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general graphs embedded in surfaces of arbitrary genus in order to explore optimal values of the encoding rate k n . A graph Γ is a collection of vertices V and edges E. Each edge joins two vertices. A graph is usually visualized flattened on the plane. Now consider a graph Γ embedded in a surface M (see Fig. 2). When M − Γ is a disjoint collection of discs, we say that the embedding FIG. 2: Left. A graph (thick lines) in the torus and its dual is a cell embedding. Let us gather these discs into a set (weak lines). Right. Several cycles in the 2-torus. a is the of faces F . boundary of A, and c is also homologous to zero because it We now introduce the notion of a dual graph, which is encloses half of the surface. b,d,e are not homologous to zero. crucial for the construction of quantum topological codes. d and e are homologous because they enclose the area B. Given a cell embedding ΓM of a graph Γ in a surface M, ∗ the dual embedding ΓM is constructed as follows. For each face f a point f ∗ is chosen to serve as a vertex for follows: the new graph Γ∗. For each edge e lying in the boundary ∗ ∗ ∗ t of the faces f1 and f2, the edge e connects f1 and f2 u v u v v u σ σ = ϕ( Ω )σ σ (2) and crosses e. Each vertex v corresponds to a face v∗. The idea is illustrated in Fig. 2. where Let us enlarge a bit the concept of surface. Take a 0 1 surface M and delete the interiors of a finite collection of Ω := (3) −1 0 non-overlapping discs. We say that the resulting space is a surface with boundary. Note that for any cell embed- πik is a 2n×2n matrix over Z2 and ϕ(k) := e , k ∈ Z2. We ding in such a surface, the boundary is a subset of the keep the minus sign in Ω because it appears for higher embedded graph. One could argue that no dual graph dimensionality or qudits [11]. The group of all the op- can be defined for this surfaces, but in fact this is not a erators generated by the set of σ-operators is the Pauli major difficulty. It is enough to think that some of the group PD(n). vertices in the dual graph are ’erased’. For us the most There exists a nice construction using the Pauli group important example of surface with boundary will be the to construct the symplectic or stabilizer codes [12], [13]. h-holed disc Dh, h ≥ 1. 2n For any subspace V ⊂ Z2 , we define the subspace V := Consider a cell embedding ΓM on a surface, with or 2n t 2n { u ∈ Z2 | ∀ v ∈ V v Ωu = 0 }. Let VC ⊂ Z2 be any without boundary. In order to construct the physical sys- b tem realizing the topological code C, we attach a qubit isotropic subspace, that is, one verifying VC ⊂ VC [11]. to each edge of the graph Γ. The study of the stabilizer Let B be one of its basis and set S = { σv | v ∈ B }. The code b and correctable errors of the code gets benefited by us- ing Z2 homology theory, which we shall now introduce. ⊗n 0λ C := { |ξi ∈ H2 | ∀ σ ∈ S σ|ξi = |ξi} (4) Consider a Z-type operator σ . A chain is a formal sum of edges c1 = j λj ej . We relate chains and Z-type op- u u erators settingPσc1 := σ0λ. Similarly, given a cochain detects the error σ iff 6∈ VC −VC and thus its distance 1 ′ ∗ is or formal sum of dual edges c = j λj ej , we relate b it to an X-type operator setting σc1P:= σλ′0. There is d(C) = min |u|, (5) a natural product between chains and cochains, namely 1 ′ u∈VC −VC (c ,c1) := λ · λ . The point is that

