Jan 6, 2008 - V of n vertices and a set of weighted edges specified by the adjacency ...... [10] A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A.. Sloane ...
Graphical Nonbinary Quantum Error-Correcting Codes 1
Dan Hu, Weidong Tang, Meisheng Zhao, and Qing Chen Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
2
Sixia Yu1,2 and C.H. Oh Physics Department, National University of Singapore, 2 Science Drive 3, Singapore 117542 (Dated: February 4, 2008)
arXiv:0801.0831v1 [quant-ph] 6 Jan 2008
In this paper, based on the nonbinary graph state, we present a systematic way of constructing good non-binary quantum codes, both additive and nonadditive, for systems with integer dimensions. With the help of computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, [[6, 2, 3]]p , [[7, 3, 3]]p , [[8, 2, 4]]p and [[8, 4, 3]]p for any odd dimension p and a family of nonadditive code ((5, p, 3))p for arbitrary p > 3. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes ((6, 2 · p2 , 3))2p for odd p, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem. I.
INTRODUCTION
Noises are inevitable and they cause errors in quantum informational processes. One active way of dealing with errors is provided by the quantum error-correcting codes (QECCs) [1, 2, 3, 4], which have found many applications in quantum computations and quantum communications, such as the fault-tolerant quantum computation [5], the quantum key distributions [6], and the entanglement purification [7, 8]. Roughly speaking, a QECC is a subspace of the Hilbert space of a system of many physical subsystems with the property that the quantum data encoded in this subspace can be recovered faithfully, even though a certain number of physical subsystems may suffer arbitrary errors, by suitable syndrome measurements followed by corresponding unitary transformations. An important family of QECCs is the stabilizer code [9, 10, 11], which is specified by the joint +1 eigenspace of a stabilizer, an Abelian group of tensor products of Pauli operators. The stabilizer formalism has also been established in the nonbinary case [12, 13, 14, 15, 16] and many good codes for systems of a prime or a power of prime dimension have been constructed [15, 16, 17, 18], including the well-known perfect code with five registers [17] for all dimensions. Though the majority of QECCs constructed so far are stabilizer codes, including the CSS codes [19], the topological codes [20], color codes [21], and also the recently introduced entanglement-assisted codes [22], there are a few exceptions called as nonadditive codes [23, 24, 25, 26]. The nonadditive code does not admit a stabilizer structure and it should not be a subcode of some larger stabilizer code with the same distance otherwise it will be a trivial nonadditive code. Though difficult to construct and identify the nonadditive codes promise a larger coding subspace since less structured than the stabilizer codes. For qubits the nonadditive error-correcting code that outperforms the stabilizer codes has been constructed [26] based on the binary graph states. A graphical approach [27], as well as a codeword stabilized code
approach [28], to the construction of binary additive and nonadditive codes has been developed based on the binary graph states.
The graph states [29, 30] are useful multipartite entangled states that are essential resources for the one-way computing [31] and can be experimentally demonstrated [32]. The binary graph state proves to an extremely effective tool [27, 30, 33] in the construction of QECCs. The nonbinary graph states were introduced first in [30] and discussed in details in the case of systems of an odd prime dimension [35]. It is also investigated in the context of universal quantum computation [36] and other applications [37]. Recently an approach to construct QECCs based on nonbinary graph states has been introduced in [38] and some new codes are found via computer search for qubits and qutrits.
Here we shall generalize the graphical construction of QECCs [27] to the nonbinary case based on the nonbinary graph states from which some analytic constructions are attainable. In Sec.II the nonbinary graph states are introduced. In Sec.III we introduce the concept of coding clique for a weighted graph and show how it is related to the construction of both stabilizer and nonadditive QECCs. In Sec.IV we present some codes found via numerical searches and the graphical versions of some known codes as illustrations. In Sec.V we construct analytically four families of optimal stabilizer codes that saturate the quantum Singleton bound for any odd dimension as well as a family of nonadditive codes ((5, p, 3))p for all p > 3. In Sec.VI we investigate the graphical codes arising from composite systems and construct a family of stabilizer codes ((6, 2 · p2 , 3))2p with p being odd. The graph states of systems composed of coprimed subsystems are in a one-to-one correspondence with the direct product of the graph states of subsystems.
2
p
V of n vertices and a set ofX weighted edges specified by 1 the adjacency matrix an|li. n × n matrix with |θj i =Γ,√which ωis−lj (2) zero diagonal entries andpthe l∈Zpmatrix element Γab ∈ Zp denotes the weight of the edge connecting vertices a and b. AAssociated with a given G =of(V, Γ), Zp -weighted graph G =weighted (V, Γ) is graph composed a set system of nand qupits are labeled by Vspecified , the graph Vforofa n vertices a setthat of weighted edges by stateadjacency is definedmatrix as [28]Γ ∈ Zn×n the , an n × n matrix with p zero diagonal entries and the matrix Y element Γab ∈ Zp 1 X 1 of V denoting the a |Γi = √then weight ω 2 s·Γ·s |si edge = connecting (Uab )Γab |θvertices (3) 0i . and b. Thepgraph state associated with a given weighted a,b∈V s∈ZV p graph G = (V, Γ) of a system of n qupits labeled with V reads [30]have denoted by ZV the set of all the vectors Here we p
s = (s1 , s2 , . . .X , sn ) with n components sa ∈ Zp (a ∈ V ), Y 1 1 2 s·Γ·s |si = of all (U by|Γi |si=the eigentsate the phase Za √ common ω )Γab |θ0shifts iV . (3) ab n p s a V (a ∈ V ) with s∈Z eigenvalue ω , and by a,b∈V p
iV = |θ0by i1 ⊗ i2 ⊗set . . . of |θ0all in the vectors Here we have|θ0denoted ZVp|θ0the s = (s1 , s2 , . . . , sn ) with n components sa ∈ Zp (a ∈ V ), the joint +1 eigenstate of all bit shifts Xa (a ∈ V ). In by |si the common eigentsate of all the phase shifts Za addition the non-binary controlled phase gate between (a ∈ V ) with eigenvalue ω sa , by two qupits a and b is defined by |θ0 iV = X |θ0 i1 ⊗ij |θ0 i2 ⊗ . . . |θ0 in (4) Uab = ω |iihi|a ⊗ |jihj|b . (4)
i,j∈Zp the joint +1 eigenstate of all bit shifts Xa (a ∈ V ), and by The non-binary graph state |Γi is also the unique (up to a global phase factor)X jointij+1 eigenstate of the following Uab = ω |iihi|a ⊗ |jihj|b (5) n vertex stabilizers i,j∈Zp Y Ga = Xa (Zb )Γab , a ∈ V. (5) the non-binary controlled phase gate between two qupits b∈V a and b. The non-binary graph state |Γi is also the unique s2 sn (upFor to sa ∈global phase factor) eigenstate of the ZVp we define X s =joint X1s1 X+1 2 . . . Xn and simis following n vertex larly for the phase stabilizers shift operator Z . Obviously Y Y 1 s·Γ·s s s·Γ sa Γab GsG≡ (6) aX∈ Z V. (6) a = Xa(Ga ) (Z= b)ω 2 , a∈V
b∈V
s2 arbitrary sn is For also sa ∈ stabilizer of the graph s ∈ ZVp , ZVp we denote X s = state X1s1 Xfor 2 . . . Xn and simii.e., Gfor |Γi. shift All the stabilizers of the graph state s |Γi larly the=phase operator Z s . Obviously belong to the generalized Pauli group for qupits Y 1 ≡ (Gsa )sta = ω 2 s·Γ·s X s Z s·Γ (7) s(p−1)s·t −iGπ V Pn = {e p X Z |s, t ∈ Z } × {ω l |l ∈ Zp }. (7) p a∈V
4
4
2
3
c b
1
5
2
4
3
(e) [[5, 1, 3]]p
1
c b b c b
5
b
c b
2 (d) [[4, 2, 2]]p
5
4
b
3
4 2
1
b c c 3bc b b c b
1 b
c bc b b c cb bb b
2
(c)
cb bc 3cb bcb
1 b
b
b c b
(b)
cb bc2 bcb 3 b1
b c b b b c c cb bb b
(a)
cb bc2 bcb 3 b1
b
cb1 bc2 bc3 b c b b c b
Zp = {0, . , p − 1} the ring with addition modulo p. II.1 . .NONBINARY GRAPH STATES Under the computational basis {|ii|i ∈ Zp } of a qupit, the generalized bit shift and phase shift operators read Here we shall consider a general system with p levels, 2π a qupitX for short, where p X is arbitrary. We denote by l = {0, 1|l. .+ 1ihl|, Z = ring ω |lihl|, (ω =modulo ei p ). (1) ZX = . , p − 1} the with addition p. p l∈Zp Under l∈Z thep computational basis {|ii|i ∈ Zp } of a qupit, the generalized bitZshift phase operators readwe p Obviously Xp = = and I and ZXshift = ωX Z. Here ¯ denote X by X the Hermitian X conjugate of X. The icompu2π ω l |lihl|, ω =eigenvalue e p . |l +|li 1ihl|, = X = basis tational is theZ eigenstate of Z with l∈Zthe p p ω l while eigenstate of Xl∈Z with eigenvalue ω j reads (1) X ZX = ωX Z. Here we Obviously X p = Z p = I1 and −lj ω |li. (2) |θj i = √ denote by X¯ the Hermitianp conjugate of X . The compul∈Zp tational basis |li is the eigenstate of Z with eigenvalue ωl A while the eigenstate eigenvalue ω j reads Z -weighted graphofGX=with (V, Γ) is composed of a set
(f) ((5, 4, 3))4
FIG. 1: Graph, graph states, and graph-state basis. (a-c) Star FIG. states, and basis; graph-state basis. (a-c) graph1:on Graph, 3 qubitsgraph and graph-state (d) Star graph on Star graph on 3 qubits and graph-state basis;Colored (d) Starvertices graph 4 vertices; (d-e) Loop graph on 5 vertices. on 4 vertices; Loop graph 5 vertices. vertices represent also(d-e) a vector in Zp :onwhite, black,Colored blue, and red represent also aweights vector 0,1,2,-1 in Zp : respectively. white, black,Black blue,edges and have red vertices having vertices having also weight 1. weights 0, 1, 2, p − 1 respectively. Black edges have weight 1.
