 3 is the (3 × 3) identity matrix, Xi and Zi are the bit The above inequality is satisfied for n ≥ 5. Therefore, a single C and phase errors respectively, and Yij = XiZj ; δ,η,µ,ξ ∈ . qutrit of information must be distributed into at least 5 qutrits in order to correct a single error. Our proposed QECC requires III. TERNARY QUANTUM ERROR CORRECTION 5 qutrits for encoding, and is, therefore, optimal in the number of qutrits. This bound, also known as Quantum Hamming A. 5-Qutrit Error Correcting Code Bound, is derivable from [14]. The quantum information in a single qutrit |ψi = α |0i + β |1i + γ |2i is distributed into five qutrits in order to protect C. Performance Analysis Of The Code the information from a single error. The encoded logical qutrit Our proposed QECC can correct a single error only and is |ψLi = α |0Li + β |1Li + γ |2Li where fails if more than one error occur on the encoded qutrit. As the set of stabilizers (shown in next section) indicates, this stabilizers which encode a single logical qutrit. Therefore, the code is not a CSS code. Hence, this code is unable to correct 5-qutrit code is a non-CSS code. a single bit and phase errors together even when they occur on different qutrits [22]. If p is the probability that a single qutrit is erroneous, then the probability that the code fails is B. Stabilizer Structure For The 5-Qutrit QECC The general stabilizer structure for higher dimensional quan- 5 5 4 1 − (1 − p) − p(1 − p) tum systems, as proposed by Gottesman [26], is 1 4 = 1 − (1+4p)(1 − p) X |ji = |j +1i mod d Z |ji = ωj |ji 2 3 = 10p + O(p ) where d is the dimension of the quantum system. If no error correcting code is applied, then the probability The proposed stabilizers for correcting bit errors in the of error on the logical qutrit p is equal to the probability of codeword are as follows: error of the physical qutrit. However, if the physical qutrit is S1 = I ⊗ X ⊗ Z ⊗ Z ⊗ X encoded using our proposed QECC, the probability of error on the logical qutrit is reduced to 10p2+O(p3). Since p ≃ 10p2+ S2 = X ⊗ I ⊗ X ⊗ Z ⊗ Z 3 1 1 O(p ) for p = 10 , for p < 10 , this technique provides an S3 = Z ⊗ X ⊗ I ⊗ X ⊗ Z improved method for preserving the coherence of the qutrits. S4 = Z ⊗ Z ⊗ X ⊗ I ⊗ X
IV. STABILIZER FORMULATION These set of stabilizers is equivalent to the 5-qubit code [9]. However, one can come up with a different set of four A. Error Correction Via Stabilizers stabilizers which leads to different codewords. In Table I we ⊗n A set of operators S1,S2,...Sm ∈ {I, σx, σz} is said show the action of the stabilizers on the errors X1,X2,Z1,Z2 to stabilize a quantum state |ψi if the following criteria are when the error occurs on the first qutrit only. The eigenvalues satisfied [23] corresponding to the stabilizers for errors occurring on other 1) Si |ψi = |ψi ∀ i. qutrits can be obtained similarly. 2) For all errors E, ∃ j such that Sj (E |ψi) = −(E |ψi). The -1 phase is for binary quantum systems. For ternary TABLE I systems, this condition will be updated as Sj (E |ψi)= ERROR CORRECTION WITH STABILIZERS FOR ERRORS ON THE FIRST 2 QUTRIT ω(E |ψi) or Sj (E |ψi)= ω (E |ψi). ′ 3) For different errors E and E , ∃ j, k such that Error State S1 S2 S3 S4 ′ Sj (E |ψi) 6= Sk(E |ψi). |ψi +1 +1 +1 +1 X1 |ψi +1 +1 ω ω 4) ∀ i, j, [Si,Sj ]=0. 