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# 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    Gavin K. Brennen Institute for Quantum Optics and Quantum Information, Techniker Str. 21a, 6020, Innsbruck, Austria
June 30^th, 2006
###### 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 for prime and , and therefore under tensor decomposition, to arbitrary finite levels. The Hamiltonian describes an exact 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.

## 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 , there exists a Hamiltonian whose topologically ordered (stabilizer-code) groundstates are parametrized by the first homology group of the complex with bit-coefficients: . The Hamiltonian is a sum of vertex and edge terms which are proportional to either tensors of Pauli operators around qubits on edges adjacent to a vertex or are proportional to tensors of operators on edges bounding a face of the complex.

For some time the existence of stabilizer codes over qudits ( 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 () level qudits (actually quits) have also been found, so that tensors provide a stabilizer formalism for all finite-level systems. In this work, we exploit the new stabilizer formalism to construct codes on a two-complex whose edges carry prime--level qudits, and we also outline the extension to -level qudits. The associated ground states are parametrized by , 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 -coefficients. Thus, after tensoring we have constructed surface codes with qudits for arbitrary finite placed on the edges of a generic orientable two-complex . 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- 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 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 be a prime number, and consider the qudit state space , with a pure state of qubits being a ket within . A possible generalization of the Pauli operators on would be to consider the group generated by the following unitary matrices:

 (1)

These are not Hermitian unless . The qudit Pauli-tensor group, say , is the group of unitary matrices generated by -fold tensors of elements of .

We might be more explicit in the description of . First, for , label the multiplication in to be a dot-product. Then . More generally, for dit-strings , we use and to abbreviate and similarly for . For the -entry dot-product with values in , we have . Thus explicitly

 P(n,d) = {ξcX⊗aZ⊗b;a,b∈(Fd)n,c∈Fd} (2)

The qudit stabilizer groups are subgroups . The code subspace of such a stabilizer group is the joint eigenspace of all .

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].

Stabilizer code projectors: The sum of unitary maps is a projection onto the code-subspace. We present the argument. First, is the identity map since for any . Second, since adjoints are inverses in the unitary group. Hence, either is a projection or is an eigenvalue of . Yet is a summand, so the complex inner product precludes for a nonzero . Now split into irreducible orthogonal unitary subrepresentations of . For each , the image under and its orthogonal complement form a decomposition of . Thus by irreducibility, either preserves a or . Clearly the former holds for any irrep (i.e. irreducible representation) within the code subspace of . On the other hand, if for some , then the latter holds.

As a remark, irreps within the code subspace of must be one-dimensional and are also known as trivial representations. As a second remark, the code subspace is nonzero iff iff .

In the Hermitian case (,) it is standard that all eigenvalues of group elements are , so that a suitable Hamiltonian for which the code space is the groundstate is . For general , the eigenvalues lie within the unit circle, so that is still the least possible real part. Also, since . Thus, one may place the qudit code subspace into the groundstate of a Hamiltonian by adjusting each summand of with a Hermitian conjugate: , so that the eigenvalues of the summands are then .

### 2.2 Quantum circuits for qudit stabilizer measurements

Given an -qudit system, it is important for purposes of error-correction to be able to test whether or not a state lies within the stabilizer code of some . It suffices to test whether is a eigenvector of each generator . We sketch quantum circuits which achieve such a measurement.

Let be the qudit Fourier transform. Considering eigenkets, . Now the number operator suffices to infer the eigenvalue of and project into the appropriate eigenstate. As a circuit, we might denote a number operator measurement with the symbol, one of several common conventions in the qubit case:

Determination of the eigenstate may be accomplished by

Similarly, there is some one-qudit unitary which will diagonalize any , usually not a Fourier transform. Yet using the diagonalization and a number operator one may infer an eigenstate.