The graph-state basis of the n-qupit Hilbert space Hn c V is also atostabilizer graph arbitrary s ∈ ZVp , refers {|Γc i ≡ of Z the |Γi|c ∈ Zstate Under the computap }. for i.e., G |Γi = |Γi. All the stabilizers of the graph state tionalsbasis a graph-state basis looks like belong to the generalized Pauli group for qupits 1 X 1 s·Γ·s+c·s √t n |Γ−i Z c |Γi = |si. (8) ω2 c iπ= (p−1)s·t s p t ∈ VZVp } × {ω l |l ∈ Zp }. (8) X Z |s, Pn = {e p s∈Z p
The graph-state of the n-qupit HVn A collection of Kbasis different vectors {c1 ,Hilbert c2 , . . . , cspace K } in Zp c V }. Under the computarefers to {|Γ i ≡ Z |Γi|c ∈ Z c p specifies a K dimensional subspace of Hn that is spanned tional basis a graph-state basis by K graph-state basis {|Γ i}Klooks . like ci
i=1
For an example the graph X state corresponding to the 1 1 s·Γ·s+c·s star graph 3 vertices weighted 1(9) as |Γc iS= Z c |Γi = √ nwith all ω 2 edges |si. 3 on p shown in Fig.1(a) represents s∈Z theVp GHZ state
1 X V |S3 iof=K√different |θp−j i1 ⊗ |ji . } in Z(9) A collection vectors {c12,⊗ c2|θ , .p−j . . ,ic3K p p specifies a K dimensional j∈Zp subspace of Hn that is spanned by K graph-state basis {|Γci i}K i=1 . AnFor edge weight will bestate represented by a black an with example the1 graph corresponding to line the and an edge with weight p−1 will be represented by star graph S3 on 3 vertices with all edges weighted a1red as thick line as in Fig.2. In addition we state will also indicate shown in Fig.1(a) represents the GHZ a vector in ZVp via colored vertices with white, black, blue, and red vertices 1 X representing weights 0, 1, 2, p − 1 |S i = |θp−j i1 ⊗graph |ji2 ⊗shown |θp−j iin (10) √ 3 3 . Fig.1(b) respectively. For example in the p 3 p indicated via the colored vertices a vector (1, 0, 2) ∈ Zj∈Z is p and therefore we have a graph-state base An edge with weight 1 will be represented by a black line X p−1 will be represented by a red 1 weight and an2 edge with Z |θInp−j−1 i1 ⊗ |ji |θp−j−2 i3 . (10) 1 Z3 |S3 i = √ thick line as in Fig.2. addition we2 ⊗ will also indicate p j∈Zp V a vector in Zp via colored vertices with white, black, blue, and red vertices representing weights 0, 1, 2, p − 1 respectively. For example in the graph shown in Fig.1(b) a vectorIII. (1, 0,CODING 2) ∈ Z3p is CLIQUES indicated via the colored vertices AND QECCS and therefore we have a graph-state base From the standard 1 X theory of the QECC we know that Z32 |S i = √errors can |θp−j−1 i1 ⊗ |ji2 ⊗ |θp−j−2 i3 .errors (11) ifZa1 set of3Pauli be corrected then all the p j∈Z p can also be corrected. The situation in non-binary case is the same, because the nonbinary error basis defined by
3 III.
CODING CLIQUES AND QECCS
From the standard theory of the QECC we know that if a set of Pauli errors can be corrected then all the errors can also be corrected. The situation in non-binary case is the same, because the nonbinary error basis defined by E = {X s Z t |s, t ∈ ZVp } is a nice basis [12, 16], i.e., the set E forms a basis for the operators acting on the n-qupit Hilbert space Hn . For a given weighted graph G = (V, Γ), from the definition the vertex stabilizers of a graph state, we have 1
X s Z t = ω − 2 s·Γ·s Gs Z t−s·Γ .
(12)
That is to say every non-binary Pauli error acting on the graph state |Γi can be equivalently replaced by some qupit phase flip errors, up to some phase factors. For convenience we refer that the vector t − s · Γ is covered by the error X s Z t . Given an integer d we introduce a d-uncoverable set as o n Dd = ZVp − t − s · Γ 0 < | sup(s) ∪ sup(t)| < d , (13)
where we have denoted by sup(s) = {a ∈ V |sa 6= 0} the support of a vector s ∈ ZVp and by |C| the number of the elements in C ⊆ V . That is to say Dd is the set of all the d-uncoverable vectors, i.e., vectors cannot be covered by Pauli errors acting nontrivially on less then d qupits, in ZVp . In addition we define the d-purity set as n o Sd = s ∈ ZVp | sup(s) ∪ sup(s · Γ)| < d . (14)
It is obvious that if s ∈ Sd then the graph stabilizer Gs has a support less than d, i.e., act nontrivially on less than d qupits. A coding clique CK d of a given graph G = (V, Γ) is a collection of K different vectors in ZVp that satisfy: i) 0 ∈ CK d ; ii) s · c = 0 for all s ∈ Sd and every c ∈ CK d ; iii) c − c0 ∈ Dd for all c, c0 ∈ CK d . forms a group with respect to the If the coding clique addition modulo p, then it will be referred to as a coding group. In words, a coding clique CK d is a collection of K d-uncoverable vectors in ZVp that is orthogonal to the vectors in Sd and the difference between any two vectors in CK d is also d-uncoverable. We can build a super graph G with vertices being the vectors in Dd ∩ S⊥ d , i.e., duncoverable vectors that are orthogonal to all the vectors in Sd , and two vertices in the super graph are connected by an edge iff their difference is also d-uncoverable. Then the coding clique CK d is exactly a K-clique, a subset of the vertices which are pairwise connected, of the super graph G. This fact justifies the nomenclature clique. Theorem 1 Given a graph G = (V, Γ) and one of its coding clique CK d , the subspace (G, K, d)p spanned CK d
by the graph-state basis |Γc i|c ∈ CK is an ((n, K, d))p d code, which is a stabilizer code if CK is a group and a d nonadditive code if CK is neither a group nor a subset of d 0 0 a coding group CK of G with d ≥ d. d0 Proof. To prove that the subspace spanned by the graph-state basis {|Γc i|c ∈ CK d } is an ((n, K, d))p code we have only to show that the condition [3, 30] hΓc |Ed |Γc0 i = f (Ed )δcc0
(15)
s t is fulfilled for all c, c0 ∈ CK d and all the error Ed = X Z that acts nontrivially on a number of qupits that is less than d, i.e., | sup(s) ∪ sup(t)| < d. If the error is proportional to a stabilizer of the graph state, i.e, Ed = f (s)Gs for some s ∈ ZVp and phase f (s), then s ∈ Sd . Because of condition ii) of the coding clique the error acts like a constant operator on the subspace (G, K, d) so that the condition Eq.(15) is fulfilled with a f (Ed ) = f (s) that is independent of c. If the error Ed is neither one of the stabilizers of the graph state |Γi nor the identity operator, i.e., Ed = X s Z t ∝ Gs Z t−s·Γ with t 6= s·Γ, then hΓc |Ed |Γc0 i ∝ hΓ|Z e |Γi with e = c0 −c+t−s·Γ. By virtue of condition iii of the coding clique it is ensured that e 6= 0 for all c, c0 ∈ CK d which gives rise to Eq.(15) with f (Ed ) = 0. Thus we have proved that (G, K, d)p is an ((n, K, d))p code. By definition, a nonbinary stabilizer code is the joint +1 eigenspace of an Abelian subgroup of generalized Pauli group Pn given in Eq.(8). The subspace (G, K, d)p spanned by the graph-state basis {|Γc i|c ∈ CK d } is stabilized by an element S of Pn if and only if S is one of the stabilizers of the graph state Gs with s ∈ S where S = s ∈ ZVp s · c = 0, ∀ c ∈ CK . (16) d
If CK d is a coding group then it is obviously an Abelian group and can be generated by a set of vectors hc1 , c2 , . . . , ck i in ZVp with degrees [µ1 , µ2 , . . . , µk ], i.e., the minimal positive number such that µi ci = 0 (mod p) (i = 1, 2, · · · k). It is easy to see that all µi ’s divide p so Qk that K = i=1 µi divides pn . Furthermore (ω = ei2π/p ) pn =
X X
c∈CK d
s∈ZV p
ω s·c =
X X
K s∈ZV p c∈Cd
ω s·c = |S|K
because 0 ∈ CK d for the first equality and k µX i −1 K if s ∈ S, X Y ω s·c = ω mi s·ci = 0 if s ∈ 6 S, i=1 mi =0 c∈CK d
(17)
(18)
(since µi ci = 0 (mod p)) for the last equality. That is to say if CK d is a group then we can find exactly a number pn /K of stabilizers {Gs |s ∈ S} whose joint +1 eigenspace is exactly (G, K, d)p . Thus the code (G, K, d)p is a stabilizer code. If CK d is not a group then we denote by C the group generated by CK d , i.e., the smallest group that contains
4 CK d . Obviously the subspace Q spanned by {|Γc i|c ∈ C} is the joint +1 eigenspace of stabilizer {Gs |s ∈ S} and |S| < pn /K. If the subspace Q can detect d − 1 or more errors then C is a coding group of G. If the subspace Q cannot detect d−1 errors then any subset of the stabilizer cannot either. On the other hand every stabilizer code that contains (G, K, d)p must have a stabilizer that is a subset of the stabilizer of Q. Therefore if CK d is not a 0 0 subset of some coding group CK of G with d ≥ d then d0 the code (G, K, d) is a nonadditive code. Q.E.D. Similar conclusions about the stabilizer codes appeared also in [38]. If all the generators of a coding group have the maximal degree p then corresponding stabilizer code can be denoted as [[n, k, d]]p . However there are cases where the generators of the coding group are not all of maximal degree. Then we have still a stabilizer code but the dimension of the code subspace may not be a power of p and we shall denote such a stabilizer code as ((n, µ1 ·µ2 ·. . .·µk , d))p . A family of such kind of stabilizer codes will be provided in Sec. VI.
