PHYSICAL REVIEW X 5, 041040 (2015)

Parafermions in a Kagome Lattice of Qubits for Topological Quantum Computation

Adrian Hutter, James R. Wootton, and Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Received 3 June 2015; revised manuscript received 16 September 2015; published 14 December 2015) Engineering complex non-Abelian anyon models with simple physical systems is crucial for topological quantum computation. Unfortunately, the simplest systems are typically restricted to Majorana zero modes (Ising anyons). Here, we go beyond this barrier, showing that the Z4 parafermion model of non-Abelian anyons can be realized on a qubit lattice. Our system additionally contains the Abelian DðZ4Þ anyons as low-energetic excitations. We show that braiding of these parafermions with each other and with the DðZ4Þ anyons allows the entire d ¼ 4 Clifford group to be generated. The error-correction problem for our model is also studied in detail, guaranteeing fault tolerance of the topological operations. Crucially, since the non- Abelian anyons are engineered through defect lines rather than as excitations, non-Abelian error correction is not required. Instead, the error-correction problem is performed on the underlying Abelian model, allowing high noise thresholds to be realized.

DOI: 10.1103/PhysRevX.5.041040 Subject Areas: Condensed Matter Physics, Quantum Physics, Quantum Information

I. INTRODUCTION Non-Abelian anyons supported by a qubit system Non-Abelian anyons exhibit exotic physics that would typically are Majorana zero modes, also known as Ising – make them an ideal basis for topological quantum compu- anyons [12 15]. A variety of proposals for experimental tation [1–3]. It has recently become apparent that truly realization of Majorana zero modes in solid-state systems scalable quantum computation with non-Abelian anyons have also been developed [16]. These anyons can be used to can only be achieved when invoking active error correction, perform universal quantum computation when enhanced by despite the protection provided by a finite anyon gap [4–6]. nontopological operations [17,18]. However, these addi- The development of practical systems in which non- tional operations are highly resource intensive. Anyon Abelian anyons may be created, manipulated, and detected models with a richer set of topological operations would is therefore highly important. Systems in which non- therefore be much more practical for the realization of Abelian anyons arise typically suffer from one of two topological quantum computation. Here, we solve this by drawbacks: Either they are experimentally extremely chal- introducing a model composed of two-qubit Hamiltonian lenging to realize (as is the case for quantum double [1] or interactions that can realize a more complex model of non- string-net models [7]), or it is not clear how they can be Abelian anyons, known as Z4 parafermions. The error- made compatible with the active error correction required correction problem for these is studied in detail. for fault tolerance (as is the case for FQH systems). Parafermion modes are generalizations of Majoranas A particularly attractive approach for building a fault- whose fusion and braiding behavior is more complex tolerant quantum computer is to use a system of physical and computationally more powerful. This has led to a 1 qubits (spin-2 particles). A number of technologies allow quest in recent years for systems that could host them. for precise qubit control, such as superconducting qubits Numerous proposals for their experimental implementation [8], trapped atomic ions [9], spin qubits [10], or cold atoms in condensed-matter systems, such as fractional quantum or polar molecules in optical lattices [11]. A qubit lattice Hall systems, nanowires, or topological insulators, have with two-body nearest-neighbor interactions would there- recently appeared [19–34]. fore be an ideal system to realize non-Abelian anyons. Extrinsic defects in Abelian topological states can While the model we propose involves three-body inter- behave like non-Abelian anyons [35]. The idea of non- actions, we discuss how these can be obtained from Abelian anyons at the ends of defect lines, first introduced two-body interactions. for FQH states [36], has been adapted to the DðZdÞ quantum double models in Refs. [37,38]. These anyons are Majorana zero modes for d ¼ 2 and more powerful parafermions for d>2. Unfortunately, the generalized Published by the American Physical Society under the terms of Z the Creative Commons Attribution 3.0 License. Further distri- Pauli operators appearing in the Dð dÞ quantum double bution of this work must maintain attribution to the author(s) and models coincide with the physically relevant spin operators the published article’s title, journal citation, and DOI. only for d ¼ 2. Otherwise, their structure makes them

2160-3308=15=5(4)=041040(21) 041040-1 Published by the American Physical Society ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015) highly difficult to realize experimentally. The case d ¼ 4, These satisfy the Zd parafermion relations, however, allows us to combine the best of both worlds. γd 1 γ γ ωsgnðk−jÞγ γ The joint Hilbert space of two qubits allows the four- j ¼ ; j k ¼ k j: ð2Þ dimensional generalized Pauli operators to be expressed in terms of two-qubit operators. Using this, we show how Z4 The operators X, Y, and Z can be represented as parafermions can emerge in a lattice of qubits with nearest- d-dimensional matrices. It is thus natural to seek a neighbor interactions only. This allows the computational representation of these operators for the case d ¼ 4 on Z 1 power of 4 parafermions to be harnessed in a qubit the Hilbert space of two qubits (spins-2). Indeed, given system. qubits 1 and 2, one easily verifies that the operators The fact that our system is built on top of a system supporting Abelian anyons [the DðZ4Þ quantum double 1 X ¼ ðσx þ σx − iσz σy þ iσyσzÞ; model] proves very useful. The non-Abelian parafermion 2 1 2 1 2 1 2 modes can not only be braided with each other but also with 1 i3π=4 σy σy σxσz σzσx Abelian excitations of the quantum double model, allowing Y ¼ 2 e ð 1 þ i 2 þ i 1 2 þ 1 2Þ; us to generate the entire Clifford group for d ¼ 4 by 1 Z ¼ pffiffiffi eiπ=4ðσz − iσzÞð3Þ quasiparticle braiding. This extends beyond the limited set 2 1 2 of gates found using the same parafermions in previous work [21]. Furthermore, we do not have to perform non- are four-dimensional generalized Pauli operators, and Z4 Abelian error correction (a still poorly understood problem parafermions can be obtained from these via Eq. (1).We [4,5,39–42]) to guarantee fault tolerance but can correct the 2 x x 2 y y 2 z z also note that X ¼ σ1σ2, Y ¼ σ1σ2, and Z ¼ σ1σ2. underlying Abelian model. This Abelian error-correction problem is nevertheless more involved than the well- III. MODEL studied error-correction problem for the standard DðZdÞ models, and we study it in detail. We consider a two-dimensional Kagome (trihexagonal) The rest of this paper is organized as follows. In Sec. II, lattice as in Fig. 1. Each vertex of the lattice hosts one we show how Z4 parafermion operators can be expressed in four-dimensional qudit (one pair of Z4 parafermions) or, in terms of qubit operators. Section III introduces a qubit other words, two qubits. The Hamiltonian of our model Hamiltonian whose low-energetic excitations correspond to is given by the DðZ4Þ quantum double model. In Sec. IV, we discuss X X Z σx σx how 4 parafermion modes appear at the ends of defect H ¼ HΔ þ h ð i1 þ i2Þ: ð4Þ strings in our model. We demonstrate in Sec. V how the Δ i non-Abelian braiding statistics of these modes can be used to perform logical gates. Appendix D contains a proof that Here, the first term is a sum of equivalent terms for each the set of gates that can be performed this way generates the triangle in the Kagome lattice. We label the vertices around one triangle a, b, and c, and the two qubits that are present entire Clifford group C4, which may be of independent interest. In Sec. VI, we study the error-correction problem of our model in detail, and we conclude in Sec. VII.

II. Z4 PARAFERMION OPERATORS IN TERMS OF QUBIT OPERATORS We consider d-dimensional generalizations of the Pauli matrices X and Z. These are unitary operators satisfying Xd ¼ Zd ¼ 1 and ZX ¼ ωXZ, where ω ¼ e2πi=d with integer d>1. If we define Y ¼ ωðdþ1Þ=2X†Z†, we also d have Y ¼ 1, XY ¼ ωYX, and YZ ¼ ωZY. Operators Xi and Zi act on qudit i and hence ½Xi;Xj¼½Zi;Zj¼ ½X ;Z ¼0 if i ≠ j. i j FIG. 1. Two qubits are located at each vertex of a Kagome These operators are related to those of parafermions. lattice. Each pair of qubits hosts two Z4 parafermions. To unlock Given a total ordering on the qudits fig, one can obtain their potential for non-Abelian braiding, two such parafermions parafermion operators via a nonlocal transformation [43] need to become unpaired, which is achieved by adding a defect line to the lattice. These are strings of strong local operators Y Y acting on qubit pairs (encircled). They create unpaired parafer- γ γ ωðdþ1Þ=2 2i−1 ¼ Xj Zi; 2i ¼ Xj Zi: ð1Þ mion modes located at their ends (light pentagon-shaped regions j

041040-2 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015) at vertex a are called a1 and a2, etc. The triangle terms in the Hamiltonian are then given by

