Qudit surface codes and gauge theory with finite cyclic groups

arXiv:quant-ph/0609070v2 12 Sep 2006 Qudit surface codes and gauge theory with finite cyclic groups Stephen S. Bullock IDA Center for Computing Sciences, 17100 Science Drive, Bowie, MD 20715-4300 USA E-mail: [email protected] Gavin K. Brennen Institute for Quantum Optics and Quantum Information, Techniker Str. 21a, 6020, Innsbruck, Austria E-mail: [email protected] Abstract. Surface codes describe quantum memory stored as a global property of interacting spins on a surface. The state space is fixed by a complete set of quasi-local stabilizer operators and the code dimension depends on the first homology group of the surface complex. These code states can be actively stabilized by measurements or, alternatively, can be prepared by cooling to the ground subspace of a quasi-local spin Hamiltonian. In the case of spin-1/2 (qubit) lattices, such ground states have been proposed as topologically protected memory for qubits. We extend these constructions to lattices or more generally cell complexes with qudits, either of prime level or of level dℓ for d prime and ℓ ≥ 0, and therefore under tensor decomposition, to arbitrary finite levels. The Hamiltonian describes an exact Zd ∼ = Z/dZ gauge theory whose excitations correspond to abelian anyons. We provide protocols for qudit storage and retrieval and propose an interferometric verification of topological order by measuring quasi-particle statistics. Qudit surface codes and gauge theory with finite cyclic groups 2 1. Introduction There is a rich history to the study of topologically ordered states of matter. Such states are defined by the property that all physical correlation functions are topological invariants. In the field of condensed matter, these states have been proposed as ground states of models for high temperature superconductors and for fractional quantum Hall states [22]. Furthermore, it has been demonstrated that such order can arise as a low energy property of hard core bosonic spin lattice models. In contrast to the familiar situation with spontaneous symmetry breaking, here the ground states exhibit more symmetry than the microscopic equations of motion. It has been suggested that such emergent properties may model gauge fields and particles found in nature [16]. In the field of quantum information it was shown by Kitaev [12] that ground states of Hamiltonians which can be expressed as a sum over quasi-local stabilizer operators provide for topologically protected qubit memories. These states are referred to as surface codes. They are robust to arbitrary quasi-local perturbations and have abelian anyonic excitations. In order to perform universal fault-tolerant quantum processing, it is necessary to use non abelian anyonic excitations that transform under an appropriate group [12]. From the algorithmic point of view attempts have also been made to understand quantum computing in terms of nonabelian anyon operations [1]. Finding suitable microscopic lattice models that provide for universal quantum computation is an area of active research [6]. This survey attempts to exhaust the topic of surface codes for topologically protected qudit memories. While not as powerful as fault tolerant models with non abelian anyons, these models offer a new perspective on non-local encoding of quantum information and give us insight into microscopic realizations of lattice gauge theories. Surface codes for two level systems [12] are by now well understood. Their implications for error-resistant quantum computer memories have also been considered [7]. In the error-correction context, the topologically ordered eigenstates may be understood as a particular case of quantum stabilizer codes (e.g. [9].) The error lengths of the resulting stabilizer codes are not exceptional, and only rarely do anyonic systems appear in classifications of near-optimal quantum codes. (Optimality in this sense refers to minimizing the number of code-qubits against the number of errors a code may correct.) Yet all the error correction operations are local upon the lattice in which the quantum data is stored, which might improve scalability. Moreover, an aside to an argument focused on deriving a famous stabilizer code from the topology of the real projective space in fact demonstrates that a qubit lattice is not required [8]. Rather, a two-complex (see e.g. [15]) suffices, where a two-complex is a generalization of a graph in which discs are also allowed with edge boundaries. On the physical system which places a qubit on each edge of a (cellular or simplicial) two-complex Γ, there exists a Hamiltonian whose topologically ordered (stabilizer-code) groundstates are parametrized by the first homology group of the complex with bit-coefficients: H1 (Γ, F2 ). The Hamiltonian is a sum of vertex and edge terms which are proportional to either tensors of Pauli Z operators around qubits on edges adjacent to a vertex or are proportional to tensors of X operators on edges bounding a face of the complex. For some time the existence of stabilizer codes over qudits (d prime) have been known [9]. Yet only recently have results on the topic become as strong as those applicable in the bit case, including estimates of optimal code-lengths etc. ([11], see also [10].) Moreover, extensions to prime-power (dℓ ) level qudits (actually qudℓ its) have also been found, so that tensors provide a stabilizer formalism for all finite-level Qudit surface codes and gauge theory with finite cyclic groups 3 systems. In this work, we exploit the new stabilizer formalism to construct codes on a two-complex whose edges carry prime-d-level qudits, and we also outline the extension to dℓ -level qudits. The associated ground states are parametrized by H1 (Γ, Fdℓ ), where the coefficient field is viewed as an abelian group under addition. This requires few new ideas, although care must be taken with sign conventions which were vacuous in the earlier work on F2 -coefficients. Thus, after tensoring we have constructed surface codes with qudits for arbitrary finite d placed on the edges of a generic orientable two-complex Γ. Recent work by Bombin and Martin-Delgado [2] investigates classical and quantum homological error correction codes. They construct a class of surface codes for qudits which asymptotically saturates the maximum coding rate and provide several example encodings on various two complexes. Here we do not address the issue of coding efficiency. Rather we concentrate on explicit constructions of Hamiltonians that support qudit surface codes in their ground eigenstates and describe how one might encode and decode therein. The manuscript is intended to be self-contained. Thus, §2 opens by reviewing some the required facts on stabilizer codes. In order to aid readers less interested in the general case, §3 treats prime-d level encoding on surfaces separately. Methods for encoding, decoding, and stabilizer measurements are given in §4. Extensions to the case of prime power qudit encodings are given in §5. Errors in our model correspond to low lying excitations in the Hamiltonian whose superselection sectors may be viewed as massive particles on the underlying cellulation. In §6 it is shown that our model reproduces a Z/dZ gauge theory where errors are described by particle anti-particle pairs of change/flux dyons. We propose an interferometer circuit for measuring the the statistics of these quasiparticles. We conclude with a summary and some. 2. Qudit Stabilizer Codes We next review stabilizer codes [9, 11]. This section focuses on the case of qudits with a prime number of levels. The first subsection recalls the definition and a basic technique. The next subsection generalizes a well known construction from bits to dits. 2.1. Stabilizers and groundstates Let d be a prime number, and consider the qudit state space H(1, d) = C |0i ⊕ · · · ⊕ C |d − 1i, with a pure state of n qubits being a ket within H(n, d) = H(1, d)⊗n . A possible generalization of the Pauli operators on H(1, d) would be to consider the group generated by the following unitary matrices: X |ji = |j + 1 mod di (1) Z |ji = ξ j |ji , for ξ = exp(2πi/d) These are not Hermitian unless d = 2. The qudit Pauli-tensor group, say P(n, d) ( U [H(n, d)], is the group of unitary matrices generated by n-fold tensors of elements of {Id , X, Z}. We might be more explicit in the description of P(n, d). First, for n = 1, label the multiplication in Fd to be a dot-product. Then Z b X a = ξ a•b X a Z b . More generally, for dit-strings a, b ∈ (Fd )n , we use X ⊗a and Z ⊗b to abbreviate X a1 ⊗ X a2 ⊗ · · · ⊗ X an and similarly Z ⊗b for Z b1 ⊗ Z b2 ⊗ · · · ⊗ Z bn . For the n-entry dot-product with values in Fd , we have