Topological color code and symmetry-protected topological phases
We study -dimensional excitations in the -dimensional color code that are created by transversal application of the phase operators on connected subregions of qubits. We find that such excitations are superpositions of electric charges and can be characterized by fixed-point wavefunctions of -dimensional bosonic SPT phases with symmetry. While these SPT excitations are localized on -dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the non-triviality of emerging SPT wavefunctions. Moreover, these SPT-excitations can be physically realized as transparent gapped domain walls which exchange excitations in the color code. Namely, in the three-dimensional color code, the domain wall, associated with the transversal operator, exchanges a magnetic flux and a composite of a magnetic flux and loop-like SPT excitation, revealing rich possibilities of boundaries in higher-dimensional TQFTs. We also find that magnetic fluxes and loop-like SPT excitations exhibit non-trivial three-loop braiding statistics in three dimensions as a result of the fact that the phase operator belongs to the third-level of the Clifford hierarchy. We believe that the connection between SPT excitations, fault-tolerant logical gates and gapped domain walls, established in this paper, can be generalized to a large class of topological quantum codes and TQFTs.
Classification of fault-tolerantly implementable logical gates in topological quantum error-correcting codes is an important stepping stone toward far-reaching goal of universal quantum computation Gottesman (1998); Eastin and Knill (2009); Bravyi and König (2013); Pastawski and Yoshida (2015). Characterization of logical operators is also essential in understanding braiding and fusion rules of anyonic excitations arising in topologically ordered systems Kitaev (2003); Levin and Wen (2005). Although some classes of two-dimensional topological quantum codes are restricted to possess string-like logical operators only, there exist non-trivial topological quantum codes with two-dimensional logical operators. Namely, in the two-dimensional color code, transversal membrane-like phase operators lead to non-trivial action on the ground space which induce a non-trivial automorphism exchanging anyon labels Bombin and Martin-Delgado (2006); Bombin (). Also, in three or more dimensions, the color code admits transversal non-Clifford logical phase gates, which are indispensable ingredient of fault-tolerant quantum computation Bombin (); Kubica and Beverland ().
While characterization of excitations and classification of logical operators are intimately related, excitations arising from logical phase gates in the color code have not been studied. In this paper, we study excitations created by transversal phase gates in the color code. Somewhat surprisingly, we find that such excitations can be characterized by bosonic symmetry-protected topological (SPT) phases, which have been actively discussed in condensed matter physics community Chen et al. (2011a, b); Lu and Vishwanath (2012); Schuch et al. (2011); Levin and Gu (2012); Chen et al. (2013); Vishwanath and Senthil (2013); Wen (2013). Formally, the system with SPT order has certain on-site symmetry and its non-degenerate ground state does not break any of the symmetries. Studies of SPT phases have provided better understanding of various quantum phases of matter, including topological insulators, the topological gauge theory and gauge/gravitational anomalies. However, studies of SPT phases have yet to find interesting applications in quantum information science except for few instances Cai et al. (2010); Else et al. (2012a, b).
Our first result concerns an observation that -dimensional excitations, created by transversal phase operators in the -dimensional color code, can be characterized by -dimensional bosonic SPT phases with symmetry. Namely, by writing the emerging wavefunction as a superposition of excited eigenstates, we find that its expression is identical to a fixed-point wavefunction of a non-trivial SPT phase. The on-site symmetries emerge from parity conservation of electric charges in the color code. While these SPT excitations are localized on -dimensional boundaries, their creations require quantum operations acting on all qubits inside the boundaries, reflecting the non-triviality of SPT wavefunctions.
Although these SPT excitations are pseudo-excitations (not being eigenstates), they may emerge as gapped domain walls in the Heisenberg picture. Our second result concerns an observation that application of transversal logical gates to a part of the system creates a gapped domain wall. In the two-dimensional color code, there is a one-to-one correspondence between transparent domain walls and two-dimensional logical gates with non-trivial automorphism of anyons. In the three-dimensional color code, gapped domain walls, created by operators, transform a loop-like magnetic flux into a composite of a magnetic flux and a loop-like SPT excitation, revealing rich possibilities of boundaries in higher-dimensional TQFTs.