J z z z z z z HΔ σ σ σ σ σ σ ¼ 2 ð a1 b1 c1 þ a2 b2 c2Þ − J σz σz σz σz σz σz σz σz σz 2 ð a1 b1 c2 þ a1 b2 c1 þ a2 b1 c1 σz σz σz σz σz σz σz σz σz þ a2 b2 c1 þ a2 b1 c2 þ a1 b2 c2Þ: ð5Þ P The second sum i in Eq. (4) runs over all vertices in the ~ † † ~ 1 FIG. 2. Two sets of logical operators X1 ¼ XX XX …, Z1 ¼ lattice. The two qubits located at vertex i are called i and … ~ † †… ~ † †… i2. This second sum thus represents a uniform magnetic ZZZZ (left figure) and X2 ¼ XX XX , Z2 ¼ ZZ ZZ (right figure) that satisfy the commutation relations of four- field in the x direction. dimensional generalized Pauli operators. Our Hamiltonian involves three-qubit terms of the form σz σz σz a b c. It is, in principle, straightforward to generate these from one-body terms and two-body interactions by use of Note that all terms in the first sum in Eq. (8) commute, so perturbative gadgets [44–46]. Consider a “mediator qubit” the unperturbed Hamiltonian is trivially solved. The lowest- u coupled to qubits a, b, and c. Starting from a Hamiltonian order nonvanishing terms appear in sixth-order perturbation theory. We find an effective Hamiltonian Δ X − σz α σz σz σx βσzσz γσz σz δσz Hgadget ¼ u þ ð a þ bÞ u þ c u þ a b þ c; − 2 Heff ¼ J ðZaZbZc þ H:c:Þ Δ ð6Þ 63 6 X − h † † † 8 5 ðXrXs XtXuXvXw þ H:c:Þ; ð9Þ we consider the perturbative regime Δ ≫ jαj, jβj. In this ð2JÞ ⎔ regime, it is possible to integrate out qubit u. Taking up to third-order terms into account, one finds an effective where the second sum runs over all hexagons in the Hamiltonian Kagome lattice and r, s, t, u, v, w label the six vertices around each hexagon. The effective Hamiltonian in Eq. (9) α2 α2β β δ σz −2 γ σz σz − 4 σz σz σz is derived in Appendix A. Heff ¼ð þ Þ c þ þ a b 2 a b c: Δ Δ We note that all summands in Heff commute, so the system is exactly solvable. The excitations of this system ð7Þ are Abelian anyons corresponding to the DðZ4Þ quantum double model. The topological degeneracy of the model δ −β γ 2 α2 Δ Choosing ¼ and ¼ ð = Þ produces the desired can be made manifest by studying nonlocal loop degrees three-qubit term without any undesired one- or two-qubit of freedom that commute with all stabilizers ZaZbZc, terms. † † † X X X X X X , and their Hermitian conjugates, and fulfill The generation of three-body interactions in optical r s t u v w Z4 relations themselves. A possible choice of operators is lattices has been discussed in detail in Refs. [47,48]. illustrated in Fig. 2. These proposals would make the perturbative gadgets In passing, we note that the Z2 version of Eq. (8),in unnecessary. A “toolbox” for generating spin-lattice mod- which the Z4 operators X and Z are replaced by Pauli els such as ours in optical lattices has also been developed operators σx and σz, leads to an effective Hamiltonian [49]. Generating non-Abelian anyons other than Majorana analogous to Eq. (9) and thus provides a very simple model zero modes by use of perturbative gadgets from two-body with topological order. While this model requires three- interactions has previously been discussed in Refs. [50,51]. body operators σzσzσz as opposed to Kitaev’s honeycomb The spin-Hamiltonian in Eq. (4) can be exactly Hamiltonian [12] which involves two-body interactions rewritten as only, all of these interactions connect the same spin X X component, which may provide a significant practical − † H ¼ J ðZaZbZc þ H:c:Þþh ðXi þ Xi Þ: ð8Þ simplification over the honeycomb model. Δ i IV. PARAFERMION MODES AND DEFECT LINES Here, again, the first sum runs over all triangles in the lattice and the corners of a triangle are labeled a, b, and c. The model is constructed from the cyclic qudit operators The second sum again runs over all vertices of the lattice. Z and X, which are related to parafermion operators. It is We now consider the perturbative limit h ≪ J and regard therefore natural to seek an interpretation of the model in the second sum in Eq. (8) as a perturbation to the first term. terms of parafermionic modes.

041040-3 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015)

To do this, we must first fix the exact form of the stabilizers neighbor of Ep, while other choices become necessary next that define the anyonic charge carried by each excitation. Let to defect lines. The transformation from the old to the new us use Ep (Mp) to denote the stabilizer for a hexagonal set of stabilizers reads Sp ¼ Ep for p ∈ H and (triangular) plaquette p. For hexagonal plaquettes, we use the † † †  † convention that E ¼ X XsX XuX Xw, where r refers to M E −1 if p ∈ Im φ p r t v p φ ðpÞ ð Þ the top-right corner and the other corners are labeled in an Rp ¼ ð11Þ M if p∉ImðφÞ anticlockwise fashion. For triangular plaquettes, we use p M ¼ Z Z Z for all triangles of the form Δ and M ¼ p a b c p for p ∈ TQ. Here, ImðQφÞ denotes the image of the map φ. † † † ∇ 1 ZaZbZc for all triangles of the form . The stabilizer Since p∈HSp ¼ p∈TRp ¼ , the charges detected by operators Ep and Mp are unitary operators with eigenvalues the new stabilizers are separately conserved (modulo 4). k ω , k ∈ f0; 1; 2; 3g, where here and in the following, ω ¼ i Just like the ψ g anyons detected by the Sp stabilizers, the Rg g for d ¼ 4. An eigenvalue ω of the EP corresponds to a charges detected by the Rp stabilizers also form an anyon charge anyon of the form eg, while MP similarly detects flux model obeying Z4 fusion. However, while these two anyon anyons mh. Fusion of charge anyons forms a representation models have the same fusion rules, they are not equivalent, of Z4, as does that of fluxes. The convention for the stabilizer as they exhibit different braiding behavior. A full clockwise 2gh operators chosen before ensures that the anyonic charge of monodromy of a ψ g around a ψ h gives a phase of ω ,a both charge- and flux-type anyons is independently con- monodromy of a rg around an rh gives a phase of 1, and a served (modulo 4). A full clockwise monodromy of an e gh g monodromy of a ψ g around a rh gives a phase of ω (see around an m , or vice versa, yields a phase ωgh (see Fig. 3 for h again Fig. 3). Just like the eg and mh charges, the ψ g and rh illustration). particles correspond to a way of decomposing the DðZ4Þ Just as Majorana modes (Ising anyons) in the qubit toric model into two submodels that are closed under fusion but code [14,38], parafermions appear in our system at the have nontrivial mutual braiding behavior, ends of defect strings. For the interpretation in terms of parafermions, it will be useful to introduce a new set of DðZ4Þ¼fe0;e1;e2;e3g × fm0;m1;m2;m3g composite anyons defined as ψ ¼ e × m . These also g g g ψ ψ ψ ψ obey Z4 fusion with each other, and their braiding behavior ¼f 0; 1; 2; 3g × fr0;r1;r2;r3g; ð12Þ can be inferred from the behavior of the constituent charge where the three particle models other than ψ 0; ψ 1; ψ 2; ψ 3 and flux particles. The particles fψ 0; ψ 1; ψ 2; ψ 3g form a f g Z chiral Abelian anyon model with Chern number ν ¼ 2 [12]. correspond to the simple 4 model. ψ Note that The stabilizer operators Sp detect the presence of g anyons which are pinned to a pentagon-shaped double plaquette, made up of a neighboring pair of triangular and e × m ¼ ψ × m − : ð10Þ g h g h g hexagonal plaquettes. These anyons can be regarded as generalizations of Dirac fermions to the group Z4 (rather We now perform a local transformation from the set of than Z2). Just as Dirac modes can be decomposed into two E ;M stabilizer generators f p pg, detecting the charges on the Majorana modes, so too can the ψ modes be decomposed S ;R left-hand side of Eq. (10), to a new set f p pg which into two parafermion modes. The two parafermion modes detects the two charges on the right-hand side. Let H Pa and Pb are therefore associated with each double denote the set of hexagonal plaquettes and T the set of plaquette P. These are described using parafermion oper- 2 triangular plaquettes. Note that jTj¼ jHj. Consider an ators satisfying Eq. (2). The parity operator for the ψ mode injective map φ: H → T, which to each hexagonal operator 1 2 † associated with a pair ðj; kÞ is defined as ωðdþ Þ= γ γ for E , assigns one of the six adjacent triangular operators j k p ω5=2γ γ† j

041040-4 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015) additional single-qudit terms are added to the Hamiltonian, means that the EP and MP operators for the double plaquettes of one of the two following forms: along these lines no longer commute with the Hamiltonian and so can no longer be used as stabilizer generators. 1 Their pentagon-shaped product R is used instead. The Y þ H:c: ¼ − pffiffiffi ðσy þ σy þ σxσz þ σzσxÞ; P 2 1 2 1 2 1 2 pentagons in Fig. 4 show how, next to a defect line, the 1 mapping φ needs to pick the bottom-left triangular-shaped ω5=2XZ† þ H:c: ¼ pffiffiffi ðσy − σy þ σxσz − σzσxÞ: ð13Þ stabilizer next to a hexagon-shaped stabilizer to ensure 2 1 2 1 2 1 2 that their product still commutes with the Hamiltonian. This change of the stabilizer generators of the code has a Specific examples are shown in Fig. 4. drastic effect. Consider an e anyon that has moved towards The single-qudit terms added along defect lines are much g a point along a defect line from one direction and an m that stronger than any other interactions, and thus effectively g has moved towards the same point from the other direction. remove the qudits on which they act from the code. This Both of these are detected by RP-type stabilizers. When they meet on the same double plaquette, they will fuse to form a ψ g, and thus, they will not be detected by the RP stabilizers anymore. In fact, since the SP stabilizer is removed for double plaquettes along a defect line, they will not be detected by any stabilizer operator. The ψ g occupancy of a defect line corresponds to an increased ground-state degeneracy of the system, referred to as a synthetic topological degeneracy [38]. In the following section, the fψ g;rhg decomposition of the DðZ4Þ model will prove more convenient than the feg;mhg decomposition. A process in which a defect line converts an mg into an e−g can equivalently be described as one in which a rg passes a defect line that emits a ψ −g.