Z ⊗b X ⊗a = ξ a•b X ⊗a Z ⊗b . Thus explicitly (2) P(n, d) = {ξ c X ⊗a Z ⊗b ; a, b ∈ (Fd )n , c ∈ Fd Qudit surface codes and gauge theory with finite cyclic groups 4 The qudit stabilizer groups are subgroups G ⊆ P(n, d). The code subspace of such a stabilizer group is the joint +1 eigenspace of all g ∈ G. Of course, such joint eigenspaces might well be trivial. Yet a standard argument shows that they are nontrivial in certain cases. This technique is so fundamental to stabilizer code manipulation that we wish to highlight it; it will be used several more times in the course of the work. While actually an elementary technique from representation theory, it has also featured prominently in the quantum computing literature [14]. P Stabilizer code projectors: The sum of unitary maps π = (#G)−1 g∈G g is a projection onto the code-subspace. We present the argument. First, π 2 is the identity map since πg = π for any g ∈ G. Second, π = π † since adjoints are inverses in the unitary group. Hence, either π is a projection or −1 is an eigenvalue of π. Yet Idn is a summand, so the complex inner product precludes π |ψi = − |ψi for a nonzero |ψi. Now split H(n, d) = V1 ⊕ V2 ⊕ · · · ⊕ Vℓ into irreducible orthogonal unitary subrepresentations of G. For each Vj , the image under π and its orthogonal complement form a decomposition of Vj . Thus by irreducibility, π either preserves a Vj or πVj = 0. Clearly the former holds for any irrep (i.e. irreducible representation) within the code subspace of G. On the other hand, if hψ|g|ψi = 6 1 for some g, then the latter holds. As a remark, irreps within the code subspace of G must be one-dimensional and are also known as trivial representations. As a second remark, the code subspace is nonzero iff Trace(π) 6= 0 iff (G ∩ {ξ j Id }) = {Idn }. In the Hermitian case (d = 2,) it is standard that all eigenvalues of group elements are ±1, so that a suitable Hamiltonian for which the code space is the groundstate is −π. For general d, the eigenvalues lie within the unit circle, so that −1 is still the least possible real part. Also, g † |λi = (1/λ) |λi = λ |λi since g † = g −1 . Thus, one may place the qudit code subspace into the groundstate of Pa Hamiltonian by adjusting each summand of π with a Hermitian conjugate: H = g∈G −(g + g † ), so that the eigenvalues of the summands are then −2Re[spec(g)]. 2.2. Quantum circuits for qudit stabilizer measurements Given an n-qudit system, it is important for purposes of error-correction to be able to test whether or not a state |ψi lies within the stabilizer code of some G = h{gj }i ⊆ P(n, d). It suffices to test whether |ψi is a +1 eigenvector of each generator gj . We sketch quantum circuits which achieve such a measurement. Pd−1 Let Fd = d−1/2 j,k=0 ξ jk |ji hk| be the qudit Fourier transform. Considering Pd−1 eigenkets, Fd† XFd = Z. Now the number operator n = j=0 j |ji hj| suffices to infer the eigenvalue of Z and project into the appropriate eigenstate. As a circuit, we might denote a number operator measurement with the Z symbol, one of several common conventions in the qubit case: FE Determination of the X eigenstate may be accomplished by Fd† FE Similarly, there is some one-qudit unitary which will diagonalize any X a Z b ∈ P(1, d), usually not a Fourier transform. Yet using the diagonalization and a number operator one may infer an eigenstate. Qudit surface codes and gauge theory with finite cyclic groups 5 For Z ⊗k and X ⊗k , we suggest using addition gates along with a qudit ancilla. We will denote |j, ki 7→ |j, (j + k) mod di by a typical control bullet with the target (in the formula second) line holding a + gate. The the following construction of Z ⊗2 generalizes for Z ⊗k : |0i + + FE • • For Z ⊗k |j1 , j2 , . . . , jk i = ξ j1 +···+jk |j1 , j2 , . . . , jk i, and we have placed |(j1 + · · · + jk ) mod di on the ancilla line before the number operator is applied. Note that Z ⊗ Z −1 results by replacing one of the modular addition gates above with modular subtraction. Powers of operators are measured by multiple appications of the sum gate appropriately. Finally, (Fd† )⊗k X ⊗k Fd⊗k = Z ⊗k , so that the following diagram for X ⊗2 extends: |0i Fd† + + • Fd† FE Fd • Fd Using similarity transforms by qudit Fourier transforms, we may similarly achieve X ⊗ Z ⊗ Z ⊗ X etc. Yet more generally, the comment on existence of diagonalizations above produces circuits for arbitrary elements g ∈ P(n, d). 3. Homologically Ordered Groundstates for Prime Qudits It is typical to place topological orders on explicit planar or spacial lattices of spinj particles, e.g. square, triangular, hexagonal, Kagome, etc. An alternative was presented in Freedman and Meyer’s derivation of certain error-correcting codes of Shor and LaFlamme [8]. Namely, qubits could be placed on the edges of a twocomplex Γ, and an appropriate Hamiltonian would have the dimension of its degenerate groundstate eigenspace equal to the number of classes within H1 (Γ, F2 ). We next extend this construction to prime-level qudits; the task is mainly to keep track of sign conventions which are vacuous in F2 . We then check that the groundstate eigenspace is similarly spanned by kets associated to elements of H1 (Γ, Fd ), by applying stabilizercode techniques. 3.1. Cellular Hamiltonians Label V to be the vertices of Γ, E to be the edges, and F to be the faces. We also require properties that hold if Γ is a cellulation of an orientable, compact, connected surface. Specifically, each edge has a boundary of exactly two vertices and each face has an orientation according to which each edge lies in the boundary of two faces with the edge taking opposite orientations in the boundary of each face. Finally, Γ is finite and H2 (Γ, Fd ) is a copy of Fd spanned by [Γ], the sum of all faces with their orientation according to Γ. Qudit surface codes and gauge theory with finite cyclic groups 6 We briefly review the appropriate homology. Label the chain sets to be formal sums of vertices, edges, and faces respectively: C0 (Γ, Fd ) = spanFd (V), C1 (Γ, Fd ) = spanFd (E), and C2 (Γ, Fd ) = spanFd (F ). We generally drop the Γ and coefficient system, which should be clear from context. Since Γ is a cell complex, there exist boundary operators ∂ 2 ∂ C0 ←− C1 ←− C2 (3) with ∂ = 0 [15]. For example, if an edge e connects v1 and v2 , say e = [v1 , v2 ], then ∂e = v1 − v2 = v1 + (d − 1)v2 . Note that the definition of Γ demands that edges e ∈ E are images of [0, 1] within Γ, and hence all edges are implicitly oriented. The coefficients further allow for Fd -valued multiplicities on each edge. Since ∂ 2 = 0, we have ker(∂1 ) ⊇ image(∂2 ) for ∂j : Cj → Cj−1 . Thus we may define the Fd vector space H1 (Γ, Fd ) = ker(∂1 )/image(∂2 ). This first homology group is well known to be a topological invariant, i.e. any topological space homotopic to that underlying Γ will produce an H1 (Γ, Fd ) of the same dimension. Homology elements are represented by cycles, i.e. elements of the kernel of the boundary operator. However, several elements might represent the same class, differing by a boundary, i.e. an element of ∂(C2 ). Recall that any Hamiltonian on n-qudits may be written as a sum of tensor products of Hamiltonians (Hermitian matrices) on each factor. The degree of a summand in the tensor basis is the greatest number of non-identity factors in any term. A k-local Hamiltonian is a Hamiltonian whose degree is bounded by k in some decomposition. The topologically ordered Hamiltonians defined below are k-local for k the maximum of the valence of any vertex and the number of edges on any face. Let n = #E, and consider placing a qudit on each e ∈ E. Again, each edge is the image of [0, 1] and is oriented (by Γ) from one vertex to the other. For the qudits associated with each edge, the |1i excitation of the edge will be implicitly associated to this orientation, while the |d − 1i state corresponds to the other. On the associated physical system H(n, d), let Xe and Ze denote the operator applied to the qudit of that edge with identity operators buffered into the remainder of the tensor. For each v ∈ V, we define a Pauli-tensor and vertex Hamiltonian by Q Q −1 gv = e=[v,∗] Ze e=[∗,v] Ze (4) † Hv = −(gv + gv ) For some U > 0, we thenPdefine the potential energy term of a topologically ordered Hamiltonian by H∂ = U v∈V Hv . P The notation H∂ has been chosen for the following reason. Suppose that ω = e∈E ne e is a chain, with each ne ∈ Fd . There is an associated qudit computational basis state, say |ωi, which is local and places the qudit of each e in state |ne i. We claim that |ωi is a groundstate of H∂ iff ∂ωP = 0, i.e. ω is a cycle. To see this, one verifies that gv |ωi = ξ c |ωi where ∂ω = cv + w6=v cw w. Hence |ωi is in the stabilizer