For and , we suggest using addition gates along with a qudit ancilla. We will denote by a typical control bullet with the target (in the formula second) line holding a gate. The the following construction of generalizes for :

For , and we have placed on the ancilla line before the number operator is applied. Note that 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, , so that the following diagram for extends:

Using similarity transforms by qudit Fourier transforms, we may similarly achieve etc. Yet more generally, the comment on existence of diagonalizations above produces circuits for arbitrary elements .

## 3 Homologically Ordered Groundstates for Prime Qudits

It is typical to place topological orders on explicit planar or spacial lattices of spin- 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 two-complex , and an appropriate Hamiltonian would have the dimension of its degenerate groundstate eigenspace equal to the number of classes within . We next extend this construction to prime-level qudits; the task is mainly to keep track of sign conventions which are vacuous in . We then check that the groundstate eigenspace is similarly spanned by kets associated to elements of , by applying stabilizer-code techniques.

### 3.1 Cellular Hamiltonians

Label to be the vertices of , to be the edges, and to be the faces. We also require properties that hold if 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, is finite and is a copy of spanned by , the sum of all faces with their orientation according to .

We briefly review the appropriate homology. Label the chain sets to be formal sums of vertices, edges, and faces respectively:     , , and . We generally drop the and coefficient system, which should be clear from context. Since is a cell complex, there exist boundary operators

 C0∂⟵ C1 ∂⟵ C2 (3)

with [15]. For example, if an edge connects and , say , then . Note that the definition of demands that edges are images of within , and hence all edges are implicitly oriented. The coefficients further allow for -valued multiplicities on each edge. Since , we have for . Thus we may define the vector space . This first homology group is well known to be a topological invariant, i.e. any topological space homotopic to that underlying will produce an 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 .

Recall that any Hamiltonian on -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 -local Hamiltonian is a Hamiltonian whose degree is bounded by in some decomposition. The topologically ordered Hamiltonians defined below are -local for the maximum of the valence of any vertex and the number of edges on any face.

Let , and consider placing a qudit on each . Again, each edge is the image of and is oriented (by ) from one vertex to the other. For the qudits associated with each edge, the excitation of the edge will be implicitly associated to this orientation, while the state corresponds to the other.

On the associated physical system , let and denote the operator applied to the qudit of that edge with identity operators buffered into the remainder of the tensor. For each , we define a Pauli-tensor and vertex Hamiltonian by

 gv=∏e=[∗,v]Ze∏e=[v,∗]Z−1eHv=−(gv+g†v) (4)

For some , we then define the potential energy term of a topologically ordered Hamiltonian by .

The notation has been chosen for the following reason. Suppose that is a chain, with each . There is an associated qudit computational basis state, say , which is local and places the qudit of each in state . We claim that is a groundstate of iff , i.e. is a cycle. To see this, one verifies that where . Hence is in the stabilizer iff is an eigenstate of each of minimial (real) eigenvalue iff is in the degenerate groundstate eigenspace of .

Strictly speaking, one should not refer to the groundstate of as being topologically ordered. Admittedly, groundstates are of the form for a cycle, colloquially a loop of excited edges. For , the edges must be properly oriented, and hitting every edge of a junction is allowed if multiplicities are accounted for. Yet the cycle subspace is not a topological invariant. Indeed, should be a cell complex, subdividing by breaking each -simplex (triangle) into several subtriangles will generally increase the size of , although such a subdivision does not change the topology of the underlying manifold. Thus, we next add a kinetic energy term to the potential, splitting the degeneracy of and reducing to a final groundstate capturing homology.

For each face , the face Hamiltonian is defined as follows. Orient according to the orientation of the manifold underlying . Label edges by for . Then we define

 gf=Xo1e1Xo2e2Xo3e3…XopepHf=−(gf+g†f) (5)

With these choices, for all faces and vertices . For the two edges incident on a given vertex will be in the boundary of some face, and after correcting for orientation conventions this commutativity check reduces to . (See Figure 1.) Hence, for some constant , we might define . Due to commutativity, the kinetic energy Hamiltonian respects the groundstate degeneracy of . Label . We next show that the dimension of the groundstate degeneracy of total Hamiltonian

 H=H∂+H\footnotesize KE (6)

(over ) corresponds to the number of elements of .