V. PARAFERMIONS AS NON-ABELIAN ANYONS Since unpaired parafermion modes reside at the end- points of defect strings, it is natural to use them to explain the properties of the modified stabilizer. Parafermion modes are described by a non-Abelian anyon model with particle species fψ 0; ψ 1; ψ 2; ψ 3; σg. Here, ψ 0 ≡ 1 corre- sponds to the anyonic vacuum and σ is an unpaired parafermion mode. The fusion rules of this anyon model are

σ × σ ¼ ψ 0 þ ψ 1 þ ψ 2 þ ψ 3;

ψ g × ψ h ¼ ψ g ⊕ h;

ψ g × σ ¼ σ; ð14Þ

where ⊕ denotes addition modulo 4. A pair of parafer- mions (or the defect line between them) may therefore collectively hold any of the four types of ψ anyons. The anyon model with the fusion rules given in Eq. (14) does not allow for a nontrivial solution of the pentagon and FIG. 4. Strings of alternating single-qudit operators of the form hexagon equations. As such, it obeys only projective non- ω5=2 ω5=2 † Abelian statistics. The computational power of braiding Yi þ H:c: or XiZi þ H:c: (encircled) are added to the Hamiltonian. These effectively eliminate the qudits on which they parafermions has recently been studied in Ref. [55]. In our act from the code, leading to enlarged, pentagon-shaped stabi- setup, we cannot only braid the parafermions with each lizers along the defect string. Parafermion modes reside on the other, but we can also braid the Abelian e and m particles pentagons at the ends of the defect strings (shaded areas). around them, which provides the possibility to perform ~ A possible choice for two logical operators XL (top panel) and additional gates. In the following, we study the gate set that ~ ~ ~ ~ ~ ZL (bottom panel) satisfying ZLXL ¼ ωXLZL is illustrated. can be generated this way.

041040-5 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015)

As for Majorana fermions (Ising anyons), we use four parafermion modes (two defect strings) for which the total fusion sector is vacuum to store one logical qudit. The natural logical operators are parity operators for the pairs g of parafermions. An eigenvalue ω corresponds to a ψ g occupancy for the pair, and so the specific result is σ × σ ¼ ψ g if they are fused. We associate the Z basis of the logical qudit with the ψ occupancy of vertical pairs (connected by defect lines). Specific choices of logical operators are illustrated in FIG. 5. All generators of the single-qudit Clifford group can be ~ performed by braiding quasi-particles. The four circles corre- Fig. 4. The ZL corresponds to a clockwise loop of an e1 spond to the four parafermion modes which are used to store one around a defect line. The braiding of this e1 around the ψ g logical qudit. The left part of the figure illustrates how to perform ωg ~ held in the pair yields the required phase of . The XL the logical operators X and Z by braiding the Abelian excitations corresponds to a clockwise loop of an e−1 anyon, which is of the DðZ4Þ model around the parafermions. The right part converted to an m1 through one defect line and back to an demonstrates a logical Hadamard gate H, which is performed by e−1 through the other. Equivalently, we can describe it as a braiding the parafermion modes themselves. clockwise loop of an r1 and a transfer of a ψ 1 from the right to the left defect line. For a logical qudit stored in four parafermion modes, Let us denote a state in which the left defect line holds a let S denote a clockwise exchange of a vertical pair of mode ψ and the right defect line holds a mode ψ by g h parafermion modes and T an exchange of a horizontal pair jψ ; ψ i. Two defect lines create a 4 × 4-fold synthetic P 2 g h ωg =2 topological degeneracy. For computational purposes, we (see Fig. 6). As discussed, we have S ¼ g jgihgj, restrict ourselves to the four-dimensional subspace of states while T is diagonal in the logical X basis. In the logical Z ≡ ψ ψ basis, T reads (for d ¼ 4) of the form jgiL j g; −gi. This is the set of states which can locally be created from the anyonic vacuum. The effect X 1 2 ~ ⊕ 1 T e−iπ=4 eiðπ=4Þðg−hÞ g h : of the logical operators on these states is XLjgiL ¼jg iL ¼ 2 j ih j ð15Þ and Z~ jgi ¼ ωgjgi . gh L L L ~ ~ In addition to the logical operators XL and ZL, which can be performed in our model by braiding the Abelian DðZ4Þ Again, in contrast to Majorana fermions, parafermions anyons around the parafermion modes (ends of defect support an entangling gate between two logical qudits by strings), we can perform further topologically protected braiding operations [21]. The controlled phase gate Λ is single-qudit and two-qudit gates by braiding the parafer- defined by its action on a logical two-qudit basis state, mion modes themselves. Defect lines used for braiding are Λjg; hi¼ωghjg; hi. In our parafermion scheme, an entan- shown in Appendix B. Crucially, braiding parafermions gling gate can be performed by braiding of a pair of allows one to perform an entangling gate by topological parafermions from one qudit with a pair from the other. Let means, which is in contrast to Majorana fermions [21]. us consider, for example, the braiding of the left vertical What is more, exploiting the fact that our non-Abelian pair for both qudits. For an initial logical product state Z system is built on top of an Abelian Dð 4Þ system allows jg; hi, the process corresponds to braiding a ψ g clockwise us to generate the entire four-level Clifford group by 2gh around a ψ h, which yields a phase of ω . The resulting braiding quasiparticles, as we discuss in the following. operation is therefore the squared controlled phase gate Λ2. For the rest of this section, X and Z refer to the logical 2 Λ2 1 ~ ~ For d ¼ , corresponding to Majorana fermions, ¼ , operators, which we called XL and ZL before, respectively. and so this operation is trivial. For d>2,however,itisa The first column in Fig. 5 illustrates how braiding of DðZ4Þ charges and fluxes can be used to perform logical X and Z gates. Whether an e1 or an m1 anyon is used to perform the logical Z gate is irrelevant. Consider two parafermion modes storing a ψ g particle. A full clockwise monodromy of one parafermion around the other can be understood as a monodromy of the g2 constituent eg around the mg, yielding an ω phase. We can thus expect a single exchange of the two parafermion modes storing a ψ to yield a square root of this phase, such g FIG. 6. Generators S and T of all gates that can be performed on 2 2 as ωg = . This is demonstrated directly by studying the a qudit stored in the fusion space of four parafermions by braiding necessary microscopic operations in Appendix C. them.

041040-6 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015)

along these lines would then be suppressed. By adiabati- cally removing the defect line that delocalizes ψ modes, its ψ g anyon occupation would be transferred to these two lines. The recombination of the defect line would be done by the reverse process. For a tensor product of d-level systems, the Clifford group Cd consists of gates that map tensor products of d-level Pauli operators to other such tensor products under conjugation. In Appendix D, we prove the following theorem. Theorem: The single-qudit gates S, T,andZ, and nearest-neighbor controlled phase gates Λ generate the entire Clifford group C4. † FIG. 7. Performance of a controlled phase gate. Panel (a) shows As an example, H~ ¼ STS ¼ TST satisfies HX~ H~ ¼ Z † a logical product state jg; hi stored in the fusion space of eight and HZ~ H~ ¼ X†, so it can be identified with the logical ψ parafermions. The defect line storing a mode g can be split into Hadamard gate, up to a phase. Indeed, using the standard Z two defect lines storing Dð 4Þ charges eg and mg, respectively. definition Braiding both endpoints of one of these lines clockwise around the defect line storing the ψ h mode, as shown in (b), produces a 1 X phase ωgh, as required. H ¼ pffiffiffi ωghjgihhj; ð16Þ d gh pffiffiffiffi one verifies that ωH H~ . nontrivial entangling gate, akin to the one proposed ¼ One possible implementation of H (up to a phase) is a in Ref. [21]. cyclic permutation of the four parafermion modes, as in Clearly, a more powerful entangling gate would be Λ Fig. 5. This can be pictorially understood as follows. An X itself. This can be achieved for Zd parafermions for odd d corresponds to a transfer of a ψ 1 from the right to the left by taking the ðd þ 1Þ=2-th power of Λ2. However, these do defect line, accompanied by a clockwise loop of an r1 not admit the simple decomposition into qubits that we around a horizontal pair. A Z corresponds to a clockwise have used in defining the model. Fortunately, we can make loop of an r1 around a vertical pair. A π=2 rotation as use of the underlying charge and flux anyons to realize Λ performed by H thus maps these two operations onto each despite the even qudit dimension. other, up to the fact that we do not perform a vertical ψ 1 The defect line may be interpreted as a hole for ψ-type transfer, as the ψ occupancy of the vertical pair is anyons: an area in which they may be placed such that their delocalized along the defect line. state becomes delocalized along the line and they are no longer detected by the stabilizers [56–58]. Similar holes can also be engineered for the constituent charge and flux VI. ERROR CORRECTION anyons. A defect line is therefore a special case of the For any system with a finite energy gap at finite combination of a charge and flux hole, in which only temperature, excitations will appear with a finite density. ψ g ¼ eg × mg-type anyons may reside rather than general This corresponds to a finite length scale on which quantum eg × mh anyons. Nevertheless, we can consider a process in computation can be performed before errors are almost which a defect line is transformed into a separate charge certain to appear. This length scale can be increased by and flux hole. Details on these holes and the transforma- increasing the gap or lowering the temperature. However, tions between them can be found in Appendix E. neither of these methods is truly scalable. Error correction When only the charge hole of one qubit is braided around is therefore required if scalable quantum computation is to the defect line of another, the process for an initial state be performed. ψ jg; hi corresponds to braiding an eg around a h, which For non-Abelian systems, the first studies of the corre- would yield the phase ωgh. The charge and flux holes can sponding error-correction problem have recently appeared then be recombined into a defect line. The net effect of the [4,5,39–42]. Error correction for non-Abelian anyons is entire process is to apply the controlled phase gate Λ. Such still poorly understood, and its feasibility has not been a process is illustrated in Fig. 7. demonstrated for the (realistic) time-continuous case. It One process that could split the defect line in this way is comes thus very welcome that while our system provides simply to intersect it with two others: One would be a line the computational power of non-Abelian parafermions, its along which charge anyons are hopped by high-strength physical excitations are still Abelian DðZ4Þ anyons, and the terms; the other would similarly hop flux anyons. The error-correction problem for DðZnÞ quantum double mod- stabilizers that detect charges and fluxes, respectively, els (including the time-continuous case) is well studied