h{gv }i ⊆ P(n, d) iff |ωi is an eigenstate of each Hv of minimial (real) eigenvalue iff |ωi is in the degenerate groundstate eigenspace of H∂ . Strictly speaking, one should not refer to the groundstate of H∂ P as being topologically ordered. Admittedly, groundstates are of the form |ψg i = αω |ωi for ω a cycle, colloquially a loop of excited edges. For d > 2, the edges must be properly oriented, and hitting every edge of a Y junction is allowed if multiplicities are accounted for. Yet the cycle subspace is not a topological invariant. Indeed, should Γ be a cell complex, subdividing Γ by breaking each 2-simplex (triangle) into several subtriangles will generally increase the size of ker(∂1 ), although such a subdivision 7 Qudit surface codes and gauge theory with finite cyclic groups Figure 1. Cellulation of an orientable surface. Each system particle (qudit) is represented by an edge. Particle interactions occur between all edges that meet at a common vertex and all edges comprising a plaquette boundary. (a) In this example, physical qudits reside on the vertices of a Kagome’ lattice on a torus such that the resultant cellulation is a honeycomb lattice on a torus. Edge and face orientations are indicated. For the vertices v0 , v1 and faces f0 , f1 the mutually −1 , gv1 = commuting operators in the Hamiltonian are gv0 = Z[v6 ,v0 ] Z[v5 ,v0 ] Z[v ,v ] 0 1 −1 −1 Z[v0 ,v1 ] Z[v Z −1 , gf0 = X[v0 ,v1 ] X[v1 ,v9 ] X[v X −1 X −1 X , ,v ] [v ,v ] ,v ] [v ,v ] [v ,v ] [v6 ,v0 ] 1 9 1 2 8 9 7 8 6 7 −1 −1 −1 gf1 = X[v X[v X[v3 ,v2 ] X[v4 ,v3 ] X[v5 ,v4 ] X[v . (b) Same cellulation with 0 ,v1 ] 1 ,v2 ] 5 ,v0 ] vertex (red) ancilla and face (green) ancilla. These can be used to perform local stabilizer checks or to mediate many body interactions between edges from physical 2-local interactions as described in §4.3. does not change the topology of the underlying manifold. Thus, we next add a kinetic energy term to the potential, splitting the degeneracy of H∂ and reducing to a final groundstate capturing homology. For each face f , the face Hamiltonian Hf is defined as follows. Orient Ppf according to the orientation of the manifold underlying Γ. Label edges by ∂f = k=1 ok ek for ok ∈ {1, d − 1}. Then we define gf Hf o = Xeo11 Xeo22 Xeo33 . . . Xepp = −(gf + gf† ) (5) With these choices, [Hf , Hv ] = 0 for all faces f and vertices v. For the two edges incident on a given vertex will be in the boundary of some face, and after correcting for −1 orientation conventions this commutativity check reduces to [X⊗X, Z⊗Z P ] = 0. (See Figure 1.) Hence, for some constant h > 0, we might define HKE = h f ∈F Hf . Due to commutativity, the kinetic energy Hamiltonian respects the groundstate degeneracy of H∂ . Label H = H∂ + HKE . We next show that the dimension of the groundstate degeneracy of total Hamiltonian H = H∂ + HKE (6) (over C) corresponds to the number of elements of H1 (Γ, Fd ). 3.2. Homology class groundstates The goal of this section is to associate the degeneracy (dimension) of this groundstate of H = H∂ + HKE to #H1 (Γ, Fd ). We accomplish this in two distinct cases for the Qudit surface codes and gauge theory with finite cyclic groups 8 manifold underlying Γ: (i) The manifold is orientable, compact, and has no boundary, so that H2 (Γ, Fd ) = Fd . (ii) The manifold is compact with boundary and has H2 (Γ, Fd ) = 0. For homology is a homotopy invariant, and such a surface retracts into its one skeleton. Assertion: Let Hloop denote the groundstate of H∂ and H[ω] = ⊕η∈[ω] C |ηi. Hloop = ⊕ω∈ker ∂ C |ωi = ⊕[ω]∈H1 (Γ,Fd ) H[ω] (7) P Throughout this section, let π = #G−1 g∈G g. Suppose either Case i or Case ii. Then for each [ω], the restriction of π to H[ω] is a rank one projector whose (nonzero) image is an element of ker (H∂ + HKE ) = ker H. To verify this, suppose |ωi is the computational basis state of some cycle ω ∈ C1 (i.e. ∂ω = 0.) Then we may also speak of [ω] ∈ H1 (Γ, Fd ), |ωi is in the groundstate of H∂ . Label X def |[ω]i = πf |ωi = (#G)−1 g |ωi . (8) g∈G It suffices for the Assertion to show the following. • If ω1 and ω2 each lie in [ω], then |[ω1 ]i and |[ω2 ]i differ by a global phase. • If |ωi = 6 0, then |[ω]i = 6 0. This suffices to see the restriction of π is a rank one projector, since the first item demands the rank ≤ 1 and the second demands the rank ≥ 1. We begin with the first item, writing ω1 − ω2 = η ∈ im ∂2 . Since the underlying manifold of Γ is orientable, Q that all faces f have positive P suppose for convenience orientation. Then for η = f ∈S(η) f we put gη = f ∈S(η) gf , implying |ω1 i = gη |ω2 i. Note that gη πf = πf gη = πf . Thus |[ω1 ]i = πf gη |ω2 i = πf |ω2 i = |[ω2 ]i. We next demonstrate that πf |H[ω] has rank ≥ 1. As discussed in §2, it suffices to show that the trace of this projection, when restricted to the subspace H[ω] which it preserves, is nonzero, and that immediately follows if ξ ℓ Idn ∈ Gf demands ξ = 1. For all other elements of P(n, d) are traceless when restricted to Hloop , since gη |ωi = |ω + ∂ηi. Case i and Case ii differ somewhat. In each case, multiples of the identity in Gf are products gη for [η] ∈ H2 (Γ, Fd ). In Case i, besides the Q empty product of the gf we also produce multiples of Idn as the full product f ∈F gfk , 0 ≤ k ≤ d − 1. This corresponds to H2 (Γ, Fd ) = Fd . Yet for these products ξ = 1, as may be verified at an individual edge. In Case ii, there is no nontrivial product of the gf which produces a multiple of the identity. This is due to the retraction demanding H2 (Γ, Fd ) = 0, the second homology of the one complex we may retract onto. Colloquially, taking a sum of all faces will force a boundary edge to be acted on nontrivially by gf for the single face it bounds. Thus in Case ii the only multiple of the identity is the trivial product of the gf , and ξ = 1 tautologically. In each case, πf |H[ω] is not traceless and hence has rank at least one. Given the last paragraph, the rank is exactly one. Retracing the argument above, we may compute the image under π of the code space of Gv is ⊕[ω]∈H1 (Γ,Fd ) C |[ω]i, which is also the code space of G. Since πf is a rank-one projector when restricted to each H[ω] , we have the following. dimC ( groundstate of H) = #H1 (Γ, Fd ) (9) Qudit surface codes and gauge theory with finite cyclic groups 9 3.3. Groundstates on a punctured disk In practice, constructing physical realizations of Hamiltonians corresponding to twocomplexes without boundary is daunting. It is possible to simply identify opposite qudits on the square fundamental domain of S 1 × S 1 = R2 /Z2 , but this would require some sort of nonlocal coupling on the boundary in addition to the standard lattice coupling. Given a lattice Hamiltonian that arises from electromagnetic coupling, one could speculate about some kind of apparatus (perhaps involving fiber-optic cabling [21]) which allows for interactions between boundary qudits. Alternately, we might modify the homological groundstates to allow for a surface with a boundary curve and punctures. Consider a cellulation Γ of a disk with k punctures. An example with k = 2 is shown in Fig. 2. Label the j-th puncture face fj′ which has the same orientation as Γ. Also label the outer boundary of the disk ∂Γ and the boundaries of the j-th puncture ∂fj′ . Analogous to the previous construction, ′ the Hamiltonian on Γ is defined H ′ = H∂ + HKE . Here the kinetic term is modified so that the set ofP face operators does not include operators on the punctured faces ′ fj′ , i.e. HKE = h f ∈F ′ Hf , where F ′ = F \ {f ∈ ∪kj=1 fj′ }. Consequently, there are edges on the boundaries ∂fj′ that are acted on by X operators from faces on one side only. Another way to see this is that all edges of the dual cellulation that cross the boundary ∂fj′ share a common vertex located at fj′ in Γ. Each edge in Γ has two vertices in V, hence the product over all vertex operators is: Y (10) gv = Idn . v∈V Not every edge in Γ borders two faces in F ′ , however, and the product over all face operators is: Y f ∈F ′ gf = C∂Γ (X) k Y Cj (X), (11) j=1 Q Q o o where C∂Γ (X) = ej ∈∂Γ Xejj and Cj (X) = ej ∈∂f ′ Xejj . The orientation oj = 1 j if the edge ej is oriented in the same direction as the boundary on which the edge resides, and ej = d − 1 if the orientations are opposite. First we argue that the code space in nonempty. Recall, the code states are defined as +1 eigenstates of the stabilizer group G′ = h{gf |f ∈ F ′ } ⊔ {gv }i. The operators h{gv }i and h{gf } commute and the only additional relations obtained from T the stabilizer group, embedded in Eqs. 10,11, guarantee that (G′ ξ j Idn ) = Idn . We next show that the code space is Hgr = H(k, d) by considering the action of operators that commute with any member of G′ but act non