### 3.2 Homology class groundstates

The goal of this section is to associate the degeneracy (dimension) of this groundstate of to . We accomplish this in two distinct cases for the manifold underlying :

1. The manifold is orientable, compact, and has no boundary, so that .

2. The manifold is compact with boundary and has . For homology is a homotopy invariant, and such a surface retracts into its one skeleton.

Assertion: Let denote the groundstate of and .

 H\footnotesize loop = ⊕ω∈% \footnotesize ker ∂C|ω⟩ = ⊕[ω]∈H1(Γ,Fd)H[ω] (7)

Throughout this section, let . Suppose either Case 1 or Case 2. Then for each , the restriction of to is a rank one projector whose (nonzero) image is an element of .

To verify this, suppose is the computational basis state of some cycle (i.e. .) Then we may also speak of , is in the groundstate of . Label

 |[ω]⟩ \scriptsize def= πf|ω⟩ = (#G)−1∑g∈Gg|ω⟩. (8)

It suffices for the Assertion to show the following.

• If and each lie in , then and differ by a global phase.

• If , then .

This suffices to see the restriction of is a rank one projector, since the first item demands the rank and the second demands the rank .

We begin with the first item, writing . Since the underlying manifold of is orientable, suppose for convenience that all faces have positive orientation. Then for we put , implying . Note that . Thus .

We next demonstrate that has . As discussed in §2, it suffices to show that the trace of this projection, when restricted to the subspace which it preserves, is nonzero, and that immediately follows if demands . For all other elements of are traceless when restricted to , since . Case 1 and Case 2 differ somewhat. In each case, multiples of the identity in are products for . In Case 1, besides the empty product of the we also produce multiples of as the full product , . This corresponds to . Yet for these products , as may be verified at an individual edge. In Case 2, there is no nontrivial product of the which produces a multiple of the identity. This is due to the retraction demanding , 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 for the single face it bounds. Thus in Case 2 the only multiple of the identity is the trivial product of the , and tautologically. In each case, 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 of the code space of is , which is also the code space of . Since is a rank-one projector when restricted to each , we have the following.

 dimC( groundstate of H) = #H1(Γ,Fd) (9)

### 3.3 Groundstates on a punctured disk

In practice, constructing physical realizations of Hamiltonians corresponding to two-complexes without boundary is daunting. It is possible to simply identify opposite qudits on the square fundamental domain of , 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 of a disk with punctures. An example with is shown in Fig. 2. Label the -th puncture face which has the same orientation as . Also label the outer boundary of the disk and the boundaries of the -th puncture . Analogous to the previous construction, the Hamiltonian on is defined . Here the kinetic term is modified so that the set of face operators does not include operators on the punctured faces , i.e. , where . Consequently, there are edges on the boundaries that are acted on by operators from faces on one side only. Another way to see this is that all edges of the dual cellulation that cross the boundary share a common vertex located at in . Each edge in has two vertices in , hence the product over all vertex operators is:

 ∏v∈Vgv=Idn. (10)

Not every edge in borders two faces in , however, and the product over all face operators is:

 ∏f∈F′gf=C∂Γ(X)k∏j=1Cj(X), (11)

where and . The orientation if the edge is oriented in the same direction as the boundary on which the edge resides, and if the orientations are opposite.