041040-7 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015)

[39,40,59–61]. However, when correcting these DðZ4Þ TABLE I. Conversion from single-qubit Pauli operators to anyons, we face a number of difficulties not considered four-dimensional generalized Pauli operators. When a syndrome in previous studies [39,59–61]. measurement is performed, a Pauli operator is converted to each (i) Our stabilizer operators are products of Z4 qudit of the generalized Pauli operators in the right-hand column with operators X, X†, Z, Z†, while an error model is equal probability. x x † 2 † 2 realistically expressed in terms of single-qubit oper- σ1, σ2 X, X , XZ , X Z σx σy σz ators , , . These do not map eigenstates of the σy, σy XZ, X†Z, XZ†, X†Z† stabilizer operators to other eigenstates, and one 1 2 σz , σz † single-qubit operator can produce a product of up to 1 2 Z, Z three qudit operators (see below). (ii) We consider quantum information stored in a syn- operators a certain single-qubit Pauli operator will translate thetic topological degeneracy, which involves a with equal probability. defect line allowing anyons to change from one As a first simple error model, which does not involve a sublattice to the other (stars to hexagons and vice notion of Hamiltonian protection, we consider depolarizing versa). Thus, we cannot decode each sublattice noise. In other words, for each qubit of the code, we apply a separately, as is usually done for the toric code Pauli operator with some probability p (the depolarization Z and other Dð dÞ quantum double models. Instead, rate), where each of the three Pauli operators is chosen with we have to correct both of them simultaneously, equal probability. taking into account the possibility of transferring More interesting from a physical perspective is a thermal anyons from one to the other. error model. We consider a quantum state stored in the (iii) Besides simplistic i.i.d. error models (such as degenerate ground states of the Hamiltonian given in depolarizing noise), we are particularly interested Eq. (9) and assume that the system is weakly coupled to in Hamiltonian protection of a quantum state subject a heat bath at some temperature T. Following, e.g., Ref. [5], to thermal errors. we assume that evolving the system according to the (iv) We do not consider a square lattice but a trihex- Metropolis algorithm provides a reasonable approximation agonal one, which makes moving anyons and of the thermalization process since the evolution obtained defining their distance more involved. by means of the Metropolis algorithm is local and Markovian and has the thermal state as its unique fixed A. Error model point. Since our four-level qudits are composed of two qubits, it During our simulation, we proceed as follows. We first 1 is natural to consider an error model in terms of single-qubit pick a spin-2 of the system at random; we then pick one operations σx, σy, and σz. For a qudit hosted in qubits 1 of the three single-qubit operators acting on that qubit at and 2, single-qubit Pauli operators can be expressed in random and convert it to a four-dimensional generalized terms of Z4 operators by inverting Eq. (3). We find Pauli operator according to Table I. We then calculate the energy cost Δ of applying that generalized Pauli operator 1 tot σx 1 − 2 (or products thereof). This energy cost is of the form 1 ¼ 2 Xð Z ÞþH:c:; 1 Δ ¼ mΔΔ þ nΔ⎔; ð18Þ σx 1 2 tot 2 ¼ 2 Xð þ Z ÞþH:c:; 1 where ΔΔ and Δ⎔ are the energy costs of creating a single σy i5π=4 1 2 1 ¼ 2 e Yð þ Z ÞþH:c:; triangle- or hexagon-type anyon with charge 1 or 3 in Δ 2 1 Eq. (9), respectively. (In other words, Δ ¼ J and σy i3π=4 1 − 2 Δ 2 63 8 6 2 5 2 ¼ 2 e Yð Z ÞþH:c:; ⎔ ¼ ð = Þ½h =ð JÞ .) Creating an anyon with charge 2 will have an energy cost of 2ΔΔ or 2Δ⎔. The coefficients σz −iπ=4 1 ¼ e Z þ H:c:; m and n are elements of f0; 2; 4g, depending on the z iπ=4 σ2 ¼ e Z þ H:c: ð17Þ change in anyonic charge. The proposed error is then −Δ accepted with probability minf1;e tot=kBTg. The noise If we start from an eigenstate of all stabilizer operators, model we apply is thus the standard classical Metropolis applying single-qubit Pauli operators will generate a super- algorithm that maps eigenstates of Eq. (9) to other position of states corresponding to different syndrome eigenstates. At any given time during our simulation, the outcomes. By measuring all stabilizer operators, we can system is “classical” in the sense that it does not involve project again into a subspace with definite syndrome superpositions of different anyon configurations. values. Each single-qubit Pauli operator thereby translates If the proposed error is accepted, we copy the current into a product of up to three qudit operators. Table I state of the system and try to correct it. If correction summarizes (up to irrelevant phases) into which qudit is successful, we continue our simulation with the

041040-8 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015) uncorrected version of the system. If correction fails (for at least one logical operator), we interpret this as the quantum information having survived for a time which is given by the number of Metropolis steps divided by the number of spins in the system. The thermal error model has three relevant energy scales kBT, Δ⎔, and ΔΔ. Since Δ⎔ appears in higher- order perturbation theory than ΔΔ, we expect ΔΔ > Δ⎔. Furthermore, effective protection requires kBT<Δ⎔, ΔΔ. We introduce a parameter λ which quantifies the separation of these three energy scales, i.e., Δ⎔ ¼ λkBT and 2 ΔΔ ¼ λ kBT. Very high values of λ are uninteresting since they exponentially suppress errors from occurring.

B. Without defects If there are no defect lines present, the anyonic charge of both types of anyons is conserved (modulo 4), and they can be corrected separately. Various techniques have been developed for correcting general DðZnÞ quantum double models [39,59–61]. However, correcting the DðZ4Þ case is particularly easy since we can exploit the relation ~ ~ Z4=Z2 ≃ Z2. Specifically, we can first fuse all oddly FIG. 8. Error rates pL of the logical operators X1 and Z1 charged anyons in pairs. In a second round, the remaining illustrated in Fig. 2 as a function of the qubit depolarization rate p 20 anyons, which are all of charge 2, are fused in pairs. In for code sizes L ¼ , 28, 36, 44, 52. Each data point represents 104 logical errors, such that error bars are negligible. We order to find these pairings, we use the library Blossom V ~ recognize a threshold error rate pc ≈ 24% for X1 and pc ≈ 10% [62], which is the latest implementation of the efficient ~ for Z1. minimum-weight perfect-matching algorithm from Edmonds [63]. The weight between two equal-type anyons is thereby defined as the minimal number of generalized decrease to an asymptotic value for large L and consid- Pauli operators that need to be applied to create a pair of erable finite-size tails. These tails correspond to the regime anyons at the two given locations. in which the breakdown of error correction is not due to the Figure 8 shows our results for the depolarizing noise density of anyons becoming so high that pairing them ~ model, i.e., the logical error rates of the logical operators X1 becomes ambiguous, but where the breakdown is caused ~ and Z1 illustrated in Fig. 2 as a function of the depolari- by one of the first pairs wandering along a topologically zation rate p. One clearly recognizes threshold error rates nontrivial path around the torus. The smaller the system, pc ≈ 24% and pc ≈ 10%, respectively. The equivalent ~ ~ figures for the logical operators Z2 and X2 look very similar and yield equivalent threshold error rates pc. These thresholds are best compared with those for an equivalent code based on Z2 anyons, with only a single qubit on each vertex. For independent bit and phase ~ ~ flips, the thresholds for X1 and Z1 are pc ≈ 16.4% and pc ≈ 6.7%, respectively [64,65]. When the hexagonal and triangular plaquettes are decoded separately, these corre- spond to thresholds of pc ≈ 24.6% and pc ≈ 10.5% for depolarizing noise. The similarity of these Z2 values with those of Z4 is remarkable. This qudit code is therefore just as adept at suppressing qubit noise as its qubit counterpart. It is well known that the finite-temperature lifetime of a FIG. 9. Average lifetimes τ of the logical qudit with logical two-dimensional quantum memory with local interactions ~ ~ operators X1 and Z1 as a function of code size L for λ ¼ 1,2,3,4, only is upper bounded by a constant independent of the 5. Each data point represents 104 experiments. The lifetime is system size (see, e.g., Ref. [66]). Figure 9 shows the defined as the number of Metropolis steps until the first logical lifetime of a logical qudit with logical operators X1 and Z1 operator detects an error, divided by the number of spins in the subject to the thermal error model. We notice lifetimes that code.

041040-9 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015) the longer it takes to produce an anyon pair, leading to the observed tails for small enough L and T (large enough λ).