trivially on Hgr . One such set of operators are non trivial Fd valued cycles on Γ generated by {Cj (X)}. We do not include the non trivial cycles generated by C∂Γ (X) because by Eq. 11 their action on the code subspace is not independent but can be generated by the cycles around the boundaries of the punctures. A non trivial cycle on Γ̃ is generated by a string of Z operations along a path P ath(j) that begins on an edge of ∂fj′ and ends on an edge of ∂Γ without touching otherQ edges on puncture boundaries. We denote the generator o of a such a cycle Cj (Z) = ej ∈P ath(j) Zejj where ok = 1 at the edge ek ∈ ∂fj′ if ′ ek and ∂fj share the same orientation and ok = d − 1 otherwise. The other oj are chosen in a consistent way such that [H ′ , C(Z)fj′ ] = 0. The operators on cycles satisfy the commutation relations C(Z)aj C(X)bj = ξ ab Cj (X)b Cj (Z)a for a, b ∈ Fd as is easily verified by considering the action on the one intersecting edge e ∈ ∂fj′ . As such the set Qudit surface codes and gauge theory with finite cyclic groups 10 Figure 2. An oriented two complex Γ, which is a cellulation of a two punctured disk encoding two logical qudits in n physical qudits. Vertex operators Hv are k local where k is the valence of the vertex whereas all face operators Hf are 4 local in this example. Ground states are +1 eigenstates of the stabilizer group G′ , but not all the stabilizer generators are independent. There are two independent non trivial cycles on Γ which can be generated by closed loops of X operators around the boundaries ∂fa′ and ∂fb′ . Similarly, there are two independent non trivial cycles on the dual Γ̃ which can be generated by strings of Z operators that connect two independent pairs of boundaries of the complex. Shown are the Pauli group operations Z k X j on qudit a and Z s X r on qudit b. Rj = {Cj (Z)a Cj (X)b )}d−1 a,b=0 generates a representation of the Pauli group P(1, d). For sufficiently spaced punctures, all paths P ath(j) exist and the group R = h{Rj }kj=1 i forms a representation of P(k, d). We then find that the ground subspace of H ′ encodes k qudits and the set Rj performs local Pauli group operations on the j-th qudit. In a lattice implementation of our model Hamiltonian, the punctures may arise as physical defects in the system. Coding operations that correspond to cycles around defects vividly illustrate the fact that even short ranged correlators (short relative to the system size) in a topologically ordered state can have non-trivial values. 4. Quantum Memory: Input/Output and Error Detection We next describe how one might exploit abelian anyons as quantum memories; the qubit case has been studied thoroughly [7]. In the new setting of prime level qudits, we must treat storage and retrieval of quantum data. It is also possible to generalize earlier discussions of stabilizer operations on topologically stored data while in code, but we will not treat that topic here. Qudit surface codes and gauge theory with finite cyclic groups 11 4.1. Storing qudits Placing quantum data into such a |[ω]i is difficult. For large lattices, this would be a special case of the qudit state-synthesis problem. Universal circuits of two-qudit operators capable of reaching arbitrary n qudit states are known to scale exponentially with the number of qudits [4]. In this section, we propose an alternative which requires a number of stabilizer measurements that is linear in the size of the lattice and also a sublinear number of entangling gates. For an orientable, connected, compact surface of genus g, it is well known that H1 (Γ, Fd ) = (Fd )2g . (See e.g. [15].) We next describe how one might transfer a qudit |ψi stored within an ancilliary copy of H(1, d) to the topologically ordered groundstate 2g eigenspace of H, say Hgr ∼ = Cd . Pd−1 The suggestion for encoding is as follows. We begin with |ψi = j=0 αj |ji. Choose a copy of Fd ⊆ H1 (Γ, Fd ), and let [ω] correspond to 1 ∈ Fd . Choose ω ∈ [ω], preferably with as few nonzero (excited) edges as possible. Now jω is also a cycle for 0 ≤ j ≤ d − 1, and by our choice {[jω] = j[ω]}d−1 j=0 contains distinct homology classes. Using whatever unitaries are convenient, we form ψ̃ E = d−1 X j=0 αj |jωi (12) For example, on a toric Γ one might have n-sites and choose a vertical or horizontal cycle unitary would cost √ on a square fundamental domain. Then thePappropriate d−1 O( n) gates. Our goal is to construct ψstorage = j=0 αj |[jω]i. For the remainder of the construction, note that all intermediate states are in the code space of the stabilizer Gv = h{gv }i ⊆ P(n, d). Hence, we may correct for errors in this code at any time. Also, the scheme below might be thought of as arising from an error correction to the stabilizer Gf = h{gf }i. Nonetheless, only ψstorage is in the code space of the full stabilizer G. Arbitrary local errors are correctible in the code space of G since the normalizer of G contains {Ze , Xe ; e ∈ E} [9, 11]. Since this is clearly false for Gv , one should perform the initialization above as quickly as possible. We suppose an ordering of the faces f ∈ F, say f1 , f2 , . . . , fL , such that for each fixed ℓ the boundary of fℓ contains some edge eℓ which (i) is not within the boundaries of f1 , f2 , . . . , fℓ−1 and (ii) does not intersect the support of ω. This is not possible for the last face fL , but E we only require this condition for 1 ≤ ℓ ≤ L − 1. To store the qudit beginning with ψ̃ , we apply the following steps for each fℓ . • Measure the eigenvalue of gfℓ , e.g. using an ancillary qudit. (See §2.2.) The eigenvalue λ will be an element of {ξ j }d−1 j=0 . • If λ = 1, then the state has collapsed onto the stabilizer h{gv } ⊔ {gfk ; 1 ≤ k ≤ ℓ}i (by induction.) Else, measuring ξ j accidentally performed the collapse Pd−1 Pj = (1/d) k=0 ξ jk gfkℓ , which is in fact a projection ‡ Let eℓ be the isolated edge as above. Since Zekℓ gfℓ = ξ k gfℓ Zℓk , we see that Zejℓ Pj = P0 Zejℓ . Thus an appropriate power of Zeℓ will fix the projection onto the unwanted eigenvalue so that the final state lies within the the +1 eigenspace of Hfℓ . Applying the process of the last paragraph clearly produces an element of Hgr . The applications of Hf , Pj , and also Ze all respect H[jω] for 0 ≤ j ≤ d − 1. Note ‡ Why is this a projection? Consider the unitary h = ξ j gfℓ and consider projection onto the stabilizer of h{h}i. 12 Qudit surface codes and gauge theory with finite cyclic groups that Hgr ∩ H[jω] = C |[jω]i. If S denotes the superoperator of the above sequence of measurements and unitary maps, then equivalently we have shown S(Hloop ) ⊆ Hgr ∩ H[jω] . Equality is immediate after noting S |[jω]i = |[jω]i. However, the effect of the superoperator on relative phases is still unclear. Given the global phase on |ωi, there is a natural global phase on |[ω]i = π |ωi. With the argument above, we have actually verified that S |0i = eiϕ0 |[0]i, S |ωi = eiϕ1 |[ω]i, Pd−1 S |2ωi = eiϕ2 |[2ω]i, etc. Thus perhaps ψstorage = j=0 eiϕj αj |[jω]i. We argue that all of these relative phases are in fact equal. For in terms of the observed eigenvalues, S = L−1 Y j=1 Ze±j Pj (fℓ ) = ℓ L−1 Y j=1 P0 (fℓ )Ze±j = π ℓ L−1 Y Ze±j ℓ (13) ℓ=1 QL−1 ±j By choice of the support of |ωi, also the support E of |jωi, we have ℓ=1PZeℓ |jωi = 1. d−1 Thus, the applying the superoperator S to ψ̃ produces ψstorage = j=0 αj |[jω]i, L−1 given that we may choose the {eℓ }ℓ=1 to be disjoint from the support of ω. 4.2. Retrieval Thus we next consider retrieval of a qudit stored as in the last subsection, i.e. swapping the data in a topological qudit with that encoded in some ancilla qudit. Physically, this is more intricate than encoding, which amounts to creating a cycle class |ωi and then applying stabilizer corrections for {gf }f ∈F generating Gf . For retrieval, the central point is that we may applyPa logical X operation to the √ encoded qudit using O( n) gates. To see this, for ω = e ne e let X ⊗ω = ⊗e∈E Xene . This might be thought of as a creation operator of an excitation of the loop ω, and moreover X ⊗ω is an element of the centralizer of G not contained within G. As such, it preserves the code space, and one readily verifies that it must map |[jω]i 7→ |[(j + 1)ω]i, up to global phase. Hence, √we may apply controlledX operations targetting the topological qudit using O( n) physical controlled-X operations. We next consider a controlled-X operation controlled on the topological qudit and targetting an ancilla. One must choose a cycle in the dual complex to Γ according to ω, say η. For example, an earlier work [7, Fig.3] depicts a picket fence dual to a loop generator of the first homology group of a torus. In order to perform the required controlled-X, follow these steps. • Prepare a second ancilla. Then prepare this second ancilla so that the Z eigenstate of the ancilla measures Z ⊗η . • Perform the controlled-X contingent on this second ancilla. • Disentangle, i.e. reverse the qudit gates of the first step. Consquently, we can perform either controlled-X to or from the topologically encoded qudit. The ability to perform a two-qudit controlled-X gate implies the ability to perform controlled modular addition. The composition begins with a single controlled increment triggering when the control carries |1i, continues with two controlled increments when the control carries |2i, etc. The entire circuit thus realizes a controlled modular addition in a number of controlled-X gates roughly the triangular number of d. Controlled modular subtraction is similar. 