IV.
GRAPHICAL NONBINARY QECCS VIA NUMERICAL SEARCH
According to Theorem 1 we can use the same systematic algorithm developed for binary case in [27] to do a systematic search for the non-binary quantum codes, i.e. i) To input a Zp -weighted graph G = (V, Γ) on n vertices; ii) To choose a distance d and compute the d-purity set Sd and the d-uncoverable set Dd so that a super G can be built; iii) To find all the K-clique CK d of the super graph G; iv) To output a (G, K, d)p , i.e., an ((n, K, d))p code that is spanned by the basis {|Γc i|c ∈ CK d }. In practice, we have used the clique finding program cliquer [39] to search for the cliques for the super graph. Within our present computation power systematic search for graphical codes can be done up to p = 6 and n = 6 and some tentative searches have been done for n = 8. In what follows a quantum code will be specified by a weighted graph together with a coding clique or the generators of a coding group. Though the search for cliques are hard, the verifications of them are relative easy.
A.
The code [[3, 1, 2]]3
The first example is the stabilizer code [[3, 1, 2]]3 , which is known and has been constructed, e.g., in [15]. We consider the star graph S3 on 3 vertices with two edges all weighted 1 as shown in Fig.1(a). A coding group generated by (1, 0, 2) ∈ Z33 as shown in Fig.1(b) provides the
code [[3, 1, 2]]3 , i.e., a 3-dimensional subspace spanned by the graph-state basis a 2a Z1 Z3 |S3 i|a ∈ Z3 . (19) The stabilizer of this code is generated by hG2 , G1 G3 i with Ga being defined in (6). It is easy to see that the stabilizer is equivalent to the stabilizer hX123 , Z123 i appeared in [8] under a local unitary transformation. It is not difficult to see that we can have a [[3, 1, 2]]p for any odd p with the same graph and the same coding group generated by (1, 0, −1) ∈ Z3p with stabilizer hG2 , G1 G3 i. B.
The code ((3, 3, 2))4
For an odd number of qupits with p being even there is no code of distance 2 that saturates the Singleton bound so far. We have found a suboptimal code ((3, 3, 2))4 instead. For the equal weighted star graph on 3 vertices as shown in Fig.1(a) we have found a coding clique: {(0, 0, 0), (1, 0, 2), (2, 0, 1)}, with corresponding graph-state basis shown in Fig.1(a-c), which span the code subspace of a nonadditive ((3, 3, 2))4 code. In fact, as we see later, we can construct a code ((3, p − 1, 2))p with even p. C.
The code [[4, 2, 2]]6
The next example we concerned is also a 1-error detecting code [[4, 2, 2]]6 which can be constructed from the star graph S4 on 4 vertices as shown in Fig.1(d). From this graph a 2 dimensional coding group can be found to be generated by vectors (1, −1, 0, 0) and (1, 0, −1, 0) in Z46 . That is to say the code [[4, 2, 2]]6 is the 36-dimensional subspace spanned by the basis −a−b a b (20) Z1 Z2 Z3 |S4 i a, b ∈ Z6 .
whose stabilizer is generated by hG1 G2 G3 , G4 i. On the same graph we also find another coding clique as (a + b, −a, −b, δa1 δb1 ) ∈ Z46 a, b ∈ Z6 (21) which can be stabilized by hG1 G2 G3 i only thus we have a nonadditive ((4, 36, 2))6 code. The nonadditive codes meeting the Singlet Bound is a very common situations in the nonbinary graphical codes, almost every stabilizer codes will have a clique set which are not a group, which means we can always construct a nonadditive code from a graphical nonbinary stabilizer code. D.
The code [[5, 1, 3]]3
The first example of 1-error-correcting code is the wellknown [[5, 1, 3]]3 code, which is the only error-correcting
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with stabilizer red in [8] t difficult d p with erated by
The code ((5, 4, 3))4
B. dimension The codeso((3, 2))4was constructed codes for arbitrary far3,and in [17] and formulated with graph state in [30]. For For dimension also found nonadditive code theFor loop L5 4onwe 5 vertices withpaall edges weighted an graph odd number of qupits with being even there is ((5, 4, 3)) that saturates the quantum Singleton bound. 4 1nowe find a 1-dimensional coding group generated by code of distance 2 that saturates the Singleton bound 5 The 1, graph we considered is stillin the loop graph Lclear on (1, 1) have ∈ Z as indicated Fig.1(e). so 1, far. 1, We a suboptimal code ((3,It3, is 2))45 in3found 5 vertices with all edgescan weighted 1. The coding clique that this coding group be generalized to arbitrary stead. For the equal weighted star graph on 3 vertices as contains the following 4 vectors in Z54 p the [[5, 1, 3]]p dimension and we we obtain arbitrary shown in Fig.1(a) have for found a coding clique: that is spanned by graph-state basis code (0, 0, 0, 0, 0), (1, 1, 1, 1, 1), (2, 3, 3, 2, 3), (3, 2, 2, 3, 2) 0, 1)}, {(0, 2), (2, a a0, 0), a (1, a 0, a ∈ Zp and . two bases (22) Z5a |L5 iin Fig.1(e) 1 Zbasis 2 Z3 Z among which Z one is4 shown are shown in Fig.1(f). Since there only ainstabilizer of with corresponding graph-state basisisshown Fig.1(a-c), the code 43 < 44ofelements and it((3, satuates which spancontaining the code subspace a nonadditive 3, 2))4 E. The code ((5, 4, 3))4 the Singleton is not we a subcode of any a1-error code. In fact, bound as we and see later, can construct code correcting code, this code is nonadditive. In fact we can ((3, p − 1, 2)) with p being arbitrary even number. p For dimension 4 we also found a nonadditive construct a nonadditive ((5, p, 3))p for any p > 3 ascode will ((5, 4, 3)) 4 that saturates the quantum Singleton bound. shown later. The graph we considered is still the loop graph L5 on C. edges Theweighted code [[4, 1. 2, 2]] 6 5 vertices with all The coding clique contains the following 4 vectors in 12, Z54 3))4 F. The code ((6, The next example we concerned is also a 1-error detect (0, 0, 0, 1, 1, 1,can 1), (2, 3, 3, 2, 3),12, (3,3)) 2, 2,as 3, 2)star ingWe code [[4,0, 2,0), 2]](1, becode constructed from the 6a which also found nonadditive ((6, shown 4 graph S4 on verticesis as shown in Fig.1(d). From with this in Fig.2. The4 graph a loop graph on 6 vertices among one basis iscoding showngroup in Fig.1(e) and two bases graph awhich 2 weighted dimensional can be found to be one edge 3 = −1 ∈ Z represented by a thick 4 are incoding Fig.1(f). is a(1,stabilizer generated by vectorsclique (1,Since −1, 0,there 0)12and 0,reads −1, 0) of in the Z46 . red shown edge. A with vectors code 43 code < 44 [[4, elements it satuates That containing is to say the 2, 2]]6 isonly the and 36-dimensional the Singleton the subspace spanned (0 0 0bound, 0 0by 0) the (0 basis 1code 1 0 2is1)not(0a2subcode 3 0 1 3) of any 1-error correcting and nonadditive. In (1 10 3 1code 0) (1 2 1is3therefore 3 1) (1 3 3 3 2 3) b1 3 2 1 1) (2 fact we can ((5, p, 3)) for (19) any (2 0construct 3Z11−a−b 0 3)Z2aa(2 2 Znonadditive |S i a, b ∈ Z 6 2 .1 1 p2) 3 4 p > 3 as will shown later. (2 3 0 2 3 0) (3 0 1 2 0 1) (3 1 2 2 2 2) whose stabilizer is generated by hG1 G2 G3 , G4 i. On the whichgraph is represented in another Fig.2 with black vertex same we also find coding clique as having weight 1 blue vertex having weight 2 3)) and4 red vertex havF. The code ((6, 12, -1 white vertex having weight ing weight 0. (a + b, −a, −b, δa1δb1 ) ∈ Z46 a, b ∈ Z6 (20) We also found a nonadditive code ((6, 12, 3))4 on the loop L6 onis 6generated vertices with edge weighted 3 whosegraph stabilizer hGone 1 G2 G3 i thus we have G. byThe codes [[7, 3,by 3]]as 3 and [[8, 4, 3]]3 represented a thick red edge in Fig.2. The coding a nonadditive ((4, 36, 2))6 code. The nonadditive codes 12 clique C of L with 12 vectors reads 6 saturate3the Singlet Bound is a very common situations in We found also two families of optimal qutrit codes, nonbinary graphical codes, almost every stabilizer codes namely the codes [[7, 3, 3]] and [[8, 4, 3]] , which be (0 0 0 0 0 0) (0 1 1 0 2 1) (0 2 3 0 1 3) can 3 which will have a clique set which3are not a group, means constructed from the loop graphs L and L respectively. (1 1 0 3 1 0) (1 2 1 3 3 1) (1 3 3 3 2 3) 7 8 we can always construct a nonadditive code from a graphThenonbinary generators in (2 0 3 stabilizer 1 of 0 3)the(2coding 1 3 2 1 groups 1) (2 2are 2 1 indicated 1 2) ical code. Fig.3(a) (2 and For0 d1 = blue 3 Fig.3(b). 0 2 3 0) (3 2 30 the 1) (3 1 2and 2 2red 2) vertices
2
5
1
6
8
3
3
2
5
1
6
8
3
4
5
2
5
1
6
8
3
4
5
5
6
cb bc
4
5
bc
cb bc
bc bcb
b
3
cb1 4 bc bc8 5 bbc 6 3
cb 4 bcb c8 5 bbc bb
7 2
bc bc b
5
bc bcb2
b
cb bc 7 bc bbc2
b
bc bcb
b
7
7
6
4
5
3
cb1 4 bc bc8 5 bbc 6 3
cb 4 bcb c8 5 bbc bb
bc bcb2
b
cb bc 7 c bcb2 bb b
b1
6 3
cb 4 bcb bb c8 5 bbc
bc bcb 7 2
bc bcb
b
b1
6
3
b
6
4
1
2
bc bc
7
b1
b
(b) [[8, 4, 3]]3
b
bc
cb bc 7 c bc2 bb
b
6 3
7
4
cb bc 6 3
7
4
3
5
bc cb2
1
2
6
4
7
4
b
b
3
7
cb bc b
4
1
2
cb bc
7
b
8
cb cb bc bc
1
bc bcb
2
3
bc
(a) [[7, 3, 3]]3
cb bc b
c b
cb bcb b bc bc
bcb Thebcb code bcb[[5, 1,bcb 3]]3 bc bcb 3 2 3 2 3 2 c4 b c bc4 c bc4 c bcb 4 cb 1 b 1 b 1 b 1 b 5 6 5 6 5 5 6 6 The first example of 1-error-correcting code is the c b b c bc bbc b bc bbc b bcb bc wellknown [[5, 1, 3]] code, which is the only error-correcting b c b c 3 bbc bbc bb bc bbc b bc bbc b codes for 3arbitrary dimension so3far 2 3 2 2 and was 3 constructed 2 c4and formulated c bbc4 bc bbc4graph1 state b cb bbc4 in [28]. b c For in [16]bb 1 bb 1 bb 1with 5 6 bb 6 bb the loop bb cgraph c L5 on bc5 56 bbcverticesbbc5 with b c all edges bc5 6 bbcweighted b 1 we find coding group bb by c a bb bb c1-dimensional c b b c b b c b b c b b cgenerated c bb (1, 1, 1, 1,31)2 ∈ Z53 as3 indicated 2 3in2 Fig.1(e). 3 2It is clear c4 coding bb c group c be bc4 generalized b c bbc4 to 1arbitrary c bb bbc4 1 bb 1 bb 1 bb that this can 5 6 6 5 5 5 6 6 dimension and we obtain for arbitrary p the [[5, c bb bb c bc bbc b bc bbc b bc bbc 1, 3]]p b 3 2
E.
d-purity t a super
1 G3 i
c b
b c b bc b bcb bb bc c b c bc b b c c b bc cb cb b bb bc bcb bc c b c c bb b c c b cb b b c b b c bb b c b c bb c b c b c b b b c c b c bb bc
es where of maxcode but a power code as
bc bcb b
6 3
cb 4 bcb bb c8 5 bbc
7
cb c8 bb b1
bc bcb b
7
bc cbb2
b1
6
cb1 bc8
cb bb c8 b1
6
bc bcb 7
(c) [[8, 2, 4]]3
FIG. 3: (a) Three generators of the coding group for the code FIG. [[7, 3,3: 3]]p ;(a) (b)Three Four generators generators of of the the coding coding group group for for the the code Fourmembers generators of the coding group code [[7, [[8,3, 4,3]] 3]]3p;; (b) (c) All of the coding groups for for the the [[8,4]] 4,3 .3]]3 ; (c) All members of the coding groups for codecode [[8, 2, the code [[8, 2, 4]]3 .
coincide. However those generators of the coding groups are also valid for thein codes 3, 3]] [[8, 4, 3]] all which is represented Fig.2[[7, with white, blue, and p andblack, p for oddvertices p > 3 which be discussed in details inThere the next red havingwill weights 0,1,2,3 respectively. are section. only 4 stabilizers can be found for this code, namely {I, G32 G62 , (G1 G2 G4 G5 )2 G3 G6 , (G1 G2 G4 G5 )2 G¯3 G¯6 }, 1.
The code [[8, 2, 4]]3
whose joint +1 eigenspace cannot be any 1-error correctingThe codelast because of the Singleton Therefore the example is the code [[8,bound. 2, 4]]3 which is found code is nonadditive. only recently [36] from the hype cube graph. The graph considered here is the equal-weighted wheel graph W8 as shown in Fig.3(c) and the corresponding graph state is codes [[7, 3, group 3]]3 and [[8, 4, 3]]3 by denoted G. as |WThe coding is generated 8 i. The
(0, 1, 1, 0, 2, 2, 0, 0) and (1, 0, 2, 2, 0, 1, 1, 1) (22) We found also two families of optimal qutrit codes, namely codes [[7,code 3, 3]](W [[8, 4, 3]]3 , which can be 3 and and thethe graphical by the 8 , 2, 4)3 is spanned constructed graph-state from basisthe loop graphs L7 and L8 respectively. The generators of the coding groups are indicated in a a+2b 2b 2a Fig.3(a) d =Z37b the blue and b ∈red Z3vertices } (23) Z62a+b Z8b |W {Z1b Zand Z4 Z5 For 8 i|a, 2 Z3 Fig.3(b). coincide. However those generators of the coding groups Thealso stabilizer ofthe thecodes code [[7, has3,the set3]]ofp for generare valid for 3]]pfollowing and [[8, 4, all atorsp > 3 which will be discussed in details in the next odd section. hG1 G4 , G2 G5 , G3 G6 , G4 G7 , G4 G8 , G1 G22 G3 i. (24) To conclude this section we list all the stabilizer codes H.Singleton The code [[8, 2,i.e., 4]]3 k + 2d ≤ n + 2, that saturate the bound, in Table I. It should be emphasized that the absence of some in Tableis Ithe does not[[8, imply these codes The codes last example code 2, 4]]3that which is found do not exist. [38] Tofrom find the some of these codes are only recently hype cube graph. Thebeyond graph our computers’ and some ofwheel themgraph will be considered here iscapacity the equal-weighted W8conas structed the following Furthermore, we can shown in in Fig.3(c) and thesections. corresponding graph state is
6 as shown in Fig.4(a). The p − 1 dimensional subspace spanned by the basis
TABLE I: The codes for p = 3. n=3 n=4 n=5 n=6 n=7 n=8
d=2 d=3 d=4 [[3, 1, 2]]3 [[4, 2, 2]]3 [[5, 3, 2]]3 [[5, 1, 3]]3 [[6, 4, 2]]3 [[6, 2, 3]]3 [[7, 3, 3]]3 [[8, 4, 3]]3 [[8, 2, 4]]3
q+j 2j+1 l ZA ZC2l |S3 i, ZA ZC |S3 i,
denoted as |W8 i. The coding group is generated by (0, 1, 1, 0, 2, 2, 0, 0) and (1, 0, 2, 2, 0, 1, 1, 1)
(23)
and the graphical code (W8 , 2, 4)3 is spanned by the graph-state basis {Z1b Z2a Z3a+2b Z42b Z52a Z62a+b Z7b Z8b |W8 i|a, b ∈ Z3 }
(24)
The stabilizer of the code has the following set of generators hG1 G4 , G2 G5 , G3 G6 , G4 G7 , G4 G8 , G1 G22 G3 i.
