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(CWS) ROPERTIES n ae ewill we later and S W , K Each . 6 , ))) = 2)) dimensions s λ together , nthis In . ∈ { s w oeof code | S = w l Our . } f l S cted | l K ± i | S =1 a , (1) cal a , S w 1] re 1 al is i o e i 1 l , , , 2 minimum distance d. A distance d code can also be used to correct errors up to weight (d 1)/2 . The conditions for 1 2 error correction were found in⌊ [14],− [15].⌋ The error correction conditions for a general code with basis vectors wl are that, in order to detect errors from a set , it is necessary| i and sufficient to have E

ci E cj = cEδij (2) 6 7 3 h | | i Z Z for all E . For a code of the form described above, this Z becomes ∈ E † S w Ewj S = cEδij . (3) h | i | i To correct errors on a fixed number of qubits, it is sufficient 5 4 to study errors of the form ZvXu with bounded weight since these form a basis [14]. This leads to the necessary and X Z sufficient conditions for detecting errors in that for all E E ∈E † i = j wi Ewj S (4) ∀ 6 6∈ ± Fig. 1. Example of the induced error on a graph state: The state has stabilizer and generators XZIIIZZ, ZXZIIII, IZXZIIZ, IIZXZII, IIZZXZZ, ZIIIZXI, and ZIZIZIX. An X error applied to node 5 in the lower-left † is translated by multiplying with the stabilizer element IIZZXZZ and turns i wi Ewi S or (5) Z ∀ 6∈ ±  into errors on the nodes indicated. † i wi Ewi S or (6) ∀ ∈  † through the in the error-correction conditions which can at i wi Ewi S (7) E ∀ ∈−  worst pick up a sign depending only on E. The two conditions Eq. (4) is the condition that two codewords should not be with S on the right are insensitive to this and the other two confused after an error, while the final three conditions express conditions± at most change places. that each error must either be detected (Eq. (5)), or the code This structure theorem gives rise to the following lemma, must be “immune” to it–i.e. the code is degenerate. which is at the heart of our construction:

Theorem 1 An ((n,K)) codeword stabilized code with word K Lemma 2 A single qubit Pauli error Z, X or Y = ZX operators W = wl l=1 and codeword stabilizer S is locally Clifford-equivalent{ to} a codeword stabilized code with word acting on a codeword w S of a CWS code in standard form is c | i operators w′ = Z l and codeword stabilizer S′ generated by equivalent up to a sign to another (possibly multi-qubit) error l consisting only of Zs. ′ rl Sl = XlZ . (8) Proof: Let the error E act only on the ith qubit. If it In other words, any CWS code is locally equivalent to a i is a Z error the result is immediate. Otherwise use the fact CWS code with a graph-state stabilizer and word operators r that Eiw S = EiSiw S , and take Si to be the generator consisting only of Zs. The set of ls form the adjacency matrix | i ± | i {0,1} of the graph. Moreover, the word operators can always be having X on bit i. Then since Ei = Zi Xi the X in Ei cancels with the from and we are left with the s from chosen to include the identity. We call this standard form. X Si Z Si as well as a Zi if Ei was ZiXi. Proof: First note that S is local-Clifford equivalent to Lemma 2 allows us to construct CWS codes with a sat- a graph state due to [16], [17], [18]so there is some local- isfying interpretation: X errors on any qubit are “pushed” n ′ Clifford unitary C = l=1 Cl that maps S to S of the outwards along the edges of the graph and transformed into † form (8). In the new basisN the word operators are CwlC = Zs. This is illustrated in figure 1. Similarly Y errors are pushed a b Z l X l , and we have along the edges, but also leave a Z behind at their original ± b locations. Since all errors become s, we can think of the † ′ ( l)i cl Z CwlC (S ) = Z , (9) i ± error model as classical, albeit consisting of strange multi- Yi bit errors. We define this translation to classical errors by the c ′ l n so that, letting wl = Z , we have function ClS(E ) 0, 1 : ∈E →{ } cl ′ † ′ ′ † ′ Z S = CwlC s S = CwlC S = Cwl S . n | i ± | i ± | i ± | i v u ClS(E = Z X )= v (u)lrl (10) Since C consists of local Clifford elements, we see that the ± ⊕ Ml=1 CWS code defined by S′ and w′ is locally Clifford equivalent to the original code. where rl is the lth row of the stabilizer’s adjacency matrix r Finally, to ensure the codeword operators include the iden- (recall from Eq. (8) Sl = XlZ l defines rl). The codeword c ˜ ′ ′ l c tity we can choose W = w˜l=wlw1 which always has operators wl = Z will be chosen to so that the ls are a { } ′ w˜1 = Identity. This can be seen by commuting the w1 classical code for this error model. 3