We also study braiding statistics of loop-like excitations in the three-dimensional color code. Our third result concerns an observation that loop-like magnetic fluxes and loop-like SPT excitations exhibit non-trivial three-loop braiding statistics. Namely, if a magnetic flux and a loop-like SPT excitation are braided while both of them are pierced through a magnetic flux, the resulting statistical phase is non-trivial. The non-trivial three-loop braiding statistics results from the fact that the three-dimensional color code admits a fault-tolerantly implementable logical gate from the third-level of the Clifford hierarchy. We also find that excitations, which may condense on the domain wall, exhibit trivial three-loop braiding statistics, implying that domain walls and boundaries in three-dimensional TQFTs may be classified by three-loop braiding statistics of magnetic fluxes and loop-like SPT excitations.
While the discussion in this paper is limited to a very specific model of topological quantum codes, we believe that our characterization is more generically applicable. Namely, we anticipate that in a large class of topologically ordered systems, pseudo-excitations resulting from fault-tolerantly implementable logical gates can be characterized by SPT wavefunctions. We further expect that these SPT excitations possess non-trivial multi-excitation braiding statistics and provide useful insight into classification of gapped boundaries. We thus view results in this paper as a stepping stone toward establishing the connection between characterizations of gapped domain walls, fault-tolerant logical gates and braiding statistics of SPT excitations.
This paper is organized as follows. In section II, we describe string and membrane operators in the two-dimensional color code. In section III, we show that a loop-like excitation in the two-dimensional color code can be characterized by an SPT wavefunction with symmetry. In section IV, we argue that SPT excitations can be physically realized as transparent domain walls in the Heisenberg picture. In section V. we describe string, membrane and volume operators in the three-dimensional color code. In section VI, we show that a membrane-like excitation in the three-dimensional color code can be characterized by an SPT wavefunction with symmetry. In section VII, we study the transparent domain wall in the three-dimensional color code. We also study the three-loop braiding statistics of magnetic fluxes and loop-like SPT excitations.
Ii Membrane-like operators in two-dimensional color code
We begin by considering the two-dimensional color code defined on a three-valent and three-colorable lattice where qubits live on vertices. Colors are denoted by . An example of such a lattice is a hexagonal lattice shown in Fig. 1 where plaquettes are colored in such that neighboring plaquettes do not have the same color. The Hamiltonian is given by
where represents a plaquette, and are tensor products of Pauli- operators acting on all qubits on a plaquette . Interaction terms commute with each other for all , and thus the system is a stabilizer Hamiltonian. Namely, a ground state satisfies stabilizer conditions for all .
Anyonic excitations in two-dimensional topologically ordered spin systems are characterized by string operators. To construct them in the color code, we assign color labels to edges of the lattice depending on color labels of two adjacent plaquettes. Consider a set of edges of color which form a one-dimensional line (Fig. 1). We define
If is an open line, they commute with all the interaction terms except stabilizers on plaquettes of color at the endpoints of . Thus, applications of and create magnetic fluxes and electric charges respectively. The correspondence between anyon labels and string operators may be represented as follows:
Similarly string operators can be constructed from open lines , consisting of edges of color , which lead to
There are important subtleties in the above characterization of anyonic excitations. First, anyonic excitations with three different color labels are not independent from each other since applications of Pauli operator on a single qubit create composites of anyons respectively. In other words, the following fusion channels exist:
Second, an electric charge exhibits the non-trivial braiding statistics with , but not with . This is because and always commute with each other for any choice of and . To fully capture the braiding statistics in the two-dimensional color code, it is convenient to construct an isomorphism between anyons of the color code and those of the toric code. Let and be anyons in two decoupled (i.e. non-interacting) copies of the toric code. Then the following correspondence is an isomorphism which preserves braiding and fusion rules:
In fact, it is known that, on a closed manifold, the two-dimensional color code is equivalent to two decoupled copies of the toric code under a local unitary transformation Yoshida (2011a); Bombín (2014); Kubica et al. (2015). In other words, they belong to the same topological phase Chen et al. (2010).