(25)
To conclude this section we list all the stabilizer codes that saturate the Singleton bound, i.e., k + 2d ≤ n + 2, in Table I. It should be emphasized that the absence of some codes in Table I does not imply that these codes do not exist. To find some of these codes are beyond our computers’ capacity and some of them will be constructed in the following sections. Furthermore, we can find the codes with a length n ≥ 6 via a random searching method, thus some of absent codes may still be found by using our systematic algorithm developed above.
with 0 ≤ l ≤ q − 1 and 0 ≤ j ≤ q − 2 is a nonadditive code ((3, p − 1, 2))p . Proof— It is enough to demonstrate that a subset Cp−1 2 composed of 2q − 2 vectors of forms (l, 0, 2l) and (q + j, 0, 2j + 1) satisfies all three conditions of coding clique. Since the 2-purity set is empty and l can assume value 0, we have only to show that the condition iii of coding clique is satisfied. Any single qupit error can only cover those vectors of forms (a, b, a), (b, c, 0), and (0, c, b) with a, b, c ∈ Zp being arbitrary. Considering the range of the l and j it is straightforward to show that all vectors in Cp−1 and pairwise difference cannot be covered by single 2 qubit error. Thus we have a code ((3, p − 1, 2))p . The stabilizer of the code turns out to be generated by GB . Therefore it is a nonadditive code. Via a similar construction by Rains [24] we are able to construct the code ((2n + 3, p2n (p − 1), 2))p for even p. We consider the graph on 2n + 3 vertices composed of the star graph S3 as shown in Fig.4(a) and the graph B2n as shown in Fig.4(b) and denote the corresponding graph state as |S3 i ⊗ |B2n i. Let us denote by {|vi} the basis in Eq.(26) for the ((3, p − 1, 2))p code constructed above then the code subspace is spanned by the basis Pn
(ZA ZC )
i=1
GRAPHICAL NONBINARY QECCS VIA ANALYTICAL CONSTRUCTIONS
Because the clique finding problem is intrinsically an NP-complete problem, it is not plausible to rely on the numerical search for codes with larger length or higher dimension. Here we shall provide some analytical constructions of good codes for higher dimension, which is based on the graphs and their coding cliques in lower dimensions found via computer research. In practice, we start from a graphical code found for qutrit and then generalize the subspace to arbitrary dimension by adopting the same graph and similar coding clique, and finally we prove that the subspace provides a code in arbitrary dimension. A.
The nonadditive code ((3, p − 1, 2))p with even p
Description of the code— Suppose that p = 2q. We consider the star graph S3 labeled with V = {A, B, C}
s2i
(ZB )
Pn
i=1
s2i−1
|vi ⊗ Z s |B2n i
(27)
with s ∈ Z2n p being arbitrary. We notice that the phase flips acting on |vi is a single qupit error on qupit B. B.
V.
(26)
The code [[2n, 2n − 2, 2]]
We consider the graph on 2n vertices with all edges weighted 1 as shown in Fig.4(b) and denote the corresponding graph state as |B2n i. The subspace spanned by the graph-state basis −
Z1
Pn−1 j=1
aj
−
Z2
Pn−1 j=1
bj
n−1 Y j=1
a
j Z2j+1
n−1 Y j=1
a
j+1 Z2(j+1) |B2n i (28)
with aj , bj ∈ Zp for j = 1, 2, . . . n−1 is the code [[2n, 2n− 2, 2]] whose stabilizer is generated by X1 Z2 X3 Z4 . . . X2n−1 Z2n , Z1 X2 Z3 X4 . . . Z2n−1 X2n .
(29)
It is straightforward to see from the stabilizer that every single qupit error can be detected, i.e., not commute with at least one of two generators defined above. In comparison in [38] the star graph has been used to construct the code [[n, n − 2, 2]]p .
7 C
bc A
bc
2 4 6
bc
bc bc bc
B
2 3
1
bc bc
bc bc
bc
(a)
cb bc
4 5
7 6
(d)
bc bc bc ···
bc
2n
bc
bc4
bc
6
bc
bc
(c)
bc bc
(e)
bc 1
cb5
1 3 5 2n − 1 (b)
cb4 3 2 1 bc 8 b bc5 c bc6 7 cb
3 2
3 4 5
cb bc 6
bc cb2 bc bc (f)
cb1 bc8
7
FIG. 4: (a) The star graph S3 on 3 vertices; (b) The graph B loop star graphgraph L6 onS63 vertices with 1 (b) edgeThe weighted 2n ; (c) FIG. 4: The (a) The on 3 vertices; graph -1; (e)6 The loopwith graph L8 ; weighted (f) The B2n(d) ; (c)The Theloop loopgraph graph LL76; on vertices 1 edge wheel 8 vertices. 8 on -1; (d)graph The W loop graph L7 ; (e) The loop graph L8 ; (f) The wheel graph W8 on 8 vertices. B. The nonadditive code ((5, p, 3))p with p > 3 C. The code [[2n + 3, 2n + 1, 2]]p with odd p
We consider the loop graph L5 with all edges weighted 5 2n + 3 composed of a subgraph Consider the graph 1 and the subset Cp3 ⊂ Zon p composed of p vectors including S3j,asj,shown in Fig.4(a) a subgraph B2n as shown in (j, j, j) with j 6= 2, and −1 and two additional vectors On−1, the2,first qupits 1, 2]] code aFig.4(b). = (2, −1, −1)3 and b a=[[3, (−1, 2, 2, −1,can 2). be Allconof structed for odd p with the stabilizer given by hGcannot B , GA Gbe Ci these vectors have full support and obviously as shownbypreviously. obtain the isstabilizer the covered any singleThen qupitwe error which able to of cover code [[2n3 +vertices. 3, 2n + 1,All 2]]pvectors as at most (c1 , c2 , · · · , c5 ) with full support that can be covered by a 2-qupit error has the property ZA XB ZC X1 Z2 X3 Z4 . . . X2n−1 Z2n , (30) 2 XA ZB XC Z1 X2 Z3 X4 . . . Z2n−1 X2n . cn + cn+1 = cn+3 . (30) It is obvious that all vectors in Cp3 do not have such a D. The nonadditive ((5, p,of 3))any >3 p with property. Furthermore the code difference pairp of vectors in Cp3 has full support and do not have the property Eq.(30) either. the Therefore Cp3 isLa5coding conWe consider loop graph with allclique edgessince weighted p 5 ditions i and ii C are trivially satisfied. To find all the composed of p vectors including 1 and the subset ⊂ Z p 3 stabilizer of with the code the codingvectors clique (j, j, j, j, j) j 6=constructed 2, −1 and from two additional p C a3 ,=i.e., (2,subspace −1, −1, 2,spanned −1) andbyb the = graph-state (−1, 2, 2, −1,basis 2). All of these vectors have full support and obviously cannot be j a {(Zby Z3 Zsingle |L5 i, Z |L5which i, Z b |L i}, to cover (31) covered any error is5able 1 Z2 4 Z5 ) qupit at most 3 vertices. All vectors (c1 , c2 , · · · , c5 ) with full we have to solve in by Eq.(15). It turns support that canequations be covered a 2-qupit errorout hasthat the the total number of the solutions to Eq.(15) is q 3 if p is property not a multiple of 3 and 3q 3 if p is not 3 but can be divided cn+1 = of cn+3 . (31) by 3. In either case cthe stabilizers is less than n +number p4 if p > 3 and since the code saturates the Singleton p have such a It is obvious that all vectors in C 3 do not bound we conclude that the code Eq.(31) is nonadditive. property. Furthermore the difference of any pair of vectors in Cp3 has full support and do not have the property Eq.(31) either. Therefore Cp3 is [[6, a coding C. The code 2, 3]]p clique since conditions i and ii are trivially satisfied. To find all the stabilizer of the code constructed from the coding clique Description of the code— Consider the weighted loop Cp3 , i.e., subspace spanned by the graph-state basis graph on 6 vertices as shown in Fig.4(c) and the correj 6 i. Theap2 -dimensional b sponding state |L subspace {(Zgraph (32) 1 Z2 Z3 Z4 Z5 ) |L 5 i, Z |L5 i, Z |L5 i}, spanned by the graph-state basis we have to solve equations in Eq.(16). It turns out that a a+b b −a a−b b a, b ∈ Zpis q 3 if(32) Z Z5 Z6 |Lto the total Z number of3 Z the p is 6 i Eq.(16) 1 Z2 4 solutions
is a [[6, 2, 3]]p code for all odd p > 2. A set of the generII: Theisstabilizer the code ators ofTABLE its stabilizer listed inofTable II. [[6, 2, 3]]p TABLE II: The1 stabilizer 2 3of the4code 5[[6, 2, 6 3]]p G1 G4 X Z Z X Z Z¯ ¯ Z X Z Z¯ X¯ G3 G6 Z 1 2 2 3 4 5 6 ¯ ¯ ¯ ¯ G1 G2 G3 ZX X Z ZX Z I Z¯ G1 G4 X Z Z X 2 Z Z¯ Z ZX X Z ZX Z G3 G4¯G5 I G3 G6 Z Z X Z Z¯ X¯ ¯ ¯ G1 G¯2 G3 ZX X¯ Z 2 ZX Z I Z¯ 2 Z ZX X Z ZX Z G3 G4 G5 I
not a multiple of 3 and 3q 3 if p is not 3 but can be divided by 3. In either case the number of stabilizers is less than of Theorem we have only to prove p4Proof— if p > Because 3 and since the code1saturates the Singleton that thewe subset of vectors bound conclude that the code Eq.(32) is nonadditive.