C. With defects When defect lines as in Fig. 4 are present, the error- correction problem becomes more involved. It is no longer possible to correct the two anyon types (hexagons and triangles in our case) separately, as is usually done for the DðZnÞ models [39,59–61]. Instead, error correction needs to take into account the possibility of converting between different anyon types. We thus pair all oddly charged FIG. 11. Average lifetimes τ of the logical qudit stored in the anyons of both types in a first round and all remaining defect logical operators X~ and Z~ illustrated in Fig. 4 as a function charge-2 anyons of both types in a second round. Pairings 4 of code size L for λ ¼ 1, 2, 3, 4, 5. Each data point represents 10 can involve anyons that are the same or of different types. experiments. The lifetime is defined as the number of Metropolis The weight for connecting two anyons is defined as the steps divided by the number of spins in the code. minimal number of generalized Pauli operators needed to create a pair of anyons at their respective positions from the 2 1 vacuum. For equal-type anyons, this will be an error string lines involving L= þ qudits, as shown in Fig. 4 for 20 ~ ~ that crosses an even number of defect lines, while for L ¼ . The logical operators XL and ZL then have a different-type anyons this will be an error string that crosses distance L þ 2 and L=2 þ 4, respectively. an odd number of defect lines. This result can mean, for For the depolarizing error model, we find the threshold ~ ~ instance, that connecting two equal-type anyons can have a error rates pc for both of the logical operators XL and ZL large weight despite them being geometrically nearby, if given in Fig. 4. The results are given in Fig. 10. For the ~ there is a defect line between them. defect operator XL, we find a threshold error rate pc ≈ 24%, ~ ~ For a code of linear size L in both dimensions, with as for the operators X1 and X2 in the defect-free case periodic boundary conditions and L even, we choose defect ~ (Figs. 2 and 8), while for the defect operator ZL, we find a ~ threshold error rate pc ≈ 10%, as for the operators Z1 and ~ Z2 in the defect-free case. The fact that these values coincide with the defect-free case is not unexpected. The introduction of the defects essentially corresponds to a change in the boundary conditions. However, the vast majority of errors have large support within the bulk. The value of the threshold is therefore dominated by bulk effects rather than boundary effects. Figure 11 shows the average lifetime of the qudit stored in the synthetic topological degeneracy in Fig. 4 for the thermal error model. We note that for a given parameter λ, the asymptotic lifetimes (L → ∞) are close to those in the defect-free case given in Fig. 9.

VII. CONCLUSIONS We have proposed a system which, on the physical level, involves only nearest-neighbor two-qubit interactions, allows one to perform all Clifford gates through quasipar- ticle braiding, and has a well-understood error-correction problem. ~ ~ FIG. 10. Error rates pL of the logical operators XL and ZL We have greatly benefited from the fact that our non- illustrated in Fig. 4 as a function of the qubit depolarization rate p Abelian system is built on top of a system whose excita- for code sizes L ¼ 20, 28, 36, 44, 52. Each data point represents tions correspond to an Abelian anyon model. This allows 104 logical errors, such that error bars are negligible. We us to perform the logical operators X, Z, and Λ through ~ recognize a threshold error rate pc ≈ 24% for XL and pc ≈ 10% quasiparticle braiding. It also makes our error-correction ~ for ZL. problem manageable, despite some subtleties such as the

041040-10 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015)

ˆ ˆ fact that single-spin Pauli operators generate superpositions Pdefine the superoperator A via AðOÞ¼½A; O. Let H0 ¼ between different syndrome outcomes and the ability to iEijiihij be the spectral decomposition of H0 and define convert between different anyon species during error the superoperator L via correction. X Universal quantum computation requires the ability to hijQOPjji “ ” LðOÞ¼ jiihjj − H:c: ðA2Þ perform non-Clifford gates, such as small-angle uni- E − E taries. While it is not difficult to perform a non-Clifford i;j i j operation by nontopological means in our system, this abandons fault tolerance. The technique of magic state We employ the convention that unless indicated otherwise L distillation [67] is typically used to restore fault tolerance. by use of brackets, a superoperator acts on all operators While research on magic state distillation has so far focused to its right. on prime qudit dimensions d [68,69], qudit codes with For the sixth-order effective Hamiltonian, one derives the right transversality properties to perform magic state from Ref. [75] the expression 4 1 1 distillation in nonprime dimensions, including d ¼ , also ð6Þ ˆ ˆ 2 ˆ ˆ ˆ ˆ ˆ ˆ 2 H PS5 V P − P S S3 S1S3S1 S3S exist [70]. Unfortunately, for nonprime d it is not known eff ¼ 2 ð odÞ 24 ð 1 þ þ 1 whether Clifford gates plus an arbitrary non-Clifford gate 1 ˆ 2 ˆ ˆ ˆ ˆ ˆ ˆ 2 ˆ 5 are sufficient to achieve universality [71]. It is our hope that þ S2S1 þ S2S1S2 þ S1S2ÞðVodÞP þ 240 PS1ðVodÞP; our work fuels interest in the d ¼ 4 case, which is a power of 2 and thus allows one to employ qubits, as demonstrated ðA3Þ in our work, while it is the smallest power of 2 that allows one to go beyond Majorana fermions. where Alternatively, one could imagine energetically penaliz- L ing one of the degrees of freedom of a two-qubit Hilbert S1 ¼ ðVodÞ; space to obtain a synthetic qutrit (d ¼ 3). Magic state −L ˆ S2 ¼ VdðS1Þ; distillation for qutrits is well studied [72], potentially 1 allowing one to perform fault-tolerant universal quantum −L ˆ Lˆ 3 S3 ¼ VdðS2Þþ3 S1ðVodÞ; computation with Z3 parafermions in a qubit system [73]. 1 −L ˆ L ˆ ˆ ˆ ˆ S4 ¼ VdðS3Þþ ðS1S2 þ S2S1ÞðVodÞ; ACKNOWLEDGMENTS 3 1 −L ˆ ˆ 2 ˆ ˆ ˆ ˆ The authors thank M. Barkeshli for elaborations on the S5 ¼ VdðS4Þþ3 ðS2 þ S1S3 þ S3S1ÞðVodÞ development of the idea of generating non-Abelian defects 1 in topological systems. This work was supported by the − Lˆ 4 45 S1ðVodÞ: ðA4Þ SNF, NCCR QSIT, and IARPA. In our case, the low-energy subspace onto which P APPENDIX A: SIXTH-ORDER DEGENERATE projects is given by the space in which all triangle operators PERTURBATION THEORY in Eq. (8) have minimal energy, i.e., ZaZbZc ≡ 1 for all triangles a; b; c . This subspace is fully degenerate. The For our perturbation theory, we employ a Schrieffer- ð Þ Wolff transformation [74], as formalized in Ref. [75]. lowest-energetic excitations change the eigenvalue of a stabilizer Z Z Z from 1 to i. Since the eigenvalue of Consider an unperturbed Hamiltonian H0 whose spec- a b c −ðZaZbZc þ H:c:Þ is thereby changed from −2 to 0, this trum can be separated into a low- and a high-energy Δ 2 subspace, which are energetically separated by a gap. has an energy cost of ¼ J. Note, however, that stabilizer Given a perturbation V, we want to find an effective eigenvalues can only be changed in pairs, such that the “ ” gap between the low-energetic (ground-state) subspace and Hamiltonian Heff describing the effective physics on 2Δ the low-energy subspace. The effective Hamiltonian can be the space of excited states is in fact given by . developed in a perturbative series A crucial property of our Hamiltonian is that there is no lower-than-sixth-order perturbation that acts within the ground-state space. Therefore, we are only interested in ð0Þ ð1Þ ð2Þ 4 Heff ¼ Heff þ Heff þ Heff þ ðA1Þ terms of the form PVodðVdÞ VodP, which allows us to greatly simplify the effective Hamiltonian. Namely, only in powers of some small expansion parameter. the first summand in all expressions in Eqs. (A3) and (A4) Let P denote the projector onto the low-energy is relevant in our case. We find subspace and Q ¼ 1 − P the projector onto the high- 6 1 energy subspace. We define Vd ¼ PVP þ QVQ and Hð Þ P LVˆ 4 LV ;V P: − eff ¼ 2 ½ð dÞ ð odÞ od ðA5Þ Vod ¼ V Vd ¼ PVQ þ QVP. For some operator A,we

041040-11 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015)

TABLE II. Possible routes the excitation energy above the ground state can take (left column), together with their respective multiplicities (right column). Note that the number of multiplic- ities adds up to 6! ¼ 720.

0 → 2Δ → 2Δ → 2Δ → 2Δ → 2Δ → 0 96 0 → 2Δ → 4Δ → 2Δ → 2Δ → 2Δ → 0 48 0 → 2Δ → 2Δ → 4Δ → 2Δ → 2Δ → 0 48 0 → 2Δ → 2Δ → 2Δ → 4Δ → 2Δ → 0 48 0 → 2Δ → 4Δ → 4Δ → 2Δ → 2Δ → 0 96 0 → 2Δ → 2Δ → 4Δ → 4Δ → 2Δ → 0 96 0 → 2Δ → 4Δ → 4Δ → 4Δ → 2Δ → 0 192 0 → 2Δ → 4Δ → 2Δ → 4Δ → 2Δ → 0 24 0 → 2Δ → 4Δ → 6Δ → 4Δ → 2Δ → 0 72