The two-dimensional color code possesses not only string-like operators, but also transversal membrane (two-dimensional) operators. Let be a Hadamard operator which exchanges Pauli and operators:
The Hamiltonian is symmetric under transversal conjugation by Hadamard operators
Since transforms -type string operators into -type string operators and vise versa, it exchanges electric charges and magnetic fluxes :
The color code admits another interesting transversal menbrane operator. Let us define phase operators, acting on a qubit, by . Of particular importance is the so-called phase operator
The operator exchanges Pauli and operators:
Let be a projector onto the ground state space of the color code Hamiltonian. Recall that the lattice is bipartite and qubits can be split into two complementary sets and . Let us define the following transversal (two-dimensional) phase operator
Then the ground state space is invariant under transversal application of operators: . This two-dimensional membrane operator implements the following exchanges of anyon labels (an automorphism):
In general, in two-dimensional topologically ordered spin systems described by TQFTs, transversal membrane-like (two-dimensional) operators may induce an automorphism of anyon labels which preserves braiding and fusion rules (i.e. monoidal centers of categories which define -dimensional TQFTs) Kitaev and Kong (2012); Beverland et al. (2014).
We conclude this section by recalling quantum information theoretical motivations to study transversal membrane-like operators in two-dimensional topologically ordered spin systems. In quantum information science, one hopes to perform quantum information processing tasks in a protected codeword space of some quantum error-correcting code. The gapped ground state space of topologically ordered systems is an idealistic platform for such purposes. But how do we perform quantum computations inside the protected subspace? Ideally one hopes to perform logical operations in a way which does not make local errors propagate to other spins. Namely, one hopes to perform logical operations by transversal unitary gates acting on each spin as a tensor product. Thus, it is important to classify transversally implementable logical gates in quantum error-correcting codes Eastin and Knill (2009); Bravyi and König (2013); Pastawski and Yoshida (2015); Beverland et al. (2014). Transversal membrane operators, such as and in the topological color code, are examples of fault-tolerantly implementable logical gates as they may have non-trivial action on the ground state space (if it is degenerate). Our goal is to characterize excitations arising from fault-tolerantly implementable logical operators, and the present paper is dedicated to studies of those in the topological color code.
Iii SPT excitations in two-dimensional topological color code
In this section, we study loop-like excitations created by parts of a membrane phase operator in the two-dimensional color code and show that they are characterized by a wavefunction of a one-dimensional bosonic SPT phase with symmetry. Our finding reveals that these loop-like excitations in the color code can be viewed as a path integral formulation of an SPT wavefunction, leading to a physically insightful proof that such a wavefunction cannot be prepared by symmetry-protected local unitary transformations. We note that the circuit depth of preparing SPT wavefunctions was previously studied by using different approaches Marvian (2013); Huang and Chen (2014).
In this section, for simplicity of discussion, we assume that the lattice is supported on a sphere so that the system has a unique ground state .
iii.1 Loop-like excitation from membrane operator
To begin, let be a phase operator . An application of on a qubit at a vertex creates an excited wavefunction:
Since is diagonal in the computational basis, it creates excitations which are associated with -type stabilizers on three neighboring plaquettes of three different colors. We would like to characterize this wavefunction in the excitation basis:
where and represents an eigenstate of the Hamiltonian with
In other worlds, records the presence or absence of electric charges at while there is no other excitations in the system. One finds
So the phase operator corresponds to
where act on and as Pauli operators. Here we used “” to denote the map from real physical systems to the excitation basis.