13 Qudit surface codes and gauge theory with finite cyclic groups Finally, modular addition and subtraction allow us to SWAP the topological qudit to an ancilla. For bits, the standard three CNOT swap relies on the fact that CNOT exclusive-or’s one bit to another. Thus the CNOTs perform b1 b2 7→ b1 (b1 ⊕ b2 ) 7→ b2 (b1 ⊕ b2 ) 7→ b2 b1 . In like manner, we may perform suitably controlled and targetted additions and subtractions for the following sequence of dit operations: d1 d2 7→ d1 (d1 + d2 ) 7→ (−d2 )(d1 + d2 ) 7→ (−d2 )d1 (14) Hence, modifying gates so that a control symbol with a + or − target means to add or subtract the control respectively, we have the following diagram: × × ∼ = • − • + • + −1 We have not described how to complete the gate |ji 7→ |d − ji on the topologically ordered state. Rather than do so, we claim the top line as the ancilla. This is also improves the cost of the controlled additions. 4.3. Modified constructions using ancillary qudits In the quantum circuit model of computation ancillary particles are often used as a means to assist in gate operations and as an entropy dump during error correction cycles. In the context of surface codes it is tempting to borrow this idea and place qudits at the center of each face and on each vertex of the cellulation Γ, so that the appropriate stabilizer checks might be done in place (see Figure 1b). Recall, any state may be projected into the groundstate of the topologically ordered Hamiltonian H = H∂ +HKE using stabilizer checks to the Pauli tensors {gv }⊔{gf } ⊂ P(n, d) (§3.2.) Each individual stabilizer check may then be performed using a certain sequence of two-qudit gates and a neighboring ancilla (§2.2.) In fact, this basic observation presents an auxilliary Hamiltonian which also computes the same topological order as the original. Namely, on the face-edge-vertex qudit system, one may build a Hamiltonian which is in the groundstate iff all the stabilizer checks gv and gf are satisfied. For gv , suppose we use Σve for the sum gate targetting the qudit of vertex v and take nv to be the qudit number operator on v. Then Y Y Y Y Σve (15) (Σve )−1 (Σve )−1 nv Σve H̃v = [∗,v]=e [v,∗]=e [∗,v]=e [v,∗]=e Then |ψi is in the groundstate of H̃v iff gv |ψi = |ψi. Similarly, fix a face f ∈ F with Q P Q ∂f = ℓj=1 nj ej for nj ∈ {1, d−1}. We take Ff = ℓj=1 (Fd )ej and Uf = ℓj=1 (Σfej )nj for Σfe the sum gate targetting the f qudit. Then for nf the number operator of the qudit at the center of the face f , we label H̃f = Ff Uf Ff † nf Ff Uf† Ff † (16) As before, we see that |ψi is in the groundstate of H̃f iff gf |ψi = |ψi. Thus for h > 0 P P and U > 0, if H̃ = U v H̃v + h f H̃f , then the groundstate of H̃ is also the code space of G = h{gv } ⊔ {gf }i, i.e. the topologically ordered groundstate spanned by {|[ω]i ; [ω] ∈ H1 (Γ, Fd )}. We finish this section describing another utility for ancillary particles, namely to mediate many body interactions present in the Hamiltonian H (Eq. 6) using Qudit surface codes and gauge theory with finite cyclic groups 14 more physically motivated binary interactions. Consider the vertex constraint term Hv = −(gv + gv† ) where the valence at that vertex is k. This k-local interaction can be obtained as a perturbative limit of 2-local interactions between each d-level qudit incident at v and a k-level ancillary qudit a located at the vertex. Begin with a local Hamiltonian for the ancilla Ha = −Ea |0ia h0|, and a perturbing interaction Pk Va = Jv r=1 (Zeorr ⊗ |r − 1i hr| + h.c.), where Ea ≫ |Jv | and the edge orientations give oj = 1 if ej = [∗, v] and oj = d − 1 if ej = [v, ∗]. By construction, the lowest nontrivial, i.e. non identity, contribution to coupling in the ground subspace Hgr = |0ia h0| (H0 + V ) |0ia h0| is the effective Hamiltonian Hveff = U (Hv + O(ǫ)) where U = (−1)k Ea (Jv /Ea )k with an error term of norm ||ǫ|| ≪ 1. By judicious choice of sign(Jv ) it is possible to fix U > 0. A similar argument applies to building the face constraint Hf using a j-level ancilla b located at face f to mediate interactions between all j edges on the boundary of f . Here we choose Hb = −Eb |0ib h0| and Pj Vb = Jf r=1 (Xeorr ⊗ |r − 1i hr| + h.c.) such that Hf eff = h(Hf + O(ǫ)), where h = (−1)j Eb (Jf /Eb )j . These mediator qudits could be placed on all the vertexes and faces of Γ to build an effective Hamiltonian in the subspace spanned by states with all ancillae in the |0i state. An argument in Ref. [19] suggests that an effective Hamiltonian between spins on a two complex can be built using such mediating interactions that closely approximates a target Hamiltonian projected to its ground subspace Hgr = Pgr HPgr . In the present context this would imply that for sufficiently large energies Ea , Eb both the degeneracy of the ground subspace of H = H∂ + HKE as well as the energy gap to the excited states could be accurately approximated by a model built from a sum of effective vertex and face operators. An analysis regarding the validity of such constructions for topologically ordered states is wanting, but is outside the scope of this work. 5. Other Homological Groundstates We have originally presented the case of groundstates for H1 (Γ, Fd ) for d prime, in order to present the new orientation conventions in the simplest possible context. This section describes a construction for homological order on dits whose number of levels is not prime but rather a prime power. Homological order for arbitrary composite d follows immediately through a tensor product of the prime-power Hamiltonians. 5.1. Homology Fdℓ stabilizer codes The hypothesis in the main text has been that qudits have d levels, for d a prime so that each |ji is associated to an element of Fd . Recent work [11] extends stabilizer techniques to the finite fields of order dℓ , i.e. Fdℓ , which exist for any ℓ ≥ 1. The generic Fdℓ constitute all fields F with #F < ∞, so this is (perhaps) the most general field for which a stabilizer code makes sense. The most typical construction of Fdℓ is to consider the polynomial ring Fd [x] and divide out relations in the ideal generated by some irreducible polynomial f (x) = xℓ + aℓ−1 xℓ−1 + . . . + a0 , aj ∈ Fd . It is typical to label α ∈ Fdℓ as the adjoined root corresponding to the class of x. The Galois group of the extension Fdℓ over Fd , say K, then acts as permutations of the roots of f (x). Note that Fdℓ is a vector space over the scalars Fd . Moreover, multiplication by any fixed a ∈ Fdℓ may be viewed as a Fd linear map, with an associated matrix with entries in Fd . Computing the trace of this matrix creates a map TraceFdℓ /Fd : Fdℓ → Fd . Another characterization is that Qudit surface codes and gauge theory with finite cyclic groups 15 P TraceFdℓ /Fd (x) = κ∈K (κ · x). To ground the discussion, let us review not extensions over finite fields but rather TraceC/R (z) = z + z = 2Re(z). The complex conjugate is the Galois action that interchanges i ↔ −i, for C = R[x]/(x2 + 1). We might instead form a 2 × 2 matrix for multiplication by z = x + iy, which results in µz = x |0i h0| − y |1i h0| − y |0i h1| + x |1i h1| with trace 2x. κ For Fdℓ extending Fd the Galois group K is cyclic of order ℓ, generated by x7→xd ℓ for x ∈ Fdℓ . Now κ generates a one-qud it unitary Uκ by Uκ |xi = |κ · xi, and the corresponding diagonal unitary on the entire lattice will be denoted Ũκ . 