0 Now using the fact that, in our case, VdP ¼ , this can be further simplified to FIG. 12. A selection of double plaquettes used in a braiding operation. Each double plaquette corresponds to two parafermion 6 1 modes. For a double plaquette P, the parafermion to the right is Hð Þ ¼ PLðLðLðLðLðV ÞV ÞV ÞV ÞV ÞV P eff 2 od d d d d od labeled P1 and that to the left is P2. 1 − L 4 L 2 PVodð VdÞ ð VodÞP − L 4 L ¼ PVodð VdÞ ð VodÞP: ðA6Þ each double plaquette, we use light blue circles. The one to the right of a double plaquette P is labeled P1, and that to There are 6! ¼ 720 possibilities for applying the six the left is P2. † † † ψ factors XrXsXtXuXvXw around one hexagon, which leads Parity operators for modes are defined on pairs of the system back to the ground state. Table II lists all parafermion modes. We are primarily concerned with two possible routes the excitation energy above the ground state types of pairing: those of the two modes within the same can take, together with their numbers of possibilities. double plaquette, and those of two modes from neighboring In conclusion, we find the sixth-order effective double plaquettes. Relevant examples of the latter type are Hamiltonian shown in Fig. 12 by red, orange, and green lines connecting the corresponding modes. 6 h † † † For the two modes within each double plaquette, the − 5 2 † Heff ¼ q 5 ðXrXsXtXuXvXw þ H:c:Þ; ðA7Þ ω = γ γ Δ parity operator P1 P2 corresponds to the stabilizer SP. The orange and red lines connecting modes P2 to ðP þ 1Þ1 † where the dimensionless prefactor denote the parity operators ω5=2γ γ . For orange lines, P2 ðPþ1Þ1 96 48 48 48 96 96 these correspond to the operator Y on the vertex through which the line passes. For red lines, they correspond to the q ¼ 32 þ 64 þ 64 þ 64 þ 128 þ 128 operator X†Z. 192 24 72 þ þ þ Consider a state initially within the stabilizer space. The 256 128 384 parity operators for the pairs of parafermion modes within 63 each double plaquette are therefore part of the stabilizer. ¼ ðA8Þ 8 Thus, the state corresponds to this definite pairing of the modes. is given by the multiplicities in Table II, divided by the Let us now consider the removal of the operator SA from product of all excitation energies (in multiples of Δ) along the set of stabilizer generators (while RA remains). The the virtual process. corresponding parafermion modes are now, in some sense, unpaired. This contributes a factor of 4 to the ground-state degeneracy [76]. However, due to the fact that the APPENDIX B: MOVING UNPAIRED “unpaired” parafermions are not well separated, it is not PARAFERMION MODES difficult for local perturbations to lift the degeneracy of To consider the creation and braiding of unpaired this space. To become truly unpaired, and benefit from parafermionic modes, we must first decide on the double topological protection, they must be separated. plaquettes with which we will work. Let us consider those To do this, we can add a term KðY þ Y†Þ to the of Fig. 12. To visualize the two parafermion modes within Hamiltonian, which corresponds to the parity operator

041040-12 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015)

ω5=2γ γ† APPENDIX C: EXCHANGE OF TWO A2 B1 . This acts on the vertex through which the orange line connecting these modes passes. PARAFERMION MODES ≫ For K J, this new term will overwhelm the SB term. To determine the effects of a single clockwise exchange, The pairing of B1 and B2 will then be broken, and B1 will we consider the smallest possible implementation. This become paired with A2 instead. The unpaired mode involves the double plaquettes labeled F, G, and P in originally at A2 is therefore effectively moved to B2.If Fig. 12, which are shown in more detail in Fig. 13.We the new term is introduced adiabatically, the degenerate consider a state in which the modes at P1 and F2 are subspace associated with the unpaired parafermion modes unpaired, and those at G1 and G2 are paired by the will remain in the same state during this process. Hamiltonian term SG. Using this, we determine the effects Similar processes can be used to move the unpaired † † of exchanging the unpaired modes. The method used to modes further. The Hamiltonian term KðX Z þ X ZÞ exchange the two modes is similar to previous methods ω5=2γ γ† corresponding to B2 A1 can then be used to move proposed in order to perform anyon braiding [52,53]. the parafermion at B2 to C2, for example. Unpaired The results of the exchange are most easily understood parafermion modes can therefore be separated by arbitrary in terms of the ψ mode formed by this pair. The parity distances, at the endpoints of lines on which single qudit operator Γ for this mode is an operator that creates a ψ 1, terms are added to the Hamiltonian. In terms of qubits, ψ −1 pair and places them in double plaquettes P and F, these correspond to two-body interactions between qubits respectively. Also, it must have eigenvalues of the form ωg, in the same site. so Γ4 ¼ 1. These conditions are satisfied by In order to unlock the potential of parafermions for ð1þ2aÞ=2 † quantum computation, it must be possible to braid the Γ ¼ ω Z1X2Z2Z3: ðC1Þ parafermion modes. Let us consider a specific example of this, using the system of Fig. 12. Consider an initial state We similarly require operations that can move parafermions between the relevant plaquettes. These correspond to the within the stabilizer space of all SP except A and L. At these green line between G and P and the red line between F and two points, we have the unpaired parafermion modes A1, G. These are A2, L1 and L2. Let us now consider operations such as those described above to move A1 and L2 away, beyond the ð1þ2bÞ=2 † ð1þ2cÞ=2 † Π ¼ ω X1X Z2Z3; Φ ¼ ω X Z1; ðC2Þ bottom of the figure. All four parafermion modes are then 2 1 well separated, so the ground-state degeneracy is topologi- respectively. In these relations, a, b, and c are all elements cal protected. of Z4. We now consider the exchange of A2 with L1. We do this We have freedom in choosing the values a, b, c ∈ Z4 for by first moving A2 to K2, then L1 to A2, and finally K2 these relations. The corresponding freedom also exists for to L1. The two modes have then swapped places. An all operators used to move parafermion modes, as well exchange of opposite chirality would correspond to first as the logical operators. The values used do not simply moving L1 to K1, and so on. Note that all modes are correspond to differences in a global phase. Instead, they kept well separated during the exchange, so topological determine which eigenspace of these operators has eigen- protection is always maintained. value ω0 ¼ 1, thus which one corresponds to the vacuum During the exchange, the movement of the modes is occupancy ψ 0 of the ψ mode. Therefore, these phases mostly achieved using Hamiltonian terms that correspond cannot be chosen entirely arbitrarily since the overall to the red and orange pairings in Fig. 12. These are all conservation constraint of ψ modes must be maintained. single qudit terms. At the junction, however, terms corre- However, since here we do not explicitly consider the sponding to the green pairings are used. We must therefore operations that placed unpaired parafermion modes at P1 consider these in detail. and F2, we can assume that their phases are chosen in a way For the pairing shown by the light green line, the parity ω5=2γ γ† ψ operator is P2 O1 . This has the effect of creating a g, ψ −g pair on the double plaquettes O and P. This requires † † the two-qudit operator X3X4Z4. For the dark green pairing, ω5=2γ γ† the P1 G2 parity operator similarly requires the three- 5=2 † qudit operator ω X1X2Z2Z3. These terms correspond to four- and six-body quasilocal interactions on the corre- sponding qubits, respectively. They can be realized by standard methods of perturbative gadgets. However, note that they need only be implemented while an exchange is FIG. 13. Double plaquettes used in the working example of a in progress. single exchange.

041040-13 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015) that maintains this conservation. We will therefore consider For an exchange operation that does have the required a free choice of a, b, and c. form, consider a ¼ 1 and b ¼ c ¼ 2. The phase assigned The first step in exchanging the parafermions is to move −g2=2 1=2 to ψ g is ω in this case, up to a global phase of ω . the one at P1 to G2. This is done by adiabatically changing 2 The required phases ωg =2 would therefore be obtained the Hamiltonian to one in which the term Π þ Π† is present from an anticlockwise exchange. and stronger than S . This causes the modes at G1 and P1 − 2 2 G The fact that we obtain the phase ω g = for a clockwise to pair, moving the mode once at P1 to G2. The mode at F2 2 exchange in this case means we would get ω−g for a full is then moved to that at P1 by adiabatically changing the † clockwise monodromy. This would seem to contradict the Hamiltonian to one in which the Φ þ Φ term is present and 2 † arguments of the main text, which predict a phase of ωg . stronger than S , and the Π þ Π term is removed, pairing G However, note that these phases differ only by a factor of F2 with G1. The mode at G2 is then moved to F2 by 2 ω2g ω2g Γ2 adiabatically removing the Π þ Π† term, thus allowing S ¼ and thus are equivalent up to a factor of . G Since such factors are to be expected for different choices to become dominant and G1 and G2 to pair. This process then results in the clockwise exchange of the modes. of a, b, and c, this monodromy does not contradict our expectations. The ambiguity in these phases reflects the fact The first step of this transformation takes a state that is 0 that the anyon model described by the fusion rules in initially in the ω eigenspace of S and projects it to one in G Eq. (14) obeys only projective non-Abelian statistics. the ω0 eigenspace of Π. The next step projects the state into ω0 Φ Note that the arguments above do not assume anything the eigenspace of . The final step projects back into the about the initial state of the exchanged parafermions. Only ω0 eigenspace of SG. The end effect is then PGPΦPΠPG. their initial positions and the operations used to move them ω0 Here, PG is the projector onto the eigenspace of SG, etc. are required. The effect of the braiding is expressed in terms The rightmost PG simply reflects the fact that the initial of Γ, the parity operator for their shared ψ mode, which can state lies within this eigenspace. be defined for any pair of parafermions. It therefore does Γ The parity operator for the pair of unpaired modes not matter what state the fusion space of the parafermions commutes with SG and so can be mutually diagonalized was initially in, and it does not matter whether or not they with the above operator. We can therefore interpret its exist at the end of the same defect line. The effect of the effects in terms of the phase factor assigned to each of the braiding is the same in all cases. possible ψ g eigenspaces of the ψ mode of the pair. When doing this, different values of a, b, and c will APPENDIX D: GENERATORS OF THE result in different operations. This may seem to contradict CLIFFORD GROUP the standard notion of a topological protected operation. Cd ⊗n However, these differences can be most easily understood For a tensor product of nd-level systems ð Þ , the P by considering movement of parafermion modes imple- Pauli group d is defined as the group generated by the generalized Pauli operators X and Z , and the Clifford mented by measurement rather than adiabatic Hamiltonian i i group C is defined as the normalizer of P in the unitary manipulation. This method forces pairing of parafermion d d group on ðCdÞ⊗n. In other words, elements in C map modes by measuring the occupancy of their corresponding d d ψ tensor products of -level Pauli operators to other such mode, thus forcing it to have a definite value. Ideally, this tensor products under conjugation. ψ measurement will give the vacuum result 0. The effect is We start with some general remarks on the action ⊗ ⊗ then the same as the adiabatic manipulation. If a different of C n on P n for d ¼ 4.Ford ¼ 4, an operator XaZb ψ d d g results, it must be removed by fusing it with the has eigenvalues f1g if a ¼ b ¼ 0, f1; −1g if both a unpaired parafermion mode being moved. The different and b are even and at least one of them is nonzero, values used for the phases when moving unpaired modes, fi1=2;i3=2;i5=2;i7=2g if both a and b are odd, and such as a, b, and c here, determine how the measurement f1;i;−1; −ig if a þ b is odd. Since the number of distinct results are interpreted in terms of ψ anyons and thus eigenvalues is preserved under conjugation, this implies P⊗n determine the net ψ g fused with the modes being moved. that the Pauli group d decays into distinct orbits when C⊗n As such, differences in the conventions used for an d acts on it by conjugation. This is in stark contrast to the exchange will change the resulting operation only by a case where d is an odd prime, which is studied in Ref. [77], factor of Γg, for some value of g that depends on a, b, and c. where there is only one nontrivial orbit. The orbit con- … For the standard convention used throughout this paper, taining the elements X1;Z1; ;Xn;Zn consists of elements of the form with a ¼ b ¼ c ¼ 2, the phase assigned to a ψ g occupation ωg2=2ωgðgþ1Þ by the exchange is . These are indeed all square kþp=2 a1 b1 an bn 2 ω Z1 X1 …Zn Xn ; ðD1Þ roots of ωg , as predicted in the main text. However, they ωg2=2 … ∈ Z are not of the elegant form that would be more where k; a1;b1; ;an;bn 4, at least one ofP the expo- … n conducive for the proofs of Appendix D. nents a1, b1, , an, bn is odd, and p ¼ i¼1 aibi