Theorem 3 A CWS code in standard form with stabilizer S Proof: To see that this CWS code describes the original c and codeword operators Z c detects errors from if and code, note that the stabilizer state associated with S is 0¯ ... 0¯ , { } ∈C E | i only if detects errors from ClS ( ) and in addition we have while the codeword generated by Wv acting on 0¯ ... 0¯ is C E | i for each E, (v¯) ... (v¯)k . | 1 i ClS(E) = 0 (11) c 6 c Theorem 5 If the word operators of an ((n,K)) CWS code or i Z i E = EZ i . (12) are an abelian group W (not containing I), then the code ∀ − Thus, any CWS code is completely specified by a graph state is an [n, k = log2 K] stabilizer code. stabilizer S and a classical code . C Proof: The stabilizer S of the CWS code is a maximal † abelian subgroup of the Paulis (not containing I) therefore it Proof: When i = j, wi Ewj S is satisfied ex- ′ − ci cj 6 6∈ ± is isomorphic to the group S = X1 ...Xn and the mapping actly when Z EZ S, which is in turn equivalent to ′ h i c Cl E c 6∈ ± from S to S is a Clifford operation C (not necessarily local). Z i Z S( )Z j S. In standard form, the only element of S without any X6∈is ± the identity, so that this is satisfied exactly This follows from the definition of the Clifford group as the automorphisms of the Pauli group. Because this automorphism when ci ClS(E) = cj . This is explicitly the classical error- detection⊕ condition.6 group allows one to achieve any bijective mapping that pre- serves commutation relations (see Chapter 4 of [6]), the map Similarly, when i = j, we must satisfy Eqs. (5), (6) and (7), ′ whose three possibilities translate directly to can further be chosen to map W to W = Z1 ...Zk . Here we have made use of the facts that all w hW anticommutei c c c Z EZ S (13) with at least one s S (which implies S∈ W = I ) and ∀ c 6∈c ± ′ ∈ ∩ { }′ or c Z EZ S (14) that S is maximal, which allows us to choose for W any ∀ c c ∈ order K group made only of Zs we like (since all products or c Z EZ S. (15) ∀ ∈− of X’s are in S′). Note this nonlocal Clifford mapping is not Since Zc = I for the c =0 codeword, Eq. (13) is equivalent the same as the conversion to Zs used in Theorem 1. to E S and therefore to (11). If (11) (and therefore (13)) We can now choose T ′, X¯ ′ and Z¯′ as follows: 6∈ ± c is not satisfied, E S. If any Z anticommutes with E we ′ ′ ∈± X¯ = W = Z1 ...Zk (18) have also E S. Since no s S is also in S this readily h i ∈∓ ∈ − ¯′ implies the equivalence of (12) to (14) and (15). Z = X1 ...Xk (19) ′ h i Remark A classical code expressed in quantum terms would T = Xk ...Xn (20) h +1 i traditionally comprise computational basis vectors that are † eigenstates of , and therefore the operators mapping one The inverse Clifford operation C maps these to our stabilizer Z ¯ ¯ codeword to another would be of the form Xc as these are code with stabilizer T , and logical operations X = W and Z. the only errors that have any effect. It then might seem odd It remains to show this is the same as the CWS code we ′ that standard form for CWS codes, the intuition of which is to started with. T is by construction a subgroup of S (T is ′ make everything classical, would employ word operators and explicitly generated by a subset of the generators of S ) and ¯ effective errors consisting only of Zs. This choice is arbitrary therefore stabilizes S . T also stabilizes all x¯ S , x¯ X, ¯ | i ¯ | i ∈ (one could exchange Z and X and nothing in the formalism since T and X commute. Using X = W we see these states would be affected) and is made since the usual form of a are exactly the basis states of the CWS code. graph state stabilizer is to have one X and some number of Zs rather than the reverse. We hope this historical accident III. EXAMPLES does not cause too much confusion going forward. We now give some examples of our construction and ⊓⊔ including all known nonadditive codes with good parameters. A. Relation to Stabilizer codes The CWS framework includes stabilizer codes, and allows A. The [5, 1, 3] code them to be understood in a new way. We now show that The celebrated [5, 1, 3] quantum code [14], [15] can be writ- any stabilizer code is a CWS code, and give a method for ten as a CWS code using Eqs. (16) and (17) but another way determining if a CWS code is also a stabilizer code. of writing it demonstrates the power of the CWS framework. Take generators corresponding