b 1 where the norm is just the number of qubits on which σu 1 1 σc σc1 = ϕ((c ,c1)) σc1 σc . (6) acts nontrivially. The set S is the stabilizer of the code. This way, the problem of finding good codes is reduced This expression already shows that the commutation re- to the problem of finding good isotropic subspaces VC. lations of operators are determined by the topology of Thus far we have dealt with the purely algebraic struc- the cell embedding ΓM . Compare with (2). ture of codes. Now, we turn to the connection with topol- Note that a chain is nothing but a collection of edges, ogy. For a surface we understand a compact connected those with a coefficient equal to one, and thus can be 2-dimensional manifold. Well known examples of surfaces easily visualized as lines drawn on the surface. If a chain are the sphere S, the torus T and the projective plane has an even number of edges at every vertex, we call it a P . More generally one can consider the g-torus or sphere cycle. When a cycle encloses an area of the surface, we with g handles gT , and the sphere with g crosscaps gP . say that it is a boundary. If two cycles enclose an area gT and gP are said to have genus g. In fact, it is a well altogether, they are said to be homologous. Boundaries known result of surface topology that the previous list of are homologous to zero. Figures 2 illustrate these con- surfaces is complete. cepts. Cocycles are defined analogously but in the dual We shall not be restricted to codes based on regular graph. Coboundaries are a bit different, however, at least lattices on a torus, or toric codes [7], but we shall use in the case of surface with boundary. If cutting the sur- 3 face along a cocycle divides it apart into two pieces, then ima/minima of certain graph properties. To this end, we it is a coboundary. Given a face f, we shall denote by ∂f find very useful to introduce the following concept. its boundary. Similarly, given a dual face v∗ we denote its coboundary by δv∗, see Fig. 5. Definition 2 Given a surface M and a positive integer Before stating the main result about general construc- d, we let the quantity µ(M, d) be the minimum number tions of topological quantum codes for arbitrary graphs of edges among the embeddings of graphs in M giving a embedded in surfaces, we need a pair of ingredients. code of distance d. Given a a surface M there exists always some cell em- Since calculating the value of the function µ is a hard bedding on it. The Euler characteristic of M is defined problem, we shall investigate some of its properties. The by issue of connectivity between sites suggests also the in- χ(M) := |V | − |E| + |F |, (7) troduction of a refinement of µ. The quantity µc(M, d) is defined as µ(M, d) but with the restriction that the and it does not depend upon the embedded graph. Now, graphs can have faces with at most c edges and vertices let ΓM be a cell embedding of a graph Γ in a surface lying in at most c edges. By connectivity here we mean M. We define the distance d(ΓM ) as the minimal length that vertex σδv∗ and face σ∂f operators should act over (edge amount) among those cycles which are not homol- at most c qubits. A loosely connected system simplifies ∗ ogous to zero. For surfaces with boundary, d(ΓM ) should the error correction stage. be understood as a symbol denoting the minimal length among cocycles. III. OPTIMAL ENCODING RATES

Theorem 1 Topological codes. Let ΓM be a cell embedding of a graph in a surface. The symplectic code There is an interesting result in surface topology stat- C of length n = |E| with stabilizer S = { σδv∗ | v ∈ V } ∪ ing that every surface can be obtained by combination of ∗ { σ∂f | f ∈ F } has distance d = min{d(ΓM ), d(ΓM )} and S, P and T . To perform this connection over two sur- encodes k = 2 − χ(M) qubits if M does not have any faces, say M1 and M2, one chooses two discs Di ⊂ Mi. boundary or k =1 − χ(M) qubits if it does. The connected sum of M1 and M2, denoted M1 ♯M2, is constructed by deleting the interiors of D1 and D2 and Proof. This proof involves homology theory. Following identifying its boundaries. Connecting g ≥ 0 tori to a standard notation, we denote the first homology group sphere one gets gT , and connecting g ≥ 1 projective by H1 = Z1/B1 and the first cohomology group by 1 1 1 ∗ planes one gets gP . The point is that given embeddings H = Z /B . Now, since (δv ,∂f) = 0, the space VC 1 of distance d in M1 and M2, a new embedding can be is isotropic. Note that VC ≃ B ⊕ B1. A key observation ∗ constructed in M1♯M2 in such a way that the number is that (δv ,c1) = 0 iff c1 ∈ Z1, and similarly for ∂f and of edges does not increase and the distance is preserved. 1 ˆ 1 Z . In other words, VC ≃ Z ⊕Z1, and the distance (5) is The procedure is displayed in Fig. 3. This implies the the one stated. As dim VˆC − dim VC =2k, where k is the following result, which we shall use to proof asymtoptic number of encoded qubits, it only remains to know the properties about minimal sizes of topological codes: dimension of the homology and cohomology group. But 1 2−χ we have H1 ≃ H ≃ Z2 for surfaces without boundary Proposition 3 Topological Subadditivity. Given two 1 1 −χ 1 2 and H1 ≃ H ≃ Z2 for surfaces with boundary.  surfaces M and M Since χ(gT )=2 − 2g, the g-torus yields codes with µ(M1♯M2, d) ≤ µ(M1, d)+ µ(M2, d). (8) k = 2g logical qubits. χ(gP )=2 − g, and thus codes from gP encode k = g qubits. For the h-holed disc Dh, Let us apply these tools starting with the torus T , the χ(Dh)=1 − h and k = h. The parity check matrix H simplest orientable surface with nontrivial first homology of a topological code has the diagonal form diag(H1,H2) group. In [7], a family of toric codes was presented, in where the matrices H1 and H2 are in essence the inci- ∗ the form of self-dual regular lattices on the torus. This is dence matrices of Γ and Γ . Thus, topological codes are a very simple instance of topological graph theory. One an example of generalized CSS codes [14], [15]. can consider other self-dual regular lattices embedded on Note that uncorrectable errors are related to cycles the torus. All of them share the property that vertex σδv∗ which are not homologous to zero. This is exemplified and face σ∂f operators act on c = 4 qubits. Among them, as part of Fig. 5. Therefore, the whole problem of con- we have found an optimal family of lattices that demand structing good topological codes related to a certain sur- half the number of qubits. Examples of both systems of face relies on finding embeddings of graphs in such a way lattices are depicted in Fig. 4, were the torus is repre- that both the embedded graph and its dual have a big sented as a quotient of the plane through a tessellation. distance whereas the number of edges keeps as small as The original toric codes lead to a family of [[2d2, 2, d]] possible. Thus, we find that this quantum problem can codes [7]. Our lattices give [[d2 +1, 2, d]] codes. This be mapped onto a problem of what is called extremal already shows that topological graph theory, a branch devoted to graph em- 2 beddings on surfaces [16] and the computation of max- µ(T, d) ≤ µ4(T, d) ≤ d +1. (9) 4