{c = (a, a + b, b, −a, a − b, b)|a, b ∈ Zp }
(33)
satisfies all threeE.conditions of the clique. ConThe code [[6, 2,coding 3]]p ditions i) and ii) are obviously satisfied since a, b can assume value zero andcode— the 3-purity setthe is empty for loop the Description of the Consider weighted loop graph for any dimension p. To prove condition iii) graph on 6 vertices as shown in Fig.4(c) and the corre2 covered by single and we have to show that c cannot be sponding graph state |L6 i. The p -dimensional subspace 2-qupit all a, b ∈ Z p . Obviously c cannot be spannederrors by theforgraph-state basis covered by any single vertex error, we consider only 2 a a+b b −a a−b b qupit errors Z1inZ2what Z3follows. Z4 Z5 Z6 |L6 i a, b ∈ Zp (33) i) b = 0, a 6= 0 with c = (a, a, 0, −a, a, 0). If a 2-qupit is a [[6, 2, 3]] code for all p > {1, 2. A the genererror ispsupported on odd vertices 2, set 4, 5}ofthen it can ators only of itstakes stabilizer form is listed in Table II. Proof— Because of Theorem 1 we have only to prove t X1n Z1s Xof5n Z X2n Z2s X¯4n Z4t , X3s X6t , 5, that the subset vectors
which covers (s, n, 0, n, t, 0), (n, s, 0, t, −n, 0) and (a,s,a0)+respectively b, b, −a, a − b, b)|a,n,bs,∈tZ∈p }Z being (34) (−s,{ct, = 0, t, with p arbitrary. All these 3 types of vectors cannot be satisfies all three conditions of the coding clique. Conidentified with c provided p > 2 and odd. For exditions i) and ii) are obviously satisfied since a, b can ample if (s, n, 0, n, t, 0) = c then 2a = 0 which is assume value zero and the 3-purity set is empty for the impossible for odd p. loop graph for any dimension p. To prove condition iii) weii)have covered and a =to0,show b 6= 0that withcccannot = (0, b,be b, 0, −b, b).byInsingle this case 2-qupit errors for all a, ∈ Zp . the Obviously c cannot be it can be proved in bexactly same manner as the covered anythat single vertexbeerror, we by consider 2firstbycase c cannot covered 2-qupitonly error. qupit errors in what follows. iii) a = b 6= 0 with c = (a, 2a, a, −a, 0, a). Any 2-qupit {1, 3,a, 4, 0, 6}−a, cana,only form i) error b = 0,that a 6= 0covers with c =2,(a, 0). Iftake a 2-qupit m s n t m s n Xerror Z X Z or X Z X , which covers {1, 2, 4, 5} then it can 1 1is supported 3 3 2 on 2vertices 5 only takes form (s, m + n, t, n, 0, −m) or (m, s, m, n, 0, n) n t ¯4nZZp4t being respectively To X1n Z1s X5with Z5 , m,Xn,2ns, Z2st X∈ , X3sarbitrary. X6t , cover c we have to require 2a = −2a or a = −a which are covers (s, n, 0,for n, t,odd 0), p.(n, s, 0, t, −n, 0) and which impossible (−s, t, 0, t, s, 0) respectively with n, s, t ∈ Zp being iv) aarbitrary. = −b 6= 0All with c =3 (−b, b, b, −2b, b).cannot Any 2these types0,of vectors be qupit error that covers 5 vertices {1, 3, 4, 5, 6} can identified m with c provided p > 2 and odd. For exn s m s n t only beifX(s, Z5n,ort, X which 2 X 5 0, 6 Z6 , 2a ample n, 0)4 =Z4cXthen = 0covers which is impossible fors,odd (m, 0, m, n, n) orp. (−n, 0, m, s, m + n, t)
∈ b,Z0, To ii) respectively a = 0, b 6= 0with withm, c n, = s, (0,t b, −b, b).arbitrary. In this case p being cover have in toexactly requirethe b = −b manner or 2b =as−2b it can cbewe proved same the which is impossible for odd p. first case that c cannot be covered by 2-qupit error.
8 iii) a = b 6= 0 with c = (a, 2a, a, −a, 0, a). Any 2-qupit error that covers {1, 2, 3, 4, 6} can only take form X1m Z1s X3n Z3t or X2m Z2s X5n , which covers (s, m + n, t, n, 0, −m) or (m, s, m, n, 0, n) respectively with m, n, s, t ∈ Zp being arbitrary. To cover c we have to require 2a = −2a or a = −a which are impossible for odd p. iv) a = −b 6= 0 with c = (−b, 0, b, b, −2b, b). Any 2qupit error that covers 5 vertices {1, 3, 4, 5, 6} can only be X2m X5n Z5s or X4m Z4s X6n Z6t , which covers (m, 0, m, n, s, n) or (−n, 0, m, s, m + n, t) respectively with m, n, s, t ∈ Zp being arbitrary. To cover c we have to require b = −b or 2b = −2b which is impossible for odd p. v) a 6= ±b, a, b 6= 0 with c = (a, a + b, b, −a, a − b, b). To cover all the 6 vertices a 2-qupit error can only takes form X1m Z1s X4n Z4t ,
X2m Z2s X5n Z5t ,
X3m Z3s X6n Z6t ,
which covers (s, m, n, t, n, −m), (m, s, m, n, t, n), and (−n, m, s, m, n, t) respectively. However all these vectors can not be identified with c if p is odd. For example if (s, m, n, t, n, −m) = c then 2a = 0 which is impossible for odd p. In summary, for any a, b ∈ Zp the vector c cannot be covered by any single or 2-qupit errors so that is a coding clique and corresponding subspace is the [[6, 2, 3]]p code for all odd p > 2. F.
The code [[7, 3, 3]]p
Description of the code— Consider the equal weighted loop graph on 7 vertices as shown in Fig.4(d) and the corresponding graph state |L7 i. The p3 -dimensional subspace spanned by the graph-state basis a+b+c a c b a−c −c b Z1 Z2 Z3 Z4 Z5 Z6 Z7 |L7 i a, b, c ∈ Zp (35) is a [[7, 3, 3]]p code for all odd p > 2. A set of the generators of its stabilizer is listed in Table III. TABLE III: The stabilizer of the code [[7, 3, 3]]p
G3 G6 G¯4 G7 G¯2 G3 G5 G¯1 G2 G3 G4
1 2 3 4 I Z X Z Z I Z¯ X¯ ¯ Z¯ X¯ Z ZX Z2 2 X¯ Z X X Z ZX
5 Z Z¯ X Z
6 X Z Z I
7 Z X I Z¯
Proof— Because of Theorem 1, we have to show that the subset of Z7p defined as {c = (a + b + c, a, c, b, a − c, −c, b)|a, b, c ∈ Zp }
(36)
is a coding clique. Since the 3-purity set is empty for the loop graphs with any dimension p, we need only to show that any nonzero c cannot be covered by any single or 2-qupit error. It is not difficult to see that c cannot be covered by any single qupit error, which covers a vectors with 4 consecutive components being zero. Because of the symmetry of the loop graph, all 2-qupit errors can be classified into 3 types T12 , T13 , and T14 where Tlj denotes two errors occur on qupits n + l and n + j with n ∈ Z7 . Each type of errors covers the vector (c1 , c2 , . . . , c7 ) with properties T12 : cn = cn+1 = cn+2 = 0; T13 : cn−1 + cn+3 = cn+1 and cn−2 = cn−3 = 0; T14 : cn−1 = cn+1 , cn−2 = 0, and cn−3 = cn+2 . It can be easily checked that as given in Eq.(36) for arbitrary a, b, c ∈ Zp does not belong to all these 3 types of vectors when p is odd. For example a T12 type of error covers vector with 3 consecutive components being zero, which is impossible to be identified with a nonzero c. That is to say c cannot be covered by any 2-qupit errors and therefore we have a coding clique and a [[7, 3, 3]]p code for all odd p. G.
The code [[8, 4, 3]]p
Description of the code— Consider the equal weighted loop graph on 8 vertices as shown in Fig.4(e) and the corresponding graph state |L8 i. The p4 -dimensional subspace spanned by the graph-state bases Z1e Z2b−c Z3e−c Z4e−a Z5a+b Z6a−b+c+e Z72b Z8a−c+e |L8 i (37) with a, b, c, e ∈ Zp is a [[8, 4, 3]]p code for all odd p > 2. A set of the generators of its stabilizer is listed in Table IV.