5.1.1. Fourier transforms for Fdℓ Having reviewed the machinery of finite fields, we next review what one would mean by a stabilizer code of Pauli matrices indexed by it [11]. Since our earlier qudit operators X and Z for Fd had order d, we might instead claim to have constructed an X operator and a Z operator for each a ∈ Fd , i.e. X a and Z b . For Fdℓ , we do not take operator powers. Label H(1, dℓ ) = ⊕a∈Fdℓ C{|ai}. Then suitable definitions are as follows, where we define ξ = exp(2πi/d). ( X(a) |bi = |a + bi (17) TraceF ℓ /Fd (ab) d Z(a) |bi = ξ |bi For ℓ = 1, this generalizes the powers of earlier Pauli operators. Furthermore, with these conventions we have a commutator relation: X(a)Z(b) = ξ ℓ TraceF dℓ ℓ ⊗n /Fd (ab) Z(b)X(a) (18) n Finally, let H(n, d ) = H(1, d ) . In a slight abuse of notation, for a, b ∈ (Fdℓ ) we will write a • b = TraceFdℓ /Fd (a0 b0 + a1 b1 + · · · + an−1 bn−1 ). Then we may generalize the earlier commutatator formula for Pauli tensors as [Z(b0 ) ⊗ Z(b1 ) ⊗ · · · ⊗ Z(bn−1 )] [X(a0 ) ⊗ X(a1 ) ⊗ · · · ⊗ X(an−1 )] = (19) ξ a•b [X(a0 ) ⊗ X(a1 ) ⊗ · · · ⊗ X(an−1 )] [Z(b0 ) ⊗ Z(b1 ) ⊗ · · · ⊗ Z(bn−1 )] Given this relation, one may define P(n, dℓ ) to be that group generated by products of Pauli tensors indexed by Fdℓ , as above. Continuing, we may consider sets of particular Pauli tensors X(a0 )Z(b0 )⊗· · ·⊗X(an−1 )Z(bn−1 ) and consider the stabilizer subspaces of the subgroup G ⊂ P(n, dℓ ) they generate. The error lengths of such code are studied in detail [11]. Before considering which of these stabilizer codes arise as topological orders, we add a point omitted in the original treatments. Namely, we wish to propose quantum circuits for the appropriate stabilizer checks. We suppose the existence of a number operator measurement which can output classical values in the finite P field, say abusively n = a∈F ℓ a |ai ha|. Then as with Fd , stabilizer checks would d follow given an appropriate Fourier transform Fdℓ : H(n, dℓ ) → H(n, dℓ ) which maps X(a) P eigenstates to |ai. This leads one to guess we should define Fdℓ |ai = (dℓ )−1/2 b∈F ℓ Z(a) |bi, i.e. d X Trace def F ℓ /Fd (ab) d ξ |bi ha| (20) Fdℓ = (dℓ )−1/2 a,b∈Fdℓ However, note that X(a) now has degenerate eigenspaces when ℓ ≥ 2. Thus, it is not clear the the above equation actually defines a unitary matrix. Qudit surface codes and gauge theory with finite cyclic groups 16 We briefly comment on why unitarity holds. For convenience, let us drop the subscript from the appropriate trace maps. A computation reveals that the unitarity assertion is equivalent to knowing that for any fixed a ∈ Fdℓ which is nonzero, X ? (21) ξ Trace(ab) = 0 b∈Fdℓ Since a 6= 0 has a multiplicative inverse, this amounts to X ? ξ Trace(b) = 0 (22) b∈Fdℓ Now suppose we use α to denote the formally adjoined root of f (x) in Fdℓ = Fd [x]/(f (x)). Then since every equivalence class may be written as a polynomial of degree less than ℓ, we see that {αj }ℓ−1 j=0 is a basis of Fdℓ over Fd . In terms of the last basis, we might express a generic polynomial class in coordinates as b = bℓ−1 xℓ−1 + bℓ−2 xℓ−2 + · · · + b0 for bj ∈ Fd . Then Equation 22 becomes Pd−1 Pd−1 Pd−1 bℓ−1 =0 bℓ−2 =0 · · · b0 =0 ℓ−1 ℓ−2 (23) [ξ Trace(α ) ]bℓ−1 [ξ Trace(α ) ]bℓ−1 · · · [ξ Trace(1) ]b0 ? =0 This will in fact be zero, unless all Trace(αj ) = 0 mod p, 0 ≤ j ≤ ℓ − 1. A standard construction in field extensions is to form the discriminant of a basis, for our basis P j+k ∆ = ℓ−1 Trace(α ) |ki hj|. For a given basis, it is not possible that this matrix j,k=0 ∆ have determinant zero in Fd [17, Thm2.37,pg.61]. Since the first column of ∆ can not then be zero, all Trace(αj ) may not be zero, and unitarity of Fdℓ follows. 5.1.2. Homological order for Fdℓ The chain complex for computing H1 (Γ, Fdℓ ) extends our early discussion by allowing for coefficients of the vertices, edges, and face to be within Fdℓ , which in context is ℓ copies of Fd since only the additive structure is relevant. Yet the previous section has nontrivally extended our definition of X and Z operators to account for field multiplication, and these operators may be used to form a homological order on the physical system in which qudℓ its (with dℓ levels) are associated to the edges of Γ: Q Q • For each vertex, we may again set gv = e=[v,∗] Ze (−1) and e=[∗,v] Ze (1) P Hv = −(gv + gv† ). Then again H∂ = U v∈V Hv . Pp • Again set gf = Xe1 (o1 )Xe2 (o2 )Xe3 (o3 ) . . . Xep (op ), where ∂f = ℓ=1 oℓ eℓ . Put Hf = −(gf + gf† ). Given the generalization of the commutators of P the new X and Z operators, [Hf , Hv ] = 0 for any f ,v. Then for h > 0, HKE = h f ∈F Hf . • So H = H∂ + HKE . A similar argument to that given before produces a basis |[ω]i of the groundspace of H, as [ω] runs over all elements of H1 (Γ, Fdℓ ). • These groundstates may again be viewed as a stabilizer code of G = h{gv , gf }i ( P(n, dℓ ). Stabilizer checks can be performed as before (see §2.2). The only required modifications are that the quantum circuit uses the new Fourier transform over Fdℓ to measure X(a) operators and powers thereof and the number operator measurement now takes values in Fdℓ . We close with one further comment. Recall Ũκ which act on each qudℓ it as Uκ |ai = |κ · ai for κ the generator of the cyclic Galois group of Fdℓ extending Fd . Now Ũκ H = H Ũκ , as one can verify directly using Hf and Hg . Thus we may view Qudit surface codes and gauge theory with finite cyclic groups 17 Ũκ or more generally the Galois action as a symmetry of the topologically ordered Pℓ−1 groundstate. Also, πκ = ℓ−1 j=0 Uκj will then act as a projection collapsing the groundstate associated to elements of H1 (Γ, Fdℓ ) onto the groundstate parametrized by H1 (Γ, Fd ) as constructed in §3. In terms of Hamiltonians, πκ projects onto the groundstate of Hκ = −(Uκ + Uκ† ), whose physical significance is unclear. 6. Z/dZ Gauge Theory and Anyonic Excitations In our treatment of code subspaces, we have used the isomorphism between spins on a surface and one-chains on a two complex to label the ground states of the Hamiltonian H in terms of homology equivalence classes. The language of cell complexes also carries over to describe the excited states. If we identify the ground subspace of H as the vacuum then excited states are labeled by Fd valued boundaries of one chains on the complex Γ or the dual complex Γ̃. These excitations can be viewed as massive particles with definite statistics. In this section we show by construction that our model is a Z/dZ gauge theory with quasi-particles corresponding to dyonic combinations of charge and flux. These quasi-particles have abelian anyonic statistics. We provide an algorithm in terms of an interferometer circuit for measuring components of the scattering matrix. 6.1. Stabilizer errors as abelian anyons Consider a two-complex Γ with a physical system of qudits associated to each edge and a topologically ordered Hamiltonian H as above. We have already seen how to associate a basis of the groundstate eigenspace with elements of H1 (Γ, Fd ). As stabilizer states, it is well known that the groundstates are entangled. Abelian anyons arise as entangled excitations of this system. In the qubit case, such excitations always arise in pairs [12]. In our generalization, this is also true, and an excitation |ji is always paired to an excitation |d − ji. The linear algebra for constructing a charge anyon is as follows. First, choose two vertices v1 and v2 of Γ on which the anyon should reside with charges j and d − j respectively. Choose P a chain ω with ∂ω = jv1 + (d − j)v2 . Recall from §3.2 the projection π = (#F )−1 f ∈F gf which projects onto the stabilizer code of all the face operators gf = ⊗e∈∂f (Xe )± . We set ψcharge anyon = π |ωi (24) 2iπj/d The resulting state is an excited state of H∂ whose eigenenergy is 4U (1 − Re(e )) above ground. It is not independent of the choice of ω, and this in fact allows for an interesting geometric interpretation of the error length of the associated stabilizer code [7]. For let ω1 and ω2 be two such choices, with |ψ1 i and |ψ2 i the resulting anyon states. Then ω1 − ω2 is a cycle, and def |φi = π(|ω1 i − |ω2 i) = |ψ1 i − |ψ2 i (25) is the ground state eigenket associated to [ω1 −ω2 ] ∈ H1 (Γ, Fd ). Hence, if we encounter such a charge anyon excitation which has sullied a qudit encoded in the groundstate of H, then correcting it amounts to choosing an cancelling anyon or equivalently to choosing a cycle on Γ. If the dual charges of the anyon are separated by roughly half the diameter of the two-complex, then this choice is likely to cause an error. Yet Qudit surface codes and gauge theory with finite cyclic groups 18 for nearby dual charges one might reasonably guess [ω1 − ω2 ] = [0]. In particular, if Γ were to cellulate √ the square fundamental domain of a torus using n qudits on the edges (implying Θ( n) qudits on a side,) then√we would expect an error length for the associated stabilizer code to be roughly O( n) [7]. Similar comments apply not only to charge anyons but also flux anyons [12]. Here, one chooses a path in the dual complex to Γ, i.e. a sequence of connected faces. Let |[0]i be the homological groundstate associated to [0] ∈ H1 (Γ, Fd ). A flux charge of multiplicity j on the endpoints of the face path is associated to · · · gf±j |[0]i (26) gf±j ψflux anyon = πv gf±j 2 1 ℓ P where πv = (#V)−1 v∈V gv and the path consists of faces f1 , f2 , . . . , fℓ with the signs allowing for orientation. The flux anyon theory follows quickly by considering the charge anyons of the dual two-complex to Γ, say Γ̃. Faces of Γ become vertices of Γ̃ while vertices become faces, and the graph of Γ̃ arises by connecting vertices corresponding to incident faces of Γ. Suitable hypotheses on the cellulation of the underlying two-manifold of Γ will cause this dualization procedure to be well behaved [15], and one might associate charge-anyonic observation of flux anyons and vice versa with pairings exploited in the proof of Poincaré duality. 