FIG. 3: The construction that proves the topological subaddi- tivity of µ. The first step is to perform a cut along a selected FIG. 5: Left. A graph embedding in the surface D4 yielding a code of distance 3. We display a boundary ∂f, a coboundary edge in each of the embeddings to be connected. Then the ∗ resulting boundaries must be identified. δv and two uncorrectable errors of Z and X-type: the cycle z is not a boundary, and the cocycle x is not a coboundary. Right. The self-dual embedding of K9 in the 10-torus. Thick lines represent the graph, weak ones its dual.

of codes correcting a single error. Figure 1 displays the rates for this family of codes and also for the optimized toric codes. Now we can appreciate the very different behaviour between toric codes and topological codes em- FIG. 4: Four self-dual lattices on the torus. Here the torus bedded in higher genus surfaces, the latter ones allowing is represented as a quotient of the plane and a tessellation of it. Thick lines are the border of tesserae. (a),(b): The toric us to increase the encoding rate up to its maximal value. codes introduced in [7], for d = 3 and d = 5. (c),(d): The So far we have not touched upon the question of phys- optimal regular toric codes for d = 3 and d = 5. ical implementations. Consider the system of qubits ar- ranged according to a given graph embedding, as ex- plained above. The hamiltonian Invoking topological subadditivity, we learn that µ(nT,d) 2 is O(d ) , that is, it grows at most quadratically with d. H = − σ∂f − σδv∗ (11) This analysis shows that our toric lattices yield a better fX∈F vX∈V k 2 k encoding rate n ∼ d2 than the original ones with n ∼ 1 has a degenerate ground state whose elements are the 2 . However, despite being optimal, these toric lattices d protected codewords. This system is naturally protected produce encoding rates vanishing in the limit of large n against errors [7]. A major drawback is the requirement qubits (see Fig. 1). Then, a major challenge arises: is it possible to find topological quantum codes with non- that the physical disposition of the qubits must give rise to a nontrivial topology. It is difficult to imagine an vanishing encoding rates? And what about approaching experimentalist constructing a system living in a torus, the maximum value of 1? The answer is positive in both for example. cases and we hereby show the construction. However, the formalism that we have presented allows To this end, let us introduce the complete graph K s us to use surfaces with boundary. In particular, the h- as the graph consisting of s vertices and all the possi- holed disc D gives rise to codes encoding k = h qubits ble edges among them. A closer examination of Fig. 4 h but has the advantage of being a subset of the plane. reveals that the lattice giving a [[10, 2, 3]] code is a self- Figure 5 shows an example of a very regular embedding dual embedding of K5. This suggests considering self- in D4. It is apparent how to generalize this example to a dual embeddings of Ks, since such an embedding would s s higher number of encoded qubits k and distances d. As give a [[ , − 2(s − 1), 3]] code. In fact, these embed- 2 2 in the case of surfaces without boundaries, the length of dings are possible in orientable surfaces with the suitable

the code will scale as O(kd2). genus as long as s = 1 (mod 4) [16]. In Fig.2 we show the constructions of such an embedding. Due to topological subadditivity and the fact that µ(gT,d) ≥ µ(gT, 1)=2g, IV. CONCLUSIONS the family of codes given by self-dual embeddings of com- plete graphs Ks is enough to show that for d =3 Finally, we want to emphasize that the locality of topo- k 2g logical codes is a very important issue for their imple- lim = lim =1, (10) n→∞ n g→∞ µ(gT, 3) mentation in physical systems, now more feasible after the introduction of planar topological codes. The em- that is, the ratio k/n is asymptotically one, and thus good beddings of complete graphs provide encoding rates that topological codes can be constructed, at least in the case overcome the barrier established so far by toric codes. 5

Moreover, we have introduced a measure µ that estab- from a PFI fellowship of the EJ-GV (H.B.), DGS grant lishes an interplay between quantum information and ex- under contract BFM 2003-05316-C02-01 (M.A.MD.), and tremal topological graph theory. CAM-UCM grant under ref. 910758. Acknowledgements We acknowledge financial support

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