The Toric Code

Paul Herringer (Dated: December 11, 2020) We review the toric code, a simple spin-lattice model that exhibits topological order and anyonic excitations. We derive these phenomena from a model Hamiltonian, explain their signiﬁcance, and highlight the connection to fault-tolerant quantum computation.

I. INTRODUCTION

Topology is the branch of mathematics that studies the properties of objects which do not change under smooth deformations, one classic example being the number of holes in a torus. In other words, topologists study global properties that are invariant to many local perturbations. From this perspective it is not hard to see why physicists are also in- terested in topology. In this article we re- FIG. 1. Our model is a square lattice with peri- 1 view one example of a physical model where odic boundary conditions and a spin- 2 particle the ground state degeneracy and elementary (circles) on each edge. The dashed lines show excitations are determined by topology: the the dual lattice. Av and Bp consist of Pauli op- toric code. We will take a condensed-matter erators Z (blue) and X (red) centered around a physicist’s approach to this model, start- vertex or plaquette of the primal lattice. ing from the Hamiltonian and deriving the ground state. We also discuss the low-energy spins: excitations and their unusual exchange statis- tics, and explain why the toric code is a cor- Y Y A = Z ; B = X (2) nerstone model in the study of fault-tolerant v j p j quantum computation. j∈star(v) j∈bdy(v) where star(v) is the set of edges adjacent to vertex v, and bdy(p) is the set of edges form- II. SPIN HAMILTONIAN ing the boundary of the plaquette p (Fig. 1). We can start by noting that A and B are Consider a 2D square lattice with periodic v p Hermitian and square to the identity, there- boundary conditions (i.e. a torus) and spin- fore have eigenvalues ±1. Furthermore, each 1 particles located on each edge (Fig. 1). 2 of them commutes with all the others. To see Define the following Hamiltonian: this, note that the overlap of Av with any Bp X X H = − A − B , (1) occurs at an even number of edges. The op- v p erators X and Z anticommute on any given v p edge, but on an even number of edges this ef- where the first sum is over all vertices v and fect cancels itself out. Therefore, the ground the second is over all faces or plaquettes p. state(s) will be the simultaneous +1 eigen- The vertex operators Av and plaquette oper- state of all the Av and Bp operators. ators Bp are tensor products of Pauli opera- The next step is to figure out what kind tors X = σx and Z = σz acting on individual of states satisfy the above constraints. Be- 2

fore we begin, let’s compute how many states we expect to find so that we can be confi- dent of the final answer. Each unit cell has two spins, and each spin occupies a Hilbert space of dimension 2, so one unit cell has a 4-dimensional Hilbert space. If there are N unit cells, the Hilbert space of the lattice 2N has dimension 2 . Now, think of Av and Bp as constraints on this Hilbert space; vi- olating any constraint costs energy, so the ground state should obey every constraint. FIG. 2. Occupied edges are shaded black. (a) Finally, note that while there are N each of Applying Av to the end of a string gives eigen- Av and Bp, each type gives only N − 1 inde- value −1. (b) Applying Av anywhere on a loop pendent constraints because the product of gives eigenvalue +1. all Av is the identity, and similarly for Bp. Putting everything together, we have 2N − 2 constraints, so we expect a GS manifold of on the border of a given plaquette. This has dimension 22N−(2N−2) = 4. the effect of adding a new loop of occupied edges or smoothly deforming an existing loop, It will be useful to introduce some nota- in the sense that we cannot break or join tion for constructing the ground state. We a string of occupied edges by applying any adopt the convention that |0i is the +1 eigen- combination of Bp. Notice that we can con- state of Z and |1i is the −1 eigenstate. In tract any loop that does not wind around the this basis, X acts as a spin-flip operator, i.e. torus until it disappears completely; in this X |0i = |1i, X |1i = |0i. Taking the |0i , |1i sense, such loops are equivalent to the trivial notation literally, we will say that an edge is state with no loops. However, a little thought “occupied” if the spin there is in the state should convince you that a string of occupied |1i, and “empty” if it is |0i. Recall that edges that winds all the way around the torus we are looking for +1 eigenstates of all Av in one direction cannot be contracted into and Bp, but for now let’s focus only on Av. nothing, as this would require breaking the Clearly the state with all edges unoccupied string somewhere. This realization gives us has eigenvalue +1 for all Av. If a single spin 4 classes of loops which cannot be deformed is flipped, the resulting state will have eigen- into one another: the trivial class with no value −1 for the Av immediately adjacent to loops, a loop that winds around the torus in that edge (Fig. 2a). In fact, anytime a vertex either direction, and the combination of two v is surrounded by an odd number of occu- loops with one winding around in each direc- pied edges, the corresponding Av will have tion (Fig. 3). eigenvalue −1. The only states which do not The fact that there are 4 inequivalent loop have at least one such vertex are those where configurations should not be too surprising; the occupied edges form closed loops, so the as you may have guessed, they correspond to set of all such states forms the +1 eigenspace the 4-dimensional GS manifold. To see this, of the Av operators (Fig. 2b). note that the 4 basic loops are not eigenstates Clearly there are many, many possible of Bp by themselves, but a completely sym- states with closed loops, so the ground state metric superposition of any one of them with is massively degenerate if we only satisfy the all members of the trivial class is an eigen- Av constraints. To fix this degeneracy we add state. In other words, we have a sort of “loop the Bp terms to the picture. Each Bp is made vacuum” (the trivial class), to which we can up of X operators, so it will flip all the spins add zero, one, or two non-trivial loops. Each 3

(a) (b) (c) (d)

FIG. 3. Examples of the four classes of loops which cannot be deformed into each other.