Now consider a subset of qubits and a restriction of the membrane phase operator onto , denoted by :
Consider an excited wavefunction
We hope to represent in the excitation basis. First let us be more precise with a definition of the excitation basis states. Let be the fluxless subspace of the entire Hilbert space where satisfies for all . Let be the total number of plaquettes on the lattice . We define the excitation basis states by
for . More explicitly, we define them by
Importantly, not all the basis states are physically allowed since there are certain constraints on values of . Observe that
where represent sets of plaquettes of color respectively. Let , , be the total number of excitations on plaquettes of color respectively. Then,
In other words, an excitation basis state is physically allowed if and only if modulo . (Indeed, if does not satisfy this condition, then the righthand side of Eq. (22) becomes zero). Because the system is not degenerate, basis states with Eq. (24) span the fluxless subspace completely. Thus, the excited wavefunction can be characterize as follows
in the excitation basis where is a complex number with proper normalization and satisfy Eq. (24).
Let us then study a loop-like excitation created by a part of the membrane-like operator. Consider a set of plaquettes of color which form a contractable connected region of qubits with a single boundary (see Fig. 2). Consider an excited wavefunction . Since , the phase operator creates excitations only around the boundary of . A key observation is that creates excitations only on plaquettes of color because can be viewed as a set of plaquettes of color . Namely, let be a set of plaquettes of color on the boundary (Fig. 2). Observe that, on the boundary , plaquettes of color and appear in an alternating way. We denote boundary plaquettes by where is the total number of boundary plaquettes. Then the excitation wavefunction can be written as
where . Here, the first part represents excitations on the boundary and the second part represents the rest.
The next task is to find an expression of . From Eq. (17), the excited wavefunction is given by
where in the product corresponds to and in the phase operator respectively. Here represents a vertex shared by three neighboring plaquettes . After careful calculations, one can find that the boundary wavefunction, in the excitation basis, is given by
The boundary wavefunction can be viewed as a one-dimensional system of qubits supported on a closed loop. It is worth finding the Hamiltonian which has the boundary wavefunction as a unique ground state. Recall that for all plaquettes . Interaction terms in the Hamiltonian are then obtained by considering . One finds
One can verify that is the unique gapped ground state of the Hamiltonian . This Hamiltonian for the boundary wavefunction is identical to that of the so-called cluster state up to transversal application of the Hadamard operators to each and every qubit along the boundary:
The ground state (the cluster state) is specified by
In quantum information science community, the cluster state is known as an important resource state for realizing the measurement-based quantum computation scheme Raussendorf et al. (2003).
iii.2 SPT excitation with symmetry
The cluster state is perhaps the simplest example of one-dimensional bosonic SPT phases with symmetry. Let us define a pair of on-site symmetry operators as follows:
By on-site, we mean that symmetry operators are transversal. The cluster state is the unique ground state of the Hamiltonian (Eq. (31)) and is symmetric under and :
This can be verified by noticing and , and . Let be a trivial product state with symmetry where . Two symmetric wavefunctions and are connected by a local unitary transformation and belong to the same quantum phase in the absence of symmetries. Indeed, one has
On the other hand, in the presence of symmetry, they belong to different SPT phases. Namely, there is no symmetry-protected local unitary transformation such that and . For instance, one sees that the unitary transformation in Eq. (35) does not commute with or . In this sense, a cluster state is a non-trivial SPT wavefunction with symmetry.
It is interesting to observe that loop-like excitations in the two-dimensional color code are characterized by wavefunctions of a one-dimensional SPT phase. To understand what this observation means, let us further establish the connection between SPT phases and loop-like excitations. Indeed, on-site symmetry of the boundary wavefunction naturally emerges from parity conservation on the number of anyonic excitations in the topological color code. Since phase operators are applied on qubits supported on plaquettes of color , there is no excitation associated with plaquettes of color . This implies
which leads to the following on-site symmetries in the excitation basis
After transversal Hadamard transformation, these symmetry operators are identical to those in Eq. (33). As such, excitations arising in the two-dimensional color code are natural platforms for constructing wavefunctions with symmetry.