to a ring graph: Theorem 4 An [n, k] stabilizer code with stabilizer gener- Si = ZXZII and cyclic shifts. (21) ators S1,...,Sn−k and logical operations X¯1 ... X¯k and ¯ ¯ Z1 ... Zk, is equivalent to the CWS code defined by This induces effective errors as follows. Letting R5 be the graph state corresponding to the unique simultaneous| i +1 S = S1 ...Sn−k, Z¯1 ... Z¯k (16) eigenvector of these generators, we have and word operators Zi R5 = Zi R5 (v) (v) wv = X¯ 1 ... X¯ k (17) | i | i 1 k Xi R5 = Zi Zi R5 ⊗ ⊗ | i −1 +1| i where v is a k-bit string. Yi R5 = Zi ZiZi R5 , (22) | i −1 +1| i 4 where all additions and subtractions are taken modulo 5. The less than (n 1)/2, leading to an encoded dimension of −n−1 corresponding 15 classical errors are: n−2 ((n−1)/2) 2 1 − .  − 2n 1  Z : 10000 01000 00100 00010 00001 We now show that these are actually CWS codes. Indeed, X : 01001 10100 01010 00101 10010 (23) the codeword stabilizer of this code will be generated by Y : 11001 11100 01110 00111 10011 X Z ...Zn,Z X ,Z X ,...,Z Xn , (26) h 1 2 1 2 1 3 1 i c We then must choose wl = Z l where the cls form a classical with the corresponding stabilizer state being equivalent to a code capable of detecting pairs of these errors. Since no pair GHZ state, ( 0 + ⊗n−1 + 1 ⊗n−1)/√2. The codeword | i| i | i|−i(x) ((x) ,...(x) ) of these errors produces 11111 the codewords c0 = 00000 and operators are simply Wx = X 1 Z 2 n for each c1 = 11111 will serve, and together with the stabilizer (21) allowed x, which can immediately be seen to generate, up completely define the code. Since the ((5, 2, 3)) code is known to local unitaries, the same codewords as Eq. (25). Putting the to be unique we need not otherwise check that our construction stabilizer into standard form, we find that the graph state it is equivalent to the traditional presentation of this code. We describes corresponds to a star graph. note also that for n 7 a ring code with codeword operators n ≥ I and Zl gives a [n, 2, 3] code. ⊗l=1 D. The ((9, 12, 3)) code Like the ((5, 6, 2)) code, the codeword stabilizer is of the B. The ((5, 6, 2)) code form The first nonadditive quantum code was found in [8], Si = ZXZIIIIII and cyclic shifts (27) and encodes a six-dimensional space into five qubits with a minimum distance of two. This outperforms the best additive The associated classical code correcting the induced errors is: five qubit distance two code, which can have an encoded 000000000 100100100 010001100 110101000 dimension of at most four. The code was originally found 000110001 100010101 011001010 111101110 (28) as follows: It was known that the linear programming upper 001010011 101110111 011111111 111011011 bound was exactly 6 for a blocklength 5 distance 2 code, and in fact it was possible to completely determine what the IV. NEW CODES weight enumerator [19] of a code meeting this bound must A. Ring codes: ((10, 18, 3)) be. The authors of [8] then performed a numerical search for such a code, and managed to find one. The structure of the In light of the excellent performance of ring-stabilizers for resulting code was mysterious, and generating larger codes in CWS codes—the ((5, 6, 2)) and ((9, 12, 3)) are both of this a similar fashion seemed intractable (though [20] showed how form—we have studied larger blocklength codes based on this to construct a ((5 + 2l, 22l+13, 2)) code from this code). stabilizer. This leads to a new code that outperforms stabilizer As a CWS code the ((5, 6, 2)) code of [8] becomes simple. codes for blocklength 10. We again use the ring stabilizer (21) and will have to detect The blocklength ten code has a codeword stabilizer gener- the induced errors (23), but since we are seeking a distance-2 ated by Zi−1XiZi+1 and has 18 word operators of the form c l h c i code we need only consider single errors rather than pairs. Z , with l taken from the list c The classical codewords l, l =0 ... 