2 2 X2 |ψi +1 +1 ω ω 2 Furthermore, an n-qudit state with m stabilizers can encode Z1 |ψi +1 ω +1 +1 k = n − m logical qudits. Therefore, the task of error Z2 |ψi +1 ω +1 +1 correction becomes equivalent to finding a set of stabilizers for the encoded quantum state. A major shortcoming of the 9-qutrit code [20] is that The stabilizers for Shor [8] and Steane code [24], called it fails to provide a complete set of stabilizers for error CSS codes, can be partitioned into two disjoint sets of op- correction. Therefore, the bit error correction is two step erators, where one set of operators ({Sx}) consists only of - first to detect the presence of error, and then to de- I and σx, and the other set ({Sz}) only of I and σz. The termine its location. Similarly, for phase errors, individual corresponding circuit of such codes are easy to implement subspaces were corrected. Since there are three subspaces [22]. Furthermore, these codes can also correct a single σx ({|0i , |1i},{|1i , |2i},{|2i , |0i}), this is a three step process. and σz errors if they occur on different qubits. The circuit Our proposed QECC overcomes these shortcomings. Operat- realization of non-CSS codes (e.g. [9]) is resource-extensive ing the proposed four stabilizers on the codeword can correct [22], [25], and such codes cannot correct two errors in any a single error. Therefore, two (three) steps are not required for scenario [22]. the correction of a single bit (phase) error. However, for a 5-qutrit code, it is not possible to have a The circuit realization of Laflamme’s 5-qubit code is not CSS type stabilizer structure. From the error model, each qutrit trivial. Furthermore, the gates used in the circuit of that code can incur two types of bit (phase) errors. Furthermore, only are not mostly implementable in modern day technology. It a single error is assumed to have occurred on the codeword. was shown in [25] that the quantum cost of that circuit is This accounts for 2 × 5=10 possible single bit (phase) error significantly higher than that of Shor and Steane code. We combinations. Each of the stabilizers can have one of the three invite the readers to come up with a circuit for this 5-qutrit possible outcomes: 1,ω,ω2. Hence at least 3 stabilizers are code. To the best of our knowledge, no universal gate set is required to detect 10 bit (phase) error combinations and the available for ternary quantum systems. However, the MS gates error free state. Therefore, in order to obtain a set of operators [27] are shown to be implementable in Ion-Trap technology. It S = {Sx,Sz}, |S| ≥ 6. However, from the equation k = may be worthwhile to try to realize the circuit for this proposed n − m, it is evident that for a 5-qutrit QECC, there are four code using MS gates and Chrestenson gates [28]. V. CONCLUSION
2 2 In this paper we have proposed a 5-qutrit error correcting |1Li = |11111i + ω |12112i + ω |10110i + ω |21211i 2 2 2 code which can correct any arbitrary (3 × 3) error on a + ω |22212i + ω |20210i + ω |01011i + ω |02012i qutrit. The error model considered in this paper overcomes ω2 ω2 ω2 the shortcomings of specialized channels considered in the + |00010i + |12121i + |10122i + |11120i 2 2 previous literature for ternary QECC. Five qutrits are necessary + ω |22221i + |20222i + ω |21220i + ω |02021i for correcting a single error in a qutrit. Hence, our code is + ω |00022i + |01020i + ω |10101i + ω2 |11102i 2 optimal in the number of qutrits. The stabilizer structure of + |12100i + ω |20201i + ω |21202i + |22200i this code is similar to the 5-qubit QECC. Furthermore, our ω2 proposed code can correct both bit and phase error in single + |00001i + |01002i + |02000i + |11212i 2 2 2 steps, as compared to multi-step corrections in the previous + ω |12210i + ω |10211i + ω |21012i + |22010i 9-qutrit code. The future scope is to come up with the circuit + ω |20011i + ω2 |01112i + ω |02110i + |00111i for this code, and also to search for the minimum qutrit CSS + ω |11010i + ω2 |12011i + |10012i + ω2 |21110i code. + ω |22111i + |20112i + |01210i + |02211i + |00212i + ω2 |12222i + |10220i + ω |11221i + |22022i + ω2 |20020i + ω |21021i + ω |02122i ACKNOWLEDGEMENT + ω |00120i + ω |01121i + ω2 |12020i + ω |10021i Ritajit Majumdar would like to acknowledge Prof. Debasis + |11022i + ω |22120i + ω |20121i + |21122i Sarkar, University of Calcutta, for helpful discussions on the + |02220i + ω |00221i + ω2 |01222i + ω2 |10202i error model. + ω |11200i + |12201i + ω |20002i + ω |21000i + ω |22001i + |00102i + ω |01100i + ω2 |02101i + |10000i + |11001i + |12002i + |20100i APPENDIX + ω |21101i + ω2 |22102i + |00200i + ω2 |01201i + ω |02202i The encoded logical qutrit is |ψLi = α |0Li+β |1Li+γ |2Li where
2 |0Li = |00000i + |01001i + |02002i + |10100i |2Li = |22222i + ω |20220i + ω |21221i + ω |02022i 2 2 2 2 2 + ω |11101i + ω |12102i + |20200i + ω |22211i + |00020i + ω |01021i + ω |12122i + ω |10120i + ω |22202i + |01010i + ω |02011i + ω2 |00012i + ω2 |11121i + ω |20202i + |21200i + ω2 |22201i 2 2 + ω |11110i + |12111i + ω |10112i + ω |21210i + |00002i + |01000i + |02001i + ω |10102i + ω2 |22211i + ω2 |20212i + |02020i + ω2 |00021i + |11100i + ω |12101i + ω2 |21212i + ω2 |22210i 2 2 2 2 2 + ω |01022i + ω |12120i + ω |10121i + ω |11122i + ω |20211i + ω |01012i + |02010i + ω |00011i + ω |22220i + ω2 |20221i + |21222i + |00101i + ω2 |11112i + ω |12110i + |10111i + ω |22020i 2 2 + ω |01102i + ω |02100i + ω |10201i + |11202i + |20021i + ω |21022i + |02120i + |00121i + ω2 |12200i + ω2 |20001i + ω2 |21002i + ω2 |22000i + |01122i + ω2 |12220i + |10221i + ω |11222i + |00202i + ω2 |01200i + ω |02201i + ω2 |10002i + ω2 |22121i + ω2 |20122i + ω2 |21120i + ω2 |02221i + ω2 |11000i + ω2 |12001i + ω |20102i + ω2 |21100i + |00222i + ω |01220i + ω2 |12021i + ω |10022i + |22101i + ω |01111i + |02112i + ω2 |00110i + |11020i + |20000i + |21001i + |22002i + |11211i + |12212i + |10210i + ω2 |21011i + |00100i + ω |01101i + ω2 |02102i + |10200i + |22012i + ω |20010i + ω2 |01212i + ω2 |02210i + ω2 |11201i + ω |12202i + ω2 |20101i + |21102i + ω2 |00211i + ω2 |11012i + |12010i + ω |10011i + ω |22100i + |00201i + ω2 |01202i + ω |02200i + ω2 |21112i + ω |22110i + |20111i + ω2 |02121i + ω |10001i + ω |11002i + ω |12000i + ω2 |21010i 2 2 2 2 + ω |00122i + ω |01120i + ω |12221i + |10222i + |22011i + ω |20012i + |01110i + ω |02111i + ω |11220i + ω2 |22021i + ω |20022i + |21020i + ω |00112i + ω |11210i + ω |12211i + ω |10212i 2 2 2 + ω |02222i + ω |00220i + |01221i + ω |12022i + ω |21111i + ω |22112i + |20110i + ω |01211i + ω |10020i + |11021i + |22122i + |20120i + ω |02212i + ω |00210i + |11011i + ω |12012i 2 + |21121i + ω |10010i REFERENCES [24] A. 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