First we argue that the code space in nonempty. Recall, the code states are defined as eigenstates of the stabilizer group . The operators and commute and the only additional relations obtained from the stabilizer group, embedded in Eqs. 10,11, guarantee that . We next show that the code space is by considering the action of operators that commute with any member of but act non trivially on . One such set of operators are non trivial valued cycles on generated by . We do not include the non trivial cycles generated by 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 is generated by a string of operations along a path that begins on an edge of and ends on an edge of without touching other edges on puncture boundaries. We denote the generator of a such a cycle where at the edge if and share the same orientation and otherwise. The other are chosen in a consistent way such that . The operators on cycles satisfy the commutation relations for as is easily verified by considering the action on the one intersecting edge . As such the set generates a representation of the Pauli group . For sufficiently spaced punctures, all paths exist and the group forms a representation of . We then find that the ground subspace of encodes qudits and the set performs local Pauli group operations on the -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.

### 4.1 Storing qudits

Placing quantum data into such a 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 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 , it is well known that . (See e.g. [15].) We next describe how one might transfer a qudit stored within an ancilliary copy of to the topologically ordered groundstate eigenspace of , say .

The suggestion for encoding is as follows. We begin with . Choose a copy of , and let correspond to . Choose , preferably with as few nonzero (excited) edges as possible. Now is also a cycle for , and by our choice contains distinct homology classes. Using whatever unitaries are convenient, we form

 ∣∣~ψ⟩ = d−1∑j=0αj|jω⟩ (12)

For example, on a toric one might have -sites and choose a vertical or horizontal cycle on a square fundamental domain. Then the appropriate unitary would cost gates. Our goal is to construct . For the remainder of the construction, note that all intermediate states are in the code space of the stabilizer . 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 . Nonetheless, only is in the code space of the full stabilizer . Arbitrary local errors are correctible in the code space of since the normalizer of contains [9, 11]. Since this is clearly false for , one should perform the initialization above as quickly as possible.

We suppose an ordering of the faces , say , , …, , such that for each fixed the boundary of contains some edge which (i) is not within the boundaries of , , …, and (ii) does not intersect the support of . This is not possible for the last face , but we only require this condition for . To store the qudit beginning with , we apply the following steps for each .

• Measure the eigenvalue of , e.g. using an ancillary qudit. (See §2.2.) The eigenvalue will be an element of .

• If , then the state has collapsed onto the stabilizer (by induction.) Else, measuring accidentally performed the collapse , which is in fact a projection 111Why is this a projection? Consider the unitary and consider projection onto the stabilizer of . Let be the isolated edge as above. Since , we see that . Thus an appropriate power of will fix the projection onto the unwanted eigenvalue so that the final state lies within the the eigenspace of .

Applying the process of the last paragraph clearly produces an element of . The applications of , , and also all respect for . Note that . If denotes the superoperator of the above sequence of measurements and unitary maps, then equivalently we have shown . Equality is immediate after noting .

However, the effect of the superoperator on relative phases is still unclear. Given the global phase on , there is a natural global phase on . With the argument above, we have actually verified that , , , etc. Thus perhaps . We argue that all of these relative phases are in fact equal. For in terms of the observed eigenvalues,

 S = L−1∏j=1Z±jeℓPj(fℓ) = L−1∏j=1P0(fℓ)Z±jeℓ = πL−1∏ℓ=1Z±jeℓ (13)

By choice of the support of , also the support of , we have . Thus, the applying the superoperator to produces , given that we may choose the to be disjoint from the support of .

### 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 and then applying stabilizer corrections for generating .

For retrieval, the central point is that we may apply a logical operation to the encoded qudit using gates. To see this, for let . This might be thought of as a creation operator of an excitation of the loop , and moreover is an element of the centralizer of not contained within . As such, it preserves the code space, and one readily verifies that it must map , up to global phase. Hence, we may apply controlled- operations targetting the topological qudit using physical controlled- operations.

We next consider a controlled- operation controlled on the topological qudit and targetting an ancilla. One must choose a cycle in the dual complex to according to , say . 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-, follow these steps.

• Prepare a second ancilla. Then prepare this second ancilla so that the eigenstate of the ancilla measures .