TABLE IV: The stabilizer of the code [[8, 4, 3]]p
1 2 3 4 5 6 7 8 ¯ G¯1 G¯3 G4 G8 X¯ Z I X Z ZX Z I Z ZX 2¯ ¯ 2 2 ¯ 2 ¯ 2 ¯ ¯ ¯ G2 G3 G5 G6 Z X Z Z X Z X Z ZX Z I G¯12 G42 G52 G¯7 X¯ 2 Z¯2 Z 2 X 2 Z 2 Z 2 X 2 Z X Z¯3 G¯12 G¯2 G3 G4 G5 Z¯X¯ 2 Z¯X¯ X X Z2 X Z Z I Z¯2
Proof—Because of Theorem 1, we have to show that the subset of Z8p composed of vectors of form c = (e, b−c, e−c, e−a, a+b, a−b+c+e, 2b, a−c+e) (38)
9 which with a, b, c, e ∈ Zp is a coding clique of the loop graph. Because of the symmetry of the loop graph, all 2-qupit errors can be classified into 4 types T12 , T13 , T14 , and T15 . Each type of errors covers the vector (c1 , c2 , . . . , c8 ) with properties (n ∈ Z8 ) T12 : cn+1 = cn+2 = cn+3 = cn+4 = 0; T13 : cn+2 = cn+3 = cn+4 = 0 and cn+1 + cn−3 = cn−1 ; T14 : cn+2 = cn+3 = 0, cn+1 = cn−1 , and cn−2 = cn−4 ; T15 : cn = cn+4 = 0, cn+1 = cn+3 , and cn−1 = cn−3 . As an example we consider a T12 type of errors which covers a vector with 4 consecutive components being zero. A nonzero vector c as defined in Eq.(38) is not possible to have such a property, e.g., the last 4 components being zero. In the same manner it can be checked that c does not belong to all those 4 types of vectors above for arbitrary a, b, c, e ∈ Zp when p is odd. Since all single qupit errors cover vectors with the same property as in type T12 , we conclude that c cannot be covered by any single or 2-qupit errors so that we have a coding clique and therefore a [[8, 4, 3]]p code for all odd p. H.
Description of the code— Consider the equal weighted wheel graph on 8 vertices as shown in Fig.3(d) and the corresponding graph state |W8 i. The p2 -dimensional subspace spanned by the graph-state basis b a a−b 2b 2a b−a b b Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 |Γi a, b ∈ Zp (39)
is a [[8, 2, 4]]p code for all odd p > 2. A set of the generators of its stabilizer is listed in Table V.
5 Z Z Z X¯ Z Z Z Z
6 Z¯ I I Z X Z¯
T13 : cn = cn+3 and cn+1 = cn+4 ; T14 : cn = cn+3 and cn+1 = cn+2 = 0; T15 : cn = cn+6 , cn+2 = cn+4 and cn+1 = cn+5 = 0. It can be checked in a straightforward manner that none of the above equalities can be satisfied by c ad defined in Eq.(40). For example we consider error of type T12 with n = 0. In this case we have constraints b = 2b, a = 2a, and a−b = b−a which are impossible for nonzero c when p is odd. This is true for all n ∈ Z8 . Thus any 2-qupit error cannot cover c. All 3-qupit errors can be classified into 7 types T123 , T124 , T125 , T126 , T127 , T135 , and T136 with each type of errors covering vectors with the following properties (n ∈ Z8 ) T124 : cn = cn+3 and cn+2 = 0; T125 : cn + cn+1 = cn+3 and cn+4 = 0; T126 : cn + cn−1 = cn−3 and cn−4 = 0; T127 : cn = cn−3 and cn−2 = 0; T135 : cn + cn+1 = cn+3 and cn + cn−1 = cn−3 ; T136 : cn = cn+2 + cn+3 = cn−2 + cn−3 . It can be checked in a tedious but straightforward manner that for every n ∈ Z8 none of those equalities above can be satisfied by c as defined in Eq.(40). For an example we consider the 3-qupit error of type T126 with n = 1. In this case we have b + b = b − a and 2a = 0 which are impossible for nonzero c. To summarize, the vector c defined in Eq.(40) for all a, b ∈ Zp is 4-uncoverable. Thus we have proved that the subspace defined in is a [[8, 4, 2]]p code for all odd dimension p.
TABLE V: The stabilizer of the code [[8, 2, 4]]p
4 I Z¯ X¯ Z¯
T12 : cn+1 = cn+4 , cn+2 = cn+5 = 0, and cn+3 = cn+6 ;
T123 : cn+2 = cn+5 and cn−2 = cn−5 ;
The code [[8, 2, 4]]p
1 2 3 G1 G¯7 X Z Z¯ G1 G¯8 X Z¯ Z I 2¯ 2 2 G1 G4 X Z Z¯ 2¯ 2 G2 G5 Z X Z2 2 G3 G6 I Z X ¯ ¯ G1 G¯2 G3 ZX X¯ Z 2 ZX
A single qupit error on vertex n can cover a vector with property cn±2 = cn±3 = 0 which is impossible for c defined above. Because of symmetry all 2-qupit errors can be classified into 4 types T12 , T13 , T14 , and T15 with each type of errors covering the vector (c1 , c2 , . . . , c8 ) with properties (n ∈ Z8 )
7 8 X I Z¯ Z X¯ I Z I I Z2 I Z Z
VI.
Proof— Because of Theorem 1 we have to show that the subset of Z8p defined as {c = (b, a, a − b, 2b, 2a, b − a, b, b)|a, b ∈ Zp }.
(40)
is a coding clique of the wheel graph. We have only to prove condition iii of the coding clique since the 4-purity set is empty.
CODES FROM COMPOSITE SYSTEMS
Consider a system with pq levels whose computational bases are denoted by {|lipq }pq−1 l=0 . We can also regard this system as a composite system of a p-level system and a qlevel system, whose computational bases are denoted as p−1 {|sip }s=0 and {|tiq }q−1 t=0 respectively. If there are n copies of pq-level systems, we also have n copies of p-level and q-level systems. On the other hand if we have two groups
By relabeling of theαp bases pq-level system according + βqof=a 1. (42) to |sip ⊗ |tiq 7→ |pt + qsipq , we can define an isometry Given a Z -weighted graphand (V, Γ ) we of canp-level build subsysa Zp between npqpq-level systems npqpairs weighted graph (V,subsystems Γp ) and a Zas -weighted graph (V, Γq ) tems and q-level q whose adjacency matrices are given by X X |pt + qsipq hs|p ⊗ ht|q , (44) R= pΓpq ≡ Γq (mod q), qΓpq ≡ Γp (mod p). (43) s∈ZV t∈ZV p
q
On the other hand, given a Zp -weighted graph (V, Γp ) which is only possible when p and q are coprime, we have and a Zq -weighted graph (V, Γq ) on the same vertex set V , we can also build a |Γ Zpq -weighted graph (V, Γpq ) with (45) pq i = R|Γq i. adjacency matrix given by Accordingly the bit2 flips and 2 phase flips are related to (44) pq ≡ p α Γq + q β Γp (mod pq). each otherΓvia By relabeling ofβthe system according α † Xpq = U (X ⊗ Xbases )U †of , aZpq-level (46) pq = U (Zp ⊗ Zq )U . to |sip ⊗ |tiq 7→p |pt +q qsipq , we can define an isometry between n pq-level n pairs of p-level subsysTheorem 2 Ifsystems p, q areand coprime then the graph state tems and q-level subsystems as on Zpq -weighted graph is in a one-to-one correspondence Xproduct X with the direct of two graph states on a Zp = qsipq hs|pgraph ⊗ ht|q ,whose (45) weightedRgraph and a|pt Zq+ -weighted adjaV t∈ZV s∈Zare cency matrices related via Eqs.(42,43). p q According to the fundament theorem of arithmetics nL which is only possible p and as q are any integer p can bewhen expressed p =coprime, pn1 1 pn2 2 .we . . phave L for some primes pi (i = 1, . . . , L). Therefore a graph state |Γ i = R|Γq i. (46) on the subsystems ofpq dimension p will be in a one-toone correspondence of aand direct product of graph Accordingly the bit flips phase are relatedstates to nflips i on the subsystems of dimension p . As a consequence i each other via we have only to consider the graph states of systems of a β † or a power of prime. So dimension is αprime the Xpq = R(Xthat Zpq = R(Zp ⊗ Zq )R† . far (47) p ⊗ Xq )R , definition of graph state is unique in the case of prime Theorem and 2 Ifthere p, q are then theofgraph dimension is a coprime second definition graphstate states on[31] Zpqin -weighted is inpower a one-to-one correspondence the casegraph of prime dimension. withAs theandirect product of two the graph a 2Zpand example we consider casestates whereon q= weighted graph of and a Zq -weighted whose adjathere 6 copies 2p-level compositegraph systems and accordcency arequbits related viaqupits. Eqs.(43,44). inglymatrices 6 pairs of and A direct product of a Accordingcode to the fundament theorem of arithmetics graphical [[6, 1, 3]]2 on qubits constructed from the L pn2 2 . . . pnLand for any integer can be expressed p = pinn1 1Fig.5(a) loop graphp with a leaf [25] as as shown the
(a)
6
b c c b c b cb
1
5
bc
p+
δ
bc
(p
−
1)δ
p
p+δ
c b b c
(b)
cb c b
5
6
2
bc bc
4
4
cb 1 p+δ
˜ K, ˜obtain of two p-level and q-level we can (also n can copies graphical codessystems (G, K, d) G, d)q we conp and ˜ d))pqup of struct pq-level system byKpairing one p-level system a code ((n, K, , which is however notand necone q-leveltosystem to made up a composite essarily be another graphical code. Assystem. will be shown Given this a Zpconstruction -weighted graph Γp )aand a Zq -weighted below will (V, yield graphical code of a graph (V,dimension Γq ) on the if same vertex setcoprime, V and corresponding higher p and q are in which case ˜ K˜α, the there cliques exist two andsubspace β such that coding CK and C spanned by the d integers d basis αp + βq = 1. (41) o n ˜ K˜ ˜ , c ∈ C (41) Z c |Γp i ⊗ Z c˜ |Γq i c ∈ CK d d Given a Zpq -weighted graph (V, Γpq ) we can build a Zp weighted graph Γp ) and a Zq -weighted graph (V, Γq ) ˜ d)) (V,code is an ((n, K K, with n = |V |. This is because whose adjacencypqmatrices are given by all the direct products of Pauli errors of the p-level systems and q-level basis for the pΓpq ≡ Γq systems (mod q),formqΓapqnice ≡ Γerror (42) p (mod p). pq-level system. That is to say, via a direct product of ˜ K, ˜ d)q graph two codes (G, given K, d)paand (G, we can(V, conOngraphical the other hand, Zp -weighted Γp ) ˜ struct a code ((n, K K, d)) , which is however not nec-set and Zq -weighted graphpq(V, Γq ) on the same vertex essarily another code. graph As will(V, beΓshown V , we to canbealso build graphical a Zpq -weighted pq ) with below this construction adjacency matrix givenwill by yield a graphical code of a higher dimension if p and q are coprime, in which case ≡ p α2α Γqand + q ββ 2such Γp (mod (43) there exist twoΓpq integers that pq).