The remaining question then is why the boundary wavefunction corresponds to a non-trivial SPT phase. The key observation is that, while these loop-like excitations are localized along the boundary , they cannot be created by a local unitary transformation acting on physical qubits in the neighborhood of . To see this, let us create a pair of magnetic fluxes which are located outside of and move one of them inside , crossing the SPT excitation (Fig. 3). Since phase operators exchange Pauli and operators, the magnetic flux will be transformed into a composite of an electric charge and a magnetic flux upon crossing the SPT excitation. Let be a string operator corresponding to the propagation of a magnetic flux into in the absence of an SPT excitation. (It is a tensor product of Pauli operators). Suppose that there exists a local unitary which creates an SPT excitation by acting only on qubits in the neighborhood of . Then differs from only at the intersection with the boundary. This implies that a magnetic flux remains to be a magnetic flux inside , leading to a contradiction. Thus, to create a loop-like SPT excitation, one needs to apply a local unitary transformation on all qubits inside (or all qubits in the complement of ).
This argument enables us to show that two symmetric wavefunctions and belong to different SPT phases. This is because symmetry-protected local unitary operators in the excitation basis have some corresponding local unitary operators in the two-dimensional color code. Suppose that there exists a symmetry-protected local unitary transformation such that , which can be generically written as
where and is geometrically local and consists only of terms with bounded norms. Here represents the time-ordering. Note can be written as a sum of symmetry-protected local terms such as , , , and their products. These symmetry-protected local unitary operators correspond to some local operators in the topological color code. Namely, and correspond to string operators of color and , consisting of Pauli operators, which end at plaquettes and respectively. Moreover, and correspond to -type plaquette operators of color . As such, the existence of a symmetry-protected local unitary implies the existence of a local unitary transformation which acts only on physical qubits near the boundary, yet creates a loop-like SPT excitation. This leads to a contradiction. Thus, we can conclude that and belong to different SPT phases.
We have argued that an excited wavefunction corresponds to a non-trivial SPT phase due to the parity conservation and the non-trivial automorphism of anyons induced by the phase operator. In general, loop-like excitations may involve plaquettes of three different colors when is chosen to be an arbitrary connected region of qubits. The excited wavefunction then can be viewed as a one-dimensional system with spins of three different colors which possesses symmetry with respect to where are tensor products of Pauli operators acting on spins of color in the excitation basis. As such, our characterization of excitations by SPT phases is valid in these cases too.
For readers who are familiar with the literature of SPT phases, it may be clear that the characterization of a wavefunction in the excitation basis in the fluxless subspace is essentially equivalent to “ungauging” the on-site symmetries, opposite to the procedure of gauging the on-site symmetries of SPT wavefunctions Levin and Gu (2012); Hu et al. (2013). In this picture of ungauging, the two-dimensional color code with a loop-like excitation serves as a path integral formulation of a one-dimensional SPT wavefunction. Namely, creation of a loop-like SPT excitation via transversal operators on the bulk can be interpreted as a symmetry-protected quantum circuit preparing a non-trivial SPT wavefunction with symmetry. An interesting application of this picture is that, if one considers the two-dimensional color code embedded on a hyperbolic surface, one obtains a MERA (multi-scale entanglement renormalization ansatz) circuit for an SPT wavefunction with Vidal (2007).
Finally we remark that similar SPT excitations emerge in two decoupled copies of the toric code where the transversal control- operator preserves the ground state space. In this setting, a loop-like SPT excitation involves electric charges from two copies of the toric code, each possessing one copy of the symmetry. This conclusion also follows from the unitary equivalence of the color code and two decoupled copies of the toric code on a closed manifold Kubica et al. (2015).