041040-14 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015) determines whether integer or half-integer powers of ω verifies that there are only 48 matrices M in the matrix ring appear as phases (we sometimes write ω for i to avoid Z2×2 satisfying the requirement det M ¼ 1 (mod 4). We 4 pffiffiffiffi pffiffiffiffi confusion with indices). † ω † ω † have SXS ¼  XZ and TZT ¼ ZX , such that The following proof is an adaption of the proof in 01 −11 Appendix A of Ref. [77], where it is shown that a certain set MðSÞ¼ 11and MðTÞ¼ 10. One can verify C⊗n of gates generate the Clifford group d for the case where by brute force that products of at most 9 factors MðSÞ and d is an odd prime. The general structure of our proof is MðTÞ generate all of the aforementioned 48 matrices. identical to the one in Ref. [77], while the generating set Finally, since XZX† ¼ ω¯ Z and ZXZ† ¼ ωX, we can and individual lemmas and their proofs are different. After generate arbitrary phases compatible with Eq. (D1) (for completion of this work, we became aware of the more n ¼ 1). □ general proof in Ref. [78]. We define the controlled Pauli operators Let us define the single-qudit unitaries X3 3 j 1 X CX ¼ jjihjj ⊗ X ðD7Þ ωjk H ¼ 2 jjihkj; ðD2Þ j¼0 j;k¼0 and X3 2 S ¼ ωj =2jjihjj; ðD3Þ X3 X3 0 j jk j¼ CZ ¼ jjihjj ⊗ Z ¼ ω jjihjj ⊗ jkihkj: ðD8Þ j¼0 j;k¼0 and Note that CZ has been called Λ in the main part of this X3 ↦ † 1 2 work. We write A U B as a shorthand for UAU ¼ B. −iπ=4 iðπ=4Þðj−kÞ T ¼ 2 e e jjihkj We have j;k 0 ffiffi ¼ ffiffi 0 p p 1 ↦ i 1 − i 1 Z1 CX Z1 B pffiffi pffiffi C 1 B 1 i 1 − i C X1 ↦C X1X2 ¼ B pffiffi pffiffi C: ðD4Þ X 2 @ − 1 1 A † i i Z2 ↦ Z1Z2 pffiffi pffiffi CX 1 − i 1 i ↦ X2 CX X2 ðD9Þ

pffiffi and Lemma 1: The gates S†, T†, Z†, X, X†, and iH can all be generated from S, T, and Z. Z1 ↦C Z1 proof.—As S8 T8 1,wehaveS† S7 and T† T7. Z p¼ffiffi ¼ ¼ ¼ † X1 ↦ X1Z2 In addition, iH STS TST and X H ZH CZ pffiffi pffiffi ¼ ¼ ¼ ¼ † 4 4 1 † 3 ↦ ð iHÞ Zð iHÞ. Finally, X ¼ Z ¼ ,soX ¼ X Z2 CZ Z2 and Z† ¼ Z3. ⊗ X2 ↦ Z1X2; ðD10Þ ∈ C n CZ If for some Clifford gate U d we have showing that C , C ∈ C⊗2. a0 b0 0 0 X Z 4 a1 b1 … an bn † α 1 1 … an bn UðZ1 X1 Zn Xn ÞU ¼ Z1 X1 Zn Xn ; ðD5Þ Lemma 3: The gate CX can be generated from S, T, and CZ. with jαj¼1, we write proof.—We note that

T 0 0 0 0 T † † † MðUÞða1;b1; …;an;bnÞ ¼ða1;b1; …;an;bnÞ : ðD6Þ HXH ¼ Z and HZH ¼ X : ðD11Þ

∈ Z2n×2n C⊗n The matrices MðUÞ d form a representation of d , Thus as MðUVÞ¼MðUÞMðVÞ. ffiffi ffiffi † p † p Lemma 2: The gates S, T, and Z generate the entire CX ¼ H2CZH2 ¼ð iH2Þ CZð iH2Þ; ðD12Þ single-qudit Clifford group C⊗1. 4   □ ⊗1 ac which, together with Lemma 1, completes the proof. proof.—For some U ∈ C , let MðUÞ¼ . 4 bd Let us define a more general controlled operator as Preserving the commutation relations of the single-qudit −st s t Pauli operators requires that ad − bc ¼ 1 (mod 4). One Cst ¼ S1 ðCXÞ ðCZÞ : ðD13Þ

041040-15 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015)

It acts by conjugation as Xn ðP; QÞ¼ aidi − bici: ðD19Þ i 1 ↦ ¼ Z1 Cst Z1 ↦ ωst=2 s t X1 Cst X1X2Z2 ⊗n Lemma 5: Given P, Q ∈ P4 with ðP; QÞ¼1, we can −s ⊗ Z2 ↦ Z Z2 ∈ C n Cst 1 generate U 4 from S, T, and nearest-neighbor CZ such ↦ t that X2 Cst Z1X2: ðD14Þ

0 0 0 0 P ↦ α Za1 Xb1 …Zan Xbb ; ωst=2 U P Up to the phase , this action is identical to the one 0 0 0 0 Q ↦ α Zc1 Xd1 …Zcn Xdn ; of the conditional Pauli gate CXsZt studied in Ref. [77]. U Q ðD20Þ We point out again that such a phase is unavoidable α α 1 ∈ 1 … for Z4, as there is, for instance, no unitary U such that with j Pj¼j Qj¼ , and there exists j f ; ;ng such 0 0 − 0 0 1 X1 ↦U X1X2Z2, since these two operators are not that aj dj bj cj ¼ . isospectral. proof.—Let P and Q be as in Eq. (D18). Since Let us define the SWAP gate S via Sjjijki¼jkijji. Xn Evidently, it acts as aidi − bici ¼ 1 ðD21Þ i¼1 X1 ↦S X2;Z1 ↦S Z2;X2 ↦S X1;Z2 ↦S Z1: by assumption, there exists j such that ðD15Þ

rj ¼ ajdj − bjcj ∈ fþ1; −1g: ðD22Þ The gate S thus allows us to generate nonlocal entangling 1 −1 ≠ gates from nearest-neighbor ones. If rj ¼ , we are done. If rj ¼ , then there is k j with 2 −1 Lemma 4: The gate iS can be generated from S, T, rk ¼ or rk ¼ . From Lemma 2, we know that with Si and CZ. and Ti we can generate single-qudit unitaries that change Note that the gates S and iS act identically by ai ci ð Þ in arbitrary ways as long as ri ¼ aidi − bici is conjugation. bi di proof.—Let preserved. So up to gates that can be generated from Sj and Tj, we can assume that aj ¼ 0, bj ¼ cj ¼ 1, and dj ¼ 0.If X3 X3 rk ¼ 2 then, up to gates that can be generated from Sk and ⊗ j j ⊗ 1 0 2 CXð1;2Þ ¼ jjihjj X ;CXð2;1Þ ¼ X jjihjj: Tk, we can assume that ak ¼ , bk ¼ ck ¼ , and dk ¼ . j¼0 j¼0 Finally, if rk ¼ −1, then, up to gates that can be generated from S and T , we can assume that a 1, b 1, ðD16Þ k k k ¼ k ¼ ck ¼ −1, and dk ¼ 2. We note that application of a phase 0 − gate CZðj;kÞ changes rj to rj ¼ rj þðbkdj bjdkÞ, and One verifies that recall that nonlocal phase gates CZðj;kÞ can be generated ffiffi from nearest-neighbor ones and SWAP gates, which we can † p 2 2 C 1 2 C C 1 2 ð iH2Þ ¼ iS; ðD17Þ generate according to Lemma 4. In both cases (rk ¼ and Xð ; Þ Xð2;1Þ Xð ; Þ −1 rk ¼ ), we find that application of a phase gate CZðj;kÞ 0 1 □ gives rj ¼ . which together with Lemmas 1 and 3 completes the ⊗n Lemma 6: Given P, Q ∈ P4 with ðP; QÞ¼1,we proof. □ ⊗ can generate U ∈ C n from S, T, and nearest-neighbor C Let 4 Z such that