FIG. 4. Strings of X or Z operators applied FIG. 5. Moving an e particle around an m parti- to the ground state create pairs of “electric” or cle causes the overall state to pick up a phase of “magnetic” excitations e and m. The energy of −1, because the X and Z strings anticommute each pair is +2 because the end of any string at exactly one site. anticommutes with one Av or Bp. the X operators at its endpoints will anticom- case is a degenerate eigenstate of the Hamil- mute with one Av each, raising the energy tonian (1), and together they form a basis for above the ground state by 2. Similarly, the 0 the 4-dimensional GS manifold. Note that ends of a dual string t anticommute with Bp. this 4-fold degeneracy is directly related to Note that it is impossible to create an exci- the topology of the system (i.e. the periodic tation with unit energy. To see this, suppose boundary conditions); we say that the toric there is an operator C acting on a state |ψi in code displays topological order. the GS manifold such that C anticommutes with only one Av (or Bp, it doesn’t matter which). Recall that Q A = 1, therefore III. ANYONIC EXCITATIONS v v Y Y C |ψi = A C |ψi = −C A |ψi = −C |ψi , Now that we know the ground state, we v v v v can try to characterize the low-energy excitations. The easiest way to do this is to think i.e. C |ψi = 0. More intuitively, it is im- in terms of the operators that create an exci- possible to create an open string with one tation. These turn out to be string operators end, so the only allowed excitations come in [1]: pairs. We call the quasiparticle pairs cre- x x Y z 0 Y ated by S (t) “electric charges” e and the S (t) = Xj; S (t ) = Zj, (3) pairs created by Sz(t0) “magnetic vortices” to j∈t j∈t0 match existing literature on gauge ﬁeld mod- where t (t0) is a string of edges on the (dual) els, but this is just a convention [1]. lattice (Fig. 4). If a string t is open-ended, Perhaps the most interesting thing about 4

manifold. Recall that mapping between orthogonal states in the GS manifold requires flipping a loop of spins that winds around one dimension of the torus (see Fig. 3), therefore the first non-zero correction will not show up until the L-th order, where L is the smallest circumference of the torus, and the correc- tions vanish in the thermodynamic limit [1]. We can take advantage of this protected ground state for quantum computa- FIG. 6. Logical operators on acting on the qubits tion. From here on we will use the terms encoded by the ground state of the toric code. spin and qubit interchangeably, but this is Anticommutation relations can be inferred from just a matter of terminology. A qubit can be the intersection of the strings, and they match any 2-level quantum system; referring to it the usual Pauli operators on two qubits. as such simply indicates our intention to use this system for information processing. The Hilbert space of n qubits has dimension 2n, so these quasiparticles is their exchange statis- we can store 2 qubits worth of information in x x tics. Clearly S (t1) commutes with S (t2) for the ground state of the toric code. The prop- any strings t1 and t2, as they only involve X erties of our Hamiltonian ensure that this in- terms. The same is true of Sz, so it appears formation is distributed across the state of that both the e and m particles are hard-core the entire system rather than localized to the bosons. However, if we move an e particle states of individual spins; this protects the around an m particle or vice versa (Fig. 5), information from local errors. A scheme for it acquires a phase of −1 from the overlapping storing quantum information in this way is string operators. Quasiparticles that behave called a quantum error-correcting code [4, 5], in this way are called anyons, because they which is where the name toric code comes fall outside the usual bosonic/fermionic pic- from [1]. ture [1]. Anyons are unique to 2D systems It turns out we can do more than just store and feature in the theory of the fractional information. Let us re-label the four orthog- quantum Hall effect [2]. More recently, Bar- onal ground states (a, b, c, d) from Fig. 3 as tolemi et al published strong experimental |00i , |10i , |01i and |11i, to make the analogy evidence for their existence using an “anyon explicit for two logical qubits (as opposed to collider” [3]. physical qubits). Then a string of X operators looped around the horizontal dimension IV. FAULT-TOLERANT QUANTUM maps |00i ↔ |10i, and a loop of Z operators COMPUTATION in the vertical direction maps |00i ↔ |00i and |10i ↔ − |10i, therefore we can identify these ¯ ¯ The ground state of the Hamiltonian (1) strings as logical Pauli operators X1 and Z1 is protected from local perturbations. For acting on the first logical qubit. Similarly, instance, consider a local X perturbation the remaining loops of X and Z operators ¯ ¯ P act as logical operators X2, Z2 on the second V = λ j Xj where j runs over every spin on the lattice and λ 1. This perturbation qubit (Fig. 6). will split the degeneracy of the ground state, Note that the action of these logical op- but the splitting terms obtained from per- erators does not depend on a specific path turbation theory are of the form hψ| V n |φi as long as that path winds around the torus, for |ψi , |φi two orthogonal states in the GS and we could also interpret these operators 5

as anyon pairs which tunnel around the torus V. CONCLUSION before self-annihilating. This is just one example of the connections between topological The toric code is a canonical example of order, anyons, and quantum computation. A a system with topological order and anyonic further discussion is beyond the scope of this excitations. It also establishes a connection review, but before we conclude it should be between physical systems and quantum infor- pointed out that the toric code and its close mation, which motivates the study of anyons relative the surface code [6] underpin some of and topological order as a route to fault- the leading proposals for fault-tolerant quan- tolerant quantum computation. The quest tum computation [7]. This section is in- to better understand systems where topology tended to provide some insight into why that plays a role and to fabricate them in the lab is is the case, and to encourage further reading an active area of research, and there is much (e.g. ref. [5] is quite accessible). to discover in the coming years.

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