5, are 0000000000 1101001100 0011001010 00000 11010 01101 10110 01011 10101 (24) 0000011111 0010001001 1111100000 1000111110 1100100101 0101101101 R5 c (29) and the code generated by c and Wl = Z l is locally 0001000110 1010010010 0100110100 | i Clifford equivalent to the ((5, 6, 2)) code of [8]. The ((5 + 1001010111 1011010001 0110111000 2l, 22l+13, 2)) codes of [20] are also CWS codes whose graph 0101110010 1110100011 0111111011. state is the union of the ring graph and l Bell pair graphs, and That this code satisfies the required error correction condi- whose classical codewords can be derived straightforwardly tions can be shown by the straightforward (if tedious) tech- from the ((5, 6, 2)) classical codewords. nique of verifying that the associated classical code corrects the classical noise model induced by the ring stabilizer. C. The SSW codes A family of distance two codes was found in [9], which B. A ((10, 20, 3)) Double Ring Code outperforms the family of [20] for odd blocklengths of eleven or larger. The codes were originally described in terms of their We now consider a CWS code with a codeword stabilizer codewords as follows. If n =1 mod4, a basis of our code that is not of the ring form. In particular, our stabilizer will consists of vectors of the form correspond to the double ring, with generators S = XZIIZZIIII S = ZIIIIXZIIZ x + x¯ , (25) 1 6 | i | i S2 = ZXZIIIZIII S7 = IZIIIZXZII where x ranges over all n-bit vectors of odd weight less than S3 = IZXZIIIZII S8 = IIZIIIZXZI (30) (n 1)/2 and x¯ is the complement of x, while if n = 3 S = IIZXZIIIZI S = IIIZIIIZXZ − 4 9 mod 4, we let x range over even weight vectors of weight S5 = ZIIZXIIIIZ S10 = IIIIZZIIZX. 5

This leads to a S that is a [10, 0, 4] stabilizer state. Our new way of thinking of stabilizer codes may help to find new classical code giving| i the codeword s operators is codes with good properties. In fact, this method has since been C used [23] to systematically categorize all codes of n 8 and 0000000000 1100101101 1100000100 0010010010 to find a ((10, 24, 3)) code as well as slightly better distance-2≤ 1001100100 0111011011 1101111110 0010111011 codes. 1001101111 0111010000 1111000101 1011010100 In a future work we hope to expand our work in several 0101100000 1011011111 0101101011 0011000001 new areas. We will give algorithms for finding codes (some of 0000101001 1110010110 0001111010 1110111111. which were employed to find the new codes presented here) as well as bounds on the computational complexity of the V. ENCODING CIRCUITS algorithms. We also hope to find more new codes, especially Thus far, we have focused on the existence and structure of distance higher than three. of CWS codes. We now address a question of fundamental importance: What is the complexity of encoding a CWS code? REFERENCES The answer we find is perhaps the strongest one could hope [1] P. Shor, “Algorithms for quantum computation: discrete logarithms and for: a CWS code will have an efficient encoding circuit as factoring,” Foundations of Computer Science, pp. 124–134, 1994. long as there is an efficient encoding circuit for the classical [2] R. Feynman, “Simulating physics with computers,” Internation Journal code . of Theoretical Physics, vol. 21, pp. 467–488, 1982. C [3] ——, “Quantum mechanical computers,” Foundations of Physics, We will use the fact [22] that a graph state S whose graph vol. 16, no. 6, pp. 507–531, 1986. ⊗| ni ⊗n has edges E is equal to (j,k)∈E P(j,k)H 0 , where [4] S. 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Wootters, c ⊗n ⊗n = Z i P H 0 (33) “Mixed state entanglement and quantum error correction,” Phys.Rev. A., (j,k) | i (j,kY)∈E vol. 54, pp. 3824–3851, 1996, arXiv:quant-ph/9604024. [15] E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” ci = Z S (34) Phys. Rev. A, vol. 55, no. 2, pp. 900–911, 1997. | i [16] D. Schlingermann, “Stabilizer codes can be realized as graph codes,” Quantum Information & Computation, vol. 2, no. 4, pp. 307–323, 2002, arXiv:quant-ph/0111080. [17] M. Grassl, A. Klappenecker, and M. R¨otteler, “Graphs, quadratic form, VI. DISCUSSION and quantum codes,” 2002 IEEE International Symposium on Informa- tion Theory (ISIT), p. 45, 2002. We have presented a new framework for quantum codes [18] M. Van den Nest, J. Dehaene, and B. DE Moor, “Graphical description and shown how it encompasses stabilizer codes, elucidates the of the action of local clifford transformations on graph states,” Phys. structure of the known good nonadditive codes, as well as Rev. 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A. 68, 022312 (2003) the quantum errors to be corrected into effectively classical [23] S. Yu, Q. Chen and C.H. Oh, “Graphical Quantum Error-Correcting errors. And second, a classical code capable of correcting the Codes,” arXiv:0709.1780. induced classical error model. With a fixed stabilizer state, finding a quantum code is reduced to finding a classical code that corrects the (perhaps rather exotic) induced error model. We also show that CWS codes include all stabilizer codes. This