6.2. Quasi-particle statistics We next wish to study such anyon states, i.e. errors of the stabilizer code as above. New notation for the excitations follows. A charge a ∈ Z/dZ at vertex v is labeled by the state |(a, 0; (v, −))i such that h(a, 0; (v, −))| gv |(a, 0; (v, −))i = ξ a . Similarly, flux b ∈ Z/dZ at face f is labeled by the state |(0, b; (−, f ))i such that h(0, b; (−, f ))| gf† |(0, b; (−, f ))i = ξ b . A dyon refers to a bound state of charge and flux at vertex v and face f neighboring each other, i.e. [v, ∗] ∈ ∂f or [∗, v] ∈ ∂f and (a, b) ∈ (Z/dZ)2 . The state of such a dyon in Hilbert space will be denoted |(a, b; (v, f ))i. For simplicity we restrict our discussion to simply connected compact surfaces with boundary such that the ground (vacuum) state is nondegenerate § Pauli-group elements local to a single edge of Γ produce dyons of the topological order in particle anti-particle pairs. To see this, note that the operator Xea acting at edge e = [v1 , v2 ] creates a pair of boundaries on the vertices, one with charge a at v1 and other with charge d − a at v2 . We name the charge d − a particle an anti-charge to a. Similarly, the operator Zeb creates quasi-particles located on the two faces f1 and f2 that share the edge e on their boundaries. Let face f1 be the face with opposite orientation to e. Then the flux at f1 is b and the anti-flux at f2 has the value d − b. A product operator X a Z −b acting on edge e creates the dyon (a, b) with charge a at vertex v1 and flux b at face f1 (see Fig. 3a). When it might be clear from context, we will drop the particle location labels (v, f ), e.g. particle anti-particle pairs might be written as |(a, b); (−a, −b)i. The mass of a dyon is given by the expectation value: ma,b = h(a, b)| H |(a, b)i − E0 = 2U (1 − Re[ξ a ]) + 2h(1 − Re[ξ b ]), where E0 is the vacuum energy. The energy to create a particle antiparticle pair is twice this value. Prior work in continuum field theory has considered dyon excitations in which charges and fluxes take values in Z/dZ. The interactions described by a Z/dZ gauge § In general |(a, b); (v, f )i describes an equivalence class of pure states which results from applying Xea Ze−b to any groundstate. For a degenerate vacuum, particle creation, followed by braiding and annihilation can result in non trivial logical operations on the code subspace. Qudit surface codes and gauge theory with finite cyclic groups 19 theory are completely characterized by the following rules [20]. |(a, b; (v, f ))i × |(a′ , b′ ; (v, f ))i = |(a + a′ , b + b′ ; (v, f ))i (27) R |(a, b; (v, f ))i |(a, b; (v ′ , f ′ ))i = ξ ab |(a, b; (v, f ))i |(a, b; (v ′ , f ′ ))i (28) R2 |(a, b; (v, f ))i |(a′ , b′ ; (v ′ , f ′ ))i = ′ ′ ξ (a b+b a) |(a, b; (v, f ))i |(a′ , b′ ; (v ′ , f ′ ))i C |(a, b; (v, f ))i = |(−a, −b; (v, f ))i T |(a, b; (v, f ))i = ξ ab |(a, b; (v, f ))i . (29) (30) (31) We next review these rules and argue that the dyonic excitations of our Hamiltonian satisfy them. The first relation is the fusion rule for particles occupying the same location where addition is performed modulo d. In the context of our model this rule follows from the additivity of boundaries of one chains. Indeed, it is the ability to annihilate particle anti-particle pairs by choosing a trivial cycle on Γ or Γ̃ that makes correction of local errors possible (see Fig. 3b). The next two rules describe the action of the monodromy operator R which performs a counterclockwise exchange of one particle with another. The quantum state of n indistinguishable particles residing on a surface belongs to a Hilbert space that transforms as a unitary representation of the braid group Bn . If we order the positions of the particles {(vj , fj )}nj=1 , then the n − 1 generators of Bn correspond to the monodromy operator R acting on the particle pairs in the n−1 locations {(vj , fj ), (vj+1 , fj+1 )}j=1 . For a Z/dZ gauge theory, the irreducible unitary representation of Bn is one dimensional, meaning the particles are abelian anyons. Notice that the definition of the monodromy operator involves orientation of the path taken during particle exchange. For a non orientable surface, Z/dZ statistics for d > 2 are not allowed because the clockwise trajectory of particle around another is not uniquely defined whereas the phases ξ, ξ −1 are distinguishable except for d = 2. 4. Here The braiding of one dyon around another is shown in Fig. we begin with a state of two dyonic particle anti-particle pairs: |Ψi = |(a, b); (−a, −b)i |(a′ , b′ ); (−a′ , −b′ )i in distinct locations on the surface. The mutual statistics are determined by winding one dyon, (a, b) around the other (a′ , b′ ) in a counterclockwise sense. This action is described by the square of monodromy operator R, which exchanges two particles in a counterclockwise sense. A non trivial phase is accumulated under the action of R2 because the closed loop string operators that wind (a, b) collide with the strings connected the dyon (a′ , b′ ) with its antiparticle. In the example shown in Fig. 4a, the strings intersect at two locations ′ ′ ′ ′ ′ ′ where we have the operators Z −b X −a = ξ a b X −a Z b and X −a Z b = ξ b a Z −b X −a . Rewriting these operators with the action of the closed strings (Pauli operators with unprimed powers) first has the advantage that the closed strings act trivially provided that there are no other quasi-particles inside the closed loops. Hence we ′ ′ have that R2 |Ψi = ξ (a b+b a) |Ψi. The preceding example illustrated the Aharanov phase accumulated when winding one charge around a flux along a trajectory that was local, i.e. did not explore the global properties of the surface. Were the flux absent, then the path would be homotopic to a point. One can also define this phase for trajectories that explore the global properties of the surface, but can be continously deformed to a process where one anyon wraps around another. On a torus, for example, the following process traces out non trivial cycles for the charges and fluxes. Represent the torus as a square with opposite sides identified, labelling the Qudit surface codes and gauge theory with finite cyclic groups 20 axes of the square x1 , x2 . Pick a non trivial cycle along the x1 direction of Γ, and call it P1 . Similarly, pick a non-trivial cycle along the x2 direction of the dual Γ̃ and call it P2 . To obtain the exchange statistics, first create the dyonic particle antiparticle pair |Ψi = |(a, b; (v1 , f1 )); (−a, −b; (v2 , f2 ))i out of the vacuum state |Ψg i. Wind the charge a around P1 so that it annihilates with its anticharge partner at site v2 . Next wind the flux b around P2 so that it annihilates with its antiflux partner at face f2 . Create another dyonic particle antiparticle pair |Ψ′ i = |(a, b; (v2 , f2 )); (−a, −b; (v1 , f1 ))i with particle antiparticle positions reversed relative to |Ψi. Wind charge a around P1 in the opposite direction to the first winding so that it annihilates with the anticharge at site v1 and likewise, flux b along P2 in the opposite direction so that it annihilates with the antiflux at face f2 . These four trajectories cross at one edge e and the action on the state (for one choice of edge orientation) is |Ψg i → Ze−b Xe−a Zeb Xea |Ψg i = ξ ab |Ψg i. If we embed the torus in R3 , then the worldlines described by intersecting strings in the above process are equivalent under ambient isotopy to linked world lines on the plane which describe winding the charge a around the flux b. Identical quasi-particle statistics are determined by exchanging one dyon (a, b) counterclockwise with another. Such a process is depicted in Fig. 4b. The action on the state |Ψi = |(a, b)i |(a, b)i can be computed by annihilating particle-antiparticle pairs after exchange, creating them again, and comparing the resultant state with the initial state |Ψi. We can annihilate the charges on the left side first. Reversing the order of the operator that created the dyon there, we have X a Z −b = ξ ab Z b X a and the charges are annihilated by applying Z b . Similarly, the charges on the right side are annihilated by