• Perform the controlled- contingent on this second ancilla.

• Disentangle, i.e. reverse the qudit gates of the first step.

Consquently, we can perform either controlled- to or from the topologically encoded qudit.

The ability to perform a two-qudit controlled- gate implies the ability to perform controlled modular addition. The composition begins with a single controlled increment triggering when the control carries , continues with two controlled increments when the control carries , etc. The entire circuit thus realizes a controlled modular addition in a number of controlled- gates roughly the triangular number of . Controlled modular subtraction is similar.

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 . In like manner, we may perform suitably controlled and targetted additions and subtractions for the following sequence of dit operations:

 d1d2 ↦ d1(d1+d2) ↦ (−d2)(d1+d2) ↦ (−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:

We have not described how to complete the gate 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 , 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 using stabilizer checks to the Pauli tensors 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 and are satisfied. For , suppose we use for the sum gate targetting the qudit of vertex and take to be the qudit number operator on . Then

 ~Hv = ∏[∗,v]=eΣve∏[v,∗]=e(Σve)−1nv∏[∗,v]=e(Σve)−1∏[v,∗]=eΣve (15)

Then is in the groundstate of iff . Similarly, fix a face with for . We take and for the sum gate targetting the qudit. Then for the number operator of the qudit at the center of the face , we label

 ~Hf = FfUfFf†nfFfU†fFf† (16)

As before, we see that is in the groundstate of iff . Thus for and , if , then the groundstate of is also the code space of , i.e. the topologically ordered groundstate spanned by .

We finish this section describing another utility for ancillary particles, namely to mediate many body interactions present in the Hamiltonian (Eq. 6) using more physically motivated binary interactions. Consider the vertex constraint term where the valence at that vertex is . This -local interaction can be obtained as a perturbative limit of -local interactions between each -level qudit incident at and a -level ancillary qudit located at the vertex. Begin with a local Hamiltonian for the ancilla , and a perturbing interaction , where and the edge orientations give if and if . By construction, the lowest nontrivial, i.e. non identity, contribution to coupling in the ground subspace is the effective Hamiltonian where with an error term of norm . By judicious choice of it is possible to fix . A similar argument applies to building the face constraint using a -level ancilla located at face to mediate interactions between all edges on the boundary of . Here we choose and such that , where . These mediator qudits could be placed on all the vertexes and faces of to build an effective Hamiltonian in the subspace spanned by states with all ancillae in the 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 . In the present context this would imply that for sufficiently large energies both the degeneracy of the ground subspace of 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 for 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 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 levels, for a prime so that each is associated to an element of . Recent work [11] extends stabilizer techniques to the finite fields of order , i.e. , which exist for any .

The generic constitute all fields with , so this is (perhaps) the most general field for which a stabilizer code makes sense. The most typical construction of is to consider the polynomial ring and divide out relations in the ideal generated by some irreducible polynomial , . It is typical to label as the adjoined root corresponding to the class of . The Galois group of the extension over , say , then acts as permutations of the roots of . Note that is a vector space over the scalars . Moreover, multiplication by any fixed may be viewed as a -linear map, with an associated matrix with entries in . Computing the trace of this matrix creates a map . Another characterization is that . To ground the discussion, let us review not extensions over finite fields but rather . The complex conjugate is the Galois action that interchanges , for . We might instead form a matrix for multiplication by , which results in with trace .

For extending the Galois group is cyclic of order , generated by for . Now generates a one-quit unitary by , and the corresponding diagonal unitary on the entire lattice will be denoted .

#### 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 and for had order , we might instead claim to have constructed an operator and a operator for each , i.e. and . For , we do not take operator powers. Label . Then suitable definitions are as follows, where we define .

 (17)

For , this generalizes the powers of earlier Pauli operators. Furthermore, with these conventions we have a commutator relation:

 X(a)Z(b) = ξ\footnotesize TraceFdℓ/Fd
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