Iv Gapped domain walls and fault-tolerant logical gates
While one-dimensional SPT excitations, created by phase operators in the two-dimensional color code, are interesting from a theoretical viewpoint, they do not exist as stable objects since they are superpositions of eigenstate excitations which would decohere immediately. In this section, we argue that SPT excitations are physically realized in a certain way. Namely, we point out that SPT excitations can be viewed as transparent gapped domain walls in the color code in the Heisenberg picture. Our finding also reveals an intriguing relation between classifications of domain walls and fault-tolerantly implementable logical gates.
iv.1 Transparent domain wall
The key idea is to transform the Hamiltonian by transversal operators instead of transforming the ground state. To begin, consider the two-dimensional color code supported on the honeycomb lattice (Fig. 4). Recall that the transversal Hadamard operator, , preserves the Hamiltonian. Here we split the entire system into two parts, the left part and the right part . Consider the restriction of the transversal Hadamard operator onto the right part of the system :
and the transformed Hamiltonian . Note that the resulting Hamiltonian remains the same as before except on the boundary between and :
where and are the same as the corresponding terms in . Since is a unitary transformation, the transformed Hamiltonian remains gapped. Here, can be viewed as a transparent gapped domain wall connecting and , which swaps the electric charge and magnetic flux as follows:
By a transparent wall, we mean that no single anyon from either side of the wall may condense on the wall. Here, represents that an anyon gets transformed into by crossing the wall from the left to the right. Transparency of the wall imposes that or for is not allowed where denotes the vacuum. Similarly, the transversal phase operator on creates a gapped domain wall which changes anyon labels as follows:
This transparent domain wall corresponds to the loop-like SPT excitation created by operators in the Heisenberg picture as shown in Fig. 4. The ground state space of the topological color code is also symmetric under the so-called transformation:
Transversal application of operators leads to the following transparent domain wall:
Clearly, the aforementioned construction of transparent gapped domain walls works for arbitrary transversal membrane-like operators in -dimensional TQFTs as long as the operators induce non-trivial automorphism among anyon labels. It turns out that the construction works not only for transversal operators but also for any locality-preserving transformations Pastawski and Yoshida (2015); Beverland et al. (2014). Formally, a locality-preserving unitary transformation is defined to satisfy the following condition. For an arbitrary unitary operator supported on some region , there exists supported on such that where is the boundary of of finite width. Examples of locality-preserving transformations include local unitary transformations with removing and adding ancilla qubits. Also translating all qubits on the lattice by finite sites is a locality-preserving transformation. By applying such a transformation with non-trivial automorphism of anyon labels on the half of the system, one can create a transparent gapped domain wall which changes anyon labels according to the automorphism associated with the membrane-like operator.
To give a concrete yet non-trivial example of locality-preserving transformation, consider the two-dimensional toric code supported on a square lattice on a torus: . Observe that transversal application of Hadamard operators, followed by shifting all the lattice sites in a diagonal direction, leaves the Hamiltonian invariant. This transformation is clearly locality-preserving and swaps electric charge and magnetic flux . If one applies this transformation partially on one side of the system, one is able to create a domain wall which exchanges and upon crossing the wall. (Equivalently, one may consider the Wen’s formulation of the toric code where unit translations lead to exchange of and Wen (2002)). We note that this domain wall in the toric code was previously constructed by Bombin Bombin (2010).
iv.2 Membrane operator and domain wall
We have seen that membrane operators associated with non-trivial automorphisms among anyon labels lead to transparent gapped domain walls in -dimensional TQFTs. An interesting question is whether the presence of transparent gapped domain walls implies membrane-like locality-preserving transformations. In this subsection, we present complete classifications of transparent gapped domain walls for the two-dimensional toric code and the two-dimensional color code. Namely, we find that every transparent domain wall has a corresponding membrane operator associated with non-trivial automorphism of anyon labels.
In -dimensional TQFTs, labels of anyons, which may condense on a gapped boundary, can be characterized by maximal sets of anyons with trivial self and mutual braiding statistics Levin and Gu (2012). To characterize a gapped domain wall which connects two topologically ordered systems, we fold the system along the domain wall and view the domain wall as a gapped boundary where two systems (the left and the right) are attached together Beigi et al. (2011). We are then able to classify automorphisms of anyon labels on the domain wall as condensation of anyons from the left and the right in the folded geometry.