applying a string of Z −b operators. Finally, the fluxes are annihilated by applying X a or X −a along the remaining two connected strings. The action on the wavefunction is then R |(a, b)i |(a, b)i = ξ ab |(a, b)i |(a, b)i. The particle conjugation operator C in Eq. 30 reverses the sign of all the particles. This is realized in our microscropic spin model by reversing the orientation of the all the edges on the cellulation. Finally, the operation T in rule 31 rotates the charge component of a dyon around its own flux, generating an Aharanov-Bohm phase in the process. This is illustrated in Fig. 4b. Here the charge component of the dyon (r, s) is wrapped around its flux component in a counterclockwise sense. During this operation, there is a collision at the edge where the dyon was created. Rewriting the operation on the edge as X 2r Z −s = ξ rs X r Z −s X r so that loop operation about boundary of the face f acts trivially first, we have that T |r, s(v, f )i = ξ rs |r, s(v, f )i. 6.3. Measuring statistical phases In any physical construction of a Hamiltonian that admits topologically ordered states it will be important to verify the predicted properties. One, albeit crude, observable is to measure the energy gap from a ground state to a first excited state. This could be done by probing linear response of the ground states to a perturbing field that generates local unitary operation at a frequency ωF . For a system with the internal Hamiltonian Eq. 6, the expected resonant absorption occurs at frequencies ωF = 2ma,b /~. However, as a witness to topological order, this measure is not sufficient because there could be another spin Hamiltonian with equal gap that does not possess topologically invariant correlations functions. Another more convincing probe would be to directly compute the statistical phases in Eq. 29. Operationally, this should be done by measuring both the phase φτ accumulated when one particle (a, b) wraps around another (r, s) and the phase φ1 when the particle (a, b) traces out Qudit surface codes and gauge theory with finite cyclic groups 21 Figure 3. Quasi-particle excitations on a honeycomb cellulation. (a) Excitations appear in particle anti-particle pairs. Charges(anti-charges) appear as boundaries on vertices represented by open(filled) diamonds, and fluxes(antifluxes) as boundaries on the faces represented by open(filled) squares. The total charge and flux of any pair is zero. Shown is a flux pair |(0, c); (0, −c)i, charge pairs |(j, 0); (j, 0)i , |(k, 0); (−k, 0)i and a bound state of charge and flux pairs |(a, b); (−a, −b)i. Notice that strings of the same or different types are allowed to intersect. (b) Fusion of quasi-particles. The upper two diagrams illustrate corrective procedures to annihilate charge and flux excitations. The lower two diagrams illustrate the fusion rules |(j, 0)i × |(k, 0)i = |(j + k, 0)i and |(0, j)i × |(0, −k)i = |(0, j − k)i. Qudit surface codes and gauge theory with finite cyclic groups 22 Figure 4. Braid relations. (a) Counterclockwise braiding of the dyon (a, b) ′ ′ around the dyon (a′ , b′ ): R2 |(a, b)i |(a′ , b′ )i = ξ (a b+b a) |(a, b)i |(a′ , b′ )i. (b) Counterclockwise exchange of identical dyons: R |(a, b); (a, b)i = ξ ab |(a, b); (a, b)i. In the upper left hand side of the surface is shown the counterclockwise winding of the charge component of a dyon (r, s) about its flux component generating an Aharanov-Bohm phase according to T |(r, s)i = ξ rs |(r, s)i. Qudit surface codes and gauge theory with finite cyclic groups 23 Figure 5. Protocol for measuring quasi-particle statistics. The green circle represents an ancillary particle which performs conditional gate operations on the qudit residing on edge e = [v2 , v0 ]. The red lines indicate operations which are done adiabatically with respect to the energy gap ∆E. The inset is a simplified space-time diagram of the braid. the same path in configuration space but does not enclose the particle (r, s). The phase difference φτ − φ1 = φtop subtracts out dynamical phases and Berry’s phases, leaving only topological information. We sketch an algorithm for computing this phase using operations in accordance with the two complex illustrated in Fig. 5. Adaptation to other cellulations is straightforward. (i) Beginning from a ground state |Ψ(0)i, prepare a state with two particle antiparticle pairs in disjoint regions of the surface: |Ψ(1)i = |(a, b; (v3 , f3 )); (−a, −b; (v4 , f4 ))i |(r, s; (v0 , f0 )); (−r, −s; (v1 , f1 ))i . (ii) Prepare an ancillary qubit a in the state |+x ia = √12 (|0ia + |1ia ) and use this qubit to perform the controlled unitary operation ∧1 (Xe−r Zes ) = |0ia h0| ⊗ 1d + |1ia h1|⊗Xe−r Zes (with (r, s) 6= (0, 0)) on the qudit residing on the edge e = [v2 , v0 ]. Measure the ancilla in the x̂ basis and record the result m = ±1. The resultant state is |Ψ(2)i = √12 (|Ψ(1)i + (−1)m Xe−r Zes |Ψ(1)i), where Xe−r Zes |Ψ(1)i = is orthogonal to |Ψ(1)i. |(a, b; (v3 , f3 )); (−a, −b; (v4 , f4 ))i |(r, s; (v2 , f2 )); (−r, −s; (v1 , f1 ))i (iii) Use a sequence of local spin operations to drag the dyon at location (v2 , f2 ) to the location (v5 , f5 ). These operations should be done adiabatically, i.e. they should be done using localized control fields with frequency components much smaller than the minimum gap energy ∆E. In this way no new particles will be created, only the component of the wavefunction with the dyon located at (v2 , f2 ) will be Qudit surface codes and gauge theory with finite cyclic groups 24 changed. Instead of using control fields to perform local spin operations, another possibility is to slowly decrease the values of U and h on the vertices and faces in the path from (v2 , f2 ) to (v5 , f5 ) so that it is energetically favorable for the dyon to follow this path. The resultant state is: |Ψ(3)i = √12 (|Ψ(1)i + (−1)m |Ψ′ i), where |Ψ′ i = |(a, b; (v3 , f3 )); (−a, −b; (v4 , f4 ))i |(r, s; (v5 , f5 )); (−r, −s; (v1 , f1 ))i . (iv) Braid the dyon (a, b; (v3 , f3 )) in a counterclockwise sense around the location (v5 , f5 ) such that it returns to location (v3 , f3 ). The state is now: |Ψ(4)i = √1 (|Ψ(1)i + (−1)m ξ (sa+rb) |Ψ′ i). 2 (v) Perform the inverse of the operations in step iii, again insuring that no new quasi-particles are created during the process. The resulting state is: |Ψ(5)i = √1 (|Ψ(1)i + (−1)m ξ (sa+rb) eiχ X −r Z s |Ψ(1)i), where we have included χ, the sum e e 2 of dynamical and Berry’s phases that may have accumulated during steps ii-iv. (vi) Reprepare the ancilla in the state |+x ia and perform the controlled unitary operation ∧1 ((−1)m Ze−s Xer ). Measure the qubit in the x̂ basis. The expectation value is:  hσax iτ = 21 cos(χ + φtop )+  δ2r,0 δ2s,0 cos(χ + φtop − 2πrs/d) , where φtop = 2π(sa + rb)/d. (vii) Repeat steps i-vi but measure the ancilla in the ŷ basis. The expectation value is:  hσay iτ = 12 sin(χ + φtop )−  δ2r,0 δ2s,0 sin(χ + φtop − 2πrs/d) , (viii) Perform a similar experiment but this time using a trivial braiding operation, i.e. perform the steps in the order (i,ii,iv,iii,v,vi,vii) so that the braid is contractible. Then the expectation values are  1 hσax i1 = cos χ + δ2r,0 δ2s,0 cos(χ − 2πrs/d) , 2   1 sin χ − δ2r,0 δ2s,0 sin(χ − 2πrs/d) . hσay i1 = 2 (ix) Compute the topological phase φtop from an ensemble average obtained by repeated measurements on identically prepared systems. As a simple example, consider the computation of the mutual statistics of charge and a flux for d = 2. Setting (r, s) = (0, 1) and (a, b) = (1, 0), the expected measurement results are hσax iτ = 12 cos(χ + φtop ), hσay iτ = 0, hσax i1 = 12 cos χ, hσay i1 = 0. If desired, the phase χ could be engineered to vary in a controlled manner over different trials in order to improve the visibility of the phase shift φtop . For d > 2 it is always possible to choose the probe dyon such that δ2r,0 δ2s,0 = 0. In this case, φtop is estimated by finding the closest solution to eiφtop = (hσax iτ +ihσay iτ )/(hσax i1 +ihσay i1 ). Qudit surface codes and gauge theory with finite cyclic groups 25 7. Conclusions We have proven the existence of a microscopic spin model that provides for topologically protected qudit encodings. This model describes a Z/dZ gauge theory with abelian charge/flux dyons as excitations. The construction is quite general, allowing for arbitrary cellulations of an orientable surface and encoding qudits with any finite number of levels. Suggested adaptations to the standard spin models using ancilla for in place stabilizer checks could prove advantageous in any physical implementation of such codes. Moreover, with some limited degree of local control it is possible to measure the anyonic statistical phases. 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