cb b c
10
bc
bc p+
bc
δ
p+
δ
2
3 3 FIG. 5: An example of composite system codes. (a) The Z2p5: -weighted graphofHthe δ = (p + 1)2codes. /2. (b)(a) Inner: 6 where FIG. An example composite system The a Z -weighted graph and outer: a Z -weighted graph. 2 graph is a Z2 -weighted graph p inner and the outer graph is a Zp -weighted graph. (b) The Z2p -weighted graph H6 where δ = (p + 1)2 /2.
graphical code [[6, 2, 3]]p arising from the graph L6 as constructed Sec.V(C) will result a code ((6, 2 · p2 , 3)). some primes pin i (i = 1, . . . , L). Therefore a graph state is insubsystems fact a stabilizer code andpwe should that the onItthe of dimension will be innote a one-todimension of the coding subspace is notofa graph power states of 2p. one correspondence of a direct product ni . As a consequence on the subsystems of dimension p Via the relabeling of the basesi of the 2p-level systems weashave only to the states systems graph of a specified R consider in Eq.(44) wegraph obtain a Z2pof-weighted dimension that is prime or a power of prime. So far the a H6 as shown in Fig.5(b). From this graph we consider definition state is in the case of prime subset Cofofgraph 2p2 vectors in unique Z2p of form dimension and there is a second definition of graph states [33] in the case of prime power dimension. 1)(a, a +we b, b,consider −a, a − the b, b)case + p(c, c, c, c,q c,=0)2 and (47) As(pan+example where there 6 copies of 2p-level composite systems and accordingly 6 pairs of qubits and qupits. A direct product of a with a, bcode ∈ Zp[[6, and c ∈ Zon The subspace spanned the 2 . qubits graphical 1, 3]] constructed fromby the c2 graph-state basis {Z |H i|c ∈ C} is exactly the stabilizer 6 loop graph with a leaf [27] as shown in Fig.5(a) and the code ((6,code 2p2 , 3)) from composite systems. graphical [[6, constructed 2, 3]]p arising from the graph L6 as A set of the generators of its stabilizer is listed in Table VI. constructed in Sec.V(E) will result a code ((6, 2 · p2 , 3)). It is in fact a stabilizer code and we should note that the dimension of the coding subspace is not a power of 2p. Via the relabeling of the bases of the 2p-level systems as specified RVI: in Eq.(45) we obtain Z2p -weighted graph TABLE The stabilizer of thea code ((6, 2p2 , 3)) 2p . H6 as shown in Fig.5(b). From this graph we consider a subset C of 2p2 vectors in Z2p of form 1
2
3
4
5
6 (48) ¯p+1 Z G32 G¯62 Z p+1 Z p+1 X2 Z p+1 Z¯p+1 X¯ 2 with 2a,¯2b ∈ Z and c ∈ Z . The subspace spanned by the p 2 G1 G2 G32 Z¯p+1 X 2 cX¯ 2 Z 2 Z¯p+1 X 2 Z p+1 I Z¯p+1 graph-state basis {Z |H C} is exactly the stabilizer 6 i|c ∈p+1 2 2 2 p+1 2 2 2 p+1 2 G3((6, G4 G2p I constructed Z Z from X composite X Z Z systems. X Z p+1 5 2 , 3)) code A p p G6generators I I I Xp set of the of Iits stabilizer is listed inZTable VI. G1p G2p X p Z p Z p X p Zp I Zp I p p p p p p Because we cannot G1 G3 theXcode [[6, I 2, 3]]2 Xdoes not Z exist Z I p remove any stabilizers G1p G Xp Z pfrom the Z p lower X phalf part I of the I 4 p p p obtain p p (2p)2 -dimensional p p p space, aboveG1table to a code G5 X Z Z I Z Z X Zp 2 c, c, c, (p + +2b, b, −a, b) + p(c, 0) G121)(a, G42 a X Z p+1a − b, Z p+1 X c, Z p+1
therefore the code ((6, 2p2 , 3))2p is not a subcode of a [[6, 2, 3]]2p , if there is any.
Because the code [[6, 2, 3]]2 does not exist we cannot VII. DISCUSSION remove any stabilizers from the lower half part of the above table to obtain a (2p)2 -dimensional code space, 2 We have generalized the2pgraphical [27] of a therefore the code ((6, , 3))2p isconstruction not a subcode QECCs to2pthe case to find both additive and [[6, 2, 3]] , if nonbinary there is any.
11 TABLE VI: The stabilizer of the code ((6, 2p2 , 3))2p .
G12 G42 G32 G¯62 2 ¯2 2 G1 G2 G3 G32 G42 G52 G6p p p G1 G2 G1p G3p G1p G4p G1p G5p
1 X2 Z p+1 Z¯p+1 X 2 I I X pZ p Xp Xp X pZ p
2
3
p+1
p+1
Z Z Z p+1 X2 2 2 p+1 X¯ Z Z¯ X2 p+1 p+1 2 Z Z X I I Z pX p Zp I Xp p Z Zp p Z I
4 5 6 2 p+1 p+1 ¯ X Z Z Z p+1 Z¯p+1 X¯ 2 p+1 Z I Z¯p+1 2 2 p+1 2 X Z Z X Z p+1 I Zp Xp p I Z I Zp Zp I Xp I I Zp Z pX p Zp
nonadditive codes based on nonbinary graph states. The advantages of our graphical approach lies in the fact that we are able to construct codes on physical systems of arbitrary dimension, prime or nonprime, to construct both additive or nonadditive codes, pure or impure. In addition the basis for all codes are explicitly constructed. In principle for prime dimension our method exhausts all the stabilizer codes, which can be demonstrated in exactly the same manner as in binary case [27]. Via numerical search we have found many optimal codes including two optimal codes [[8, 4, 3]]3 and [[8, 2, 4]]3 which have been found in [38] on a hyper cube graph. Since the clique finding problem for non-binary case is even more difficult and is intrinsically a NPcomplete problem, our computational result is mainly
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limited in codes with n ≤ 8 and small distance d ≤ 4. With the help of the codes found numerically we also manage to construct analytically some families of optimal stabilizer codes that saturating the Singleton bound such as [[6, 2, 3]]p , [[7, 3, 3]]p , [[8, 4, 3]]p and [[8, 2, 4]]p for any odd p. We have also explicitly constructed the code [[n, n − 2, 2]]p except the case of even p and odd n. There exist stabilizer codes whose code subspace is not a power of the dimension of physical systems such as the code ((6, 2 · p2 , 3))2p which is constructed via a composite system approach. Furthermore we have constructed a family of nonadditive codes ((3, p − 1, 2))p with even p, a nonadditive optimal code ((5, p, 3))p for all p > 3. Finally, we briefly address some questions which are still open. Although our method can also be used to find additive and nonadditive codes for non-prime dimension, we can not exhaust all the non-prime stabilizer codes because of our definition of graph states. Because of Theorem 2 we have to consider the graph states on system of prime and prime power dimension. There are at least two different definitions of the graph state on n subsystems of a dimension p = q m being a power of prime: one is as defined in this paper and in [38] and the other one is as defined in [33]. A different definition of graph states may result in some new families of codes. Acknowledgement
The financial supports from NNSF of China (Grant No. 10675107 and Grant No. 10705025) and WBS (Project Account No): R-144-000-189-305, Quantum information and Storage (QIS) are acknowledged.
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