To begin, let us characterize transparent domain walls of the toric code. Anyons are denoted by . One finds the following two transparent domain walls:
Here represents a trivial domain wall where anyon labels are not altered and corresponds to the aforementioned domain wall in the toric code which exchanges and . One can view the domain wall as a gapped boundary which absorbs and where the subscripts and denote the left and the right. One then sees that and have trivial self and mutual braiding statistics. We note that this classification of transparent domain walls in the toric code is well known in the literature, see Lan et al. (2015) for instance.
Next, let us characterize transparent domain walls for the two-dimensional color code, which is unitarily equivalent to two copies of the toric code. The problem can be reduced to finding matrices with binary entries which satisfy certain conditions reflecting triviality of braiding statistics in the folded geometry Lan et al. (2015). We found different types of transparent domain walls in the color code by an exhaustive search. Observe that transparent domain walls form a group since a product of two walls is also a wall. As such, domain walls can be constructed from a certain complete set of generators of domain walls. Below, we list five types of domain walls which form a complete set.
These domain walls preserve the color labels of anyons while exchanging electric charges and magnetic fluxes. Here corresponds to operators which permute Pauli operators and corresponds to Hadamard operators. Here and form a subgroup which is isomorphic to the symmetric group . Thus the group of domain walls is non-abelian. The following domain walls exchange color labels of anyons:
One can construct the corresponding membrane operators by decoupling the color code into two copies of the toric code and exchanging them. Finally, we find the following domain wall
which corresponds to a domain wall in a single copy of the toric code. We find that domain walls generate all the different transparent gapped domain walls in the color code. The main conclusion is that, for every transparent domain wall in the color code, there exists a corresponding membrane-like locality-preserving operator which preserves the ground state space and induces non-trivial automorphism of anyon labels.
Classification of gapped domain walls is an important problem in condensed matter physics community as many of realistic physical systems have boundaries and domain walls. Classification of fault-tolerantly implementable logical gates is of relevance to the quantum information science community as they are indispensable building blocks for fault-tolerant quantum computation. Whether the correspondence between domain walls and membrane operators generically holds for -dimensional TQFTs is an interesting future problem to study.
V Volume operator in three-dimensional color code
Consider the three-dimensional color code defined on a four-valent and four-colorable lattice where qubits live on vertices. Colors are denoted by , and are associated with volumes. The Hamiltonian is given by
where represents a plaquette and represents a volume (Fig. 5). Here commute with each other due to the four-valence and four-colorability of the lattice. For simplicity of discussion, we assume that the lattice is supported on (homomorphic to) a three-sphere so that the ground state is unique. A systematic procedure of constructing such a lattice is known Bombin and Martin-Delgado (2007).
Three-dimensional topologically ordered spin systems may possess both point-like and loop-like excitations which are characterized by string and two-dimensional membrane operators respectively. We begin by constructing string operators. Given a four-valent and four-colorable lattice , one can assign color labels to its edges since, for a given edge, there always exist three volumes of different colors sharing the edge. Consider a set of edges of color which form a one-dimensional line such that a string connects volumes of color as shown in Fig. 6(a). We define
If is an open line, commutes with all the interaction terms except stabilizers on volumes of color at the endpoints of . Thus, one can characterize electric charges as follows
To construct membrane operators, we assign color labels to plaquettes of the lattice . Consider a set of plaquettes of color which form a two-dimensional sheet (membrane) as shown in Fig. 6(b). We define
If is an open membrane with boundaries, the operator creates excitations associated with stabilizers on plaquettes of color on boundaries of . Thus, one can characterize loop-like magnetic fluxes as follows
These excitations with different color labels are not independent from each other since the following fusion channels exist
It is convenient to construct an isomorphism between anyons of the color code and those of the three-dimensional toric code. Let be anyons in three decoupled copies of the toric code. Then one has
It is known that, on a closed manifold, the three-dimensional color code is equivalent to three decoupled copies of the toric code under a local unitary transformation Yoshida (2011b); Kubica et al. (2015).
Note that the three-dimensional color code has volume and membrane phase operators. Recall that the lattice is bipartite and qubits can be split into two sets and . For a phase operator , let us define the following transversal volume (three-dimensional) phase operator