0 0 a1 b1 an bn P ↦ Z ⊗ P and Q ↦ X ⊗ Q ; D23 P ¼ αPZ1 X1 …Zn Xn ; U U ð Þ

c1 d1 cn dn Q ¼ α Z1 X1 …Zn Xn : ðD18Þ 0 0 ⊗n−1 Q with P , Q ∈ P4 . proof.—Let P and Q be as in Eq. (D18). By Lemma 5, we can assume that there is j with ajdj − bjcj ¼ 1. All arithmetics involving the exponents aj, bj, cj, and d that follow are to be understood modulo 4. It Employing Lemma 4, we can perform a SWAP between j qudits 1 and j. Finally, we perform a single-qudit unitary L follows from the commutation relation ZX ¼ ωXZ that   d −c PQ ¼ ωðP;QÞQP, where on qudit 1, which is such that MðLÞ¼ j j .As −bj aj

041040-16 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015)

† † 0 0 det MðLÞ¼1 (mod 4), such a unitary L can be constructed 1 ¼ðZ1;X1Þ¼ðVZ1V ;VX1V Þ¼ðZ ⊗ Q ;X ⊗ P Þ from S and T according to Lemma 2. We note that Xn ¼ 1 þ ðaidi − biciÞ; ðD32Þ aj bj cj dj Z X ↦L αZZ and Z X ↦L αXX; ðD24Þ i¼2 with jαXj¼jαZj¼1, which completes the proof. □ we finally conclude that ⊗n Lemma 7: For any V ∈ C4 , we can construct U from 0 † † X1 ↦ X ⊗ P VX1V ; S, T, Z, and nearest-neighbor CZ such that UX1U ¼ U ¼ † † † VX1V and UZ1U VZ1V . 0 † ¼ Z1 ↦U Z ⊗ Q ¼ VZ1V ; ðD33Þ proof.—Clearly, as required. □ † † 1 ⊗n ðVZ1V ;VX1V Þ¼ðZ1;X1Þ¼ ; ðD25Þ Theorem 1: Any Clifford gate V ∈ C4 can be con- structed from S, T, Z, and nearest-neighbor CZ. † 0 so by Lemma 6, we can assume that VX1V ¼ X ⊗ P and proof.—The proof is done by induction over n. The case † 0 VZ1V ¼ Z ⊗ Q , up to gates that can be constructed from n ¼ 1 is given by Lemma 2. For n>1, let U be as in † S, T, and nearest-neighbor CZ. Lemma 7. Since U V commutes with X1 and Z1,wehave Now let † 0 0 ⊗n−1 U V ¼ 1 ⊗ V , where V ∈ C4 acts on qudits f2;…;ng. Assuming that the induction hypothesis holds for n − 1, V0 0 α a2 b2 … an bn P ¼ PZ2 X2 Zn Xn ; and hence V can be constructed from S, T, and nearest- 0 c2 d2 cn dn neighbor CZ. □ Q ¼ αQZ2 X2 …Zn Xn : ðD26Þ

We define APPENDIX E: DEFECT LINES AND HOLES

−st s t Quantum computation in surface codes often uses the Cst;i ¼ S1 ðCXð1;iÞÞ ðCZð1;iÞÞ ; ðD27Þ concept of “hole” defects [56–58]. These are extended with i ∈ f2; …;ng. The gate areas in which a single anyon can reside. Their large size ffiffi ffiffi makes it difficult to measure their anyon occupancy and p p † also to change it without leaving a trace nearby. This allows U ¼ð iH1ÞU ð iH1Þ U ; ðD28Þ Q P them to store an additional logical qubit in a topologically with protected manner. The code distance is given by the size of the hole (for Z errors) and the distance to its nearest Yn Yn neighbor (for X errors), and so can be made arbitrarily

UP ¼ Cbnan;i and UQ ¼ Cdncn;i; ðD29Þ large. They are primarily considered in systems without i¼2 i¼2 a background Hamiltonian, where they are created and moved using measurements [56–58]. However, they can can be constructed from S, T, and nearest-neighbor CZ also be created by adiabatic means when a Hamiltonian is according to Lemmas 1, 3, and 4. present [79–81]. We now discuss this in detail for our Using Eqs. (D11) and (D14), we find the sequences of system. mappings

0 ffiffi † 0 1. Enlarging and shrinking holes X1 ↦ X ⊗ P ↦ p † Z ⊗ P UP P ð iH1Þ n The stabilizer is generated by the plaquette operators Mp −1− ðaidi−biciÞ 0 ↦ Z i¼2 ⊗ P and E for all triangular and hexagonal plaquettes. Let us UQ P p 1 n − consider, however, removing some of these operators from ffiffi þ ðaidi biciÞ 0 ↦p X i¼2 ⊗ P ; ðD30Þ iH1 the stabilizer. In terms of the Hamiltonian, this means removing their corresponding terms. and Specifically, let us remove the plaquette operators Ep and E for two triangular plaquettes p and q. This will open ffiffi 0 ffiffi 0 q Z1 ↦ Z1 ↦ p † X1 ↦ X ⊗ Q ↦p Z ⊗ Q ; UP ð iH1Þ UQ iH1 up a new fourfold degeneracy in the stabilizer space. ðD31Þ Corresponding Z basis states jgi can be labeled by the g ω eigenstates of Ep. These states are therefore distin- † up to phases. Using again that XZX ¼ ω¯ Z and guished by the type of eg anyon residing in the hole. † ZXZ ¼ ωX, and that X can be generated according to Because of the conservation of anyons, the antiparticle e−g Lemma 1, allows us to generate arbitrary phases compatible must reside in q. A further fourfold degeneracy will arise with Eq. (D1). Since from each additional triangular plaquette removed from the

041040-17 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015) plaquette. This is because only one removed plaquette, To achieve this, recall that the combination of the such as q, needs to have its occupation determined by the † plaquettes in the hole is enforced by the strong Xj þ Xj conservation of anyons. terms on their shared sites. To remove p from the hole, the For a logical qudit encoded in the additional stabilizer † term Ep þ Ep should be added to the Hamiltonian, and the space, an X-type operation corresponds to creating a † X þ X term incident on p should be removed. Doing this particle-antiparticle pair of e anyons. One is placed on p j j adiabatically will result in a final state with the E term in q p and the other on . The number of sites on which this 1 process has support, and thus the number of sites on which its ground state, which is its þ eigenspace. Because of the conservation of anyon charge, the e anyon that was noise must act in order to cause a logical X error, is the g distance between the two plaquettes. The qudit will there- held in the larger hole must still be held in the smaller hole. The qudit state therefore remains g . fore be topologically protected against such errors as long j i In addition to this being true for each basis state jgi,we as the plaquettes are well separated. must also be sure that the process preserves coherent The logical Z of the logical qubit corresponds exactly to † superpositions. Any process that causes decoherence in the operator Ep, and Eq corresponds to Z . Since these are this basis will correspond to measurement (by the envi- three-body operators, this type of logical error requires ronment) in the Z basis. Any unitary that introduces action on only three qubit pairs. Therefore, the stored qudit unwanted relative phases can be expressed as a sum of is clearly not topologically protected against Z-type errors. powers of Z. As such, these processes must have support on To address this problem, we can deform the lattice by all sites on which Z has support, which are all sites around making the plaquettes p and q larger. By making them the hole. Since the shrinking (and expansion) of holes does arbitrarily large, logical Z errors can be arbitrarily not have such support, it cannot cause any decoherence. suppressed. Corresponding processes can also be applied to hexago- To enlarge p, consider a neighboring triangular plaquette nal plaquettes. In that case, a logical qudit can be stored 0 p . Let us use j to denote the single site shared by these. in the mg occupations of plaquettes. Such qudits can be If the stored qudit holds an arbitrary state jgi, the state of † topologically protected by using lines of Zj þ Z terms to ωg ωg j the code will be a eigenstate of Ep. It will also be an combine neighboring horizontal plaquettes. 1 eigenstate of EpEp0 since the state is a þ eigenstate of the Using the processes of extending and shrinking holes, it stabilizer Ep0. is possible to move them. Gates can then be implemented † Consider the adiabatic introduction of the term Xj þ Xj through braiding. Braiding an e-type hole of triangular to the Hamiltonian. This term should be much stronger than plaquettes in state jgi around an m-type hole of hexagonal the adjacent plaquette operators and thus will effectively ones in state jhi corresponds to braiding the eg anyon force j into an eigenstate of X and remove it from the code. held by the former around the mh of the latter, yielding a gh Since this term does not commute with Ep or Ep0, the phase ω . This is a qudit generalization of the controlled resulting state will not be an eigenstate of these operators. phase gate. However, the term does commute with the product EpEp0 since the support of the two plaquette operators on j cancels 2. Fusing holes into defect lines in this product. It therefore remains the same ωg eigenstate By considering the alternative stabilizer generators S and as it was for the initial state. Thus, the qudit state jgi has R discussed in Sec. IV, holes can also be created which hold been effectively transferred from the single plaquette p to ψ g anyons. These are formed by similarly combining the 0 the combined plaquette pp . The operator EpEp0 becomes double plaquettes. Indeed, these are exactly the defect lines the logical Z and has support on four qubit pairs rather than considered in the bulk of this paper. These have the three. This is further extended as more plaquettes are added property that anyons crossing the defect line undergo an † automorphism that preserves the structure of the underlying using more Xj þ X terms. As long as this is done for both j Abelian state: It maps e anyons to their dual, the m anyons, p and q, the qudit will become topologically protected and vice versa. This property is not shared by the e- and m- against Z errors as well as X. type holes. In these cases, the lines form a boundary along The process used to extend holes can be reversed in order which one type of anyon can condense but the other cannot to shrink them. Consider a set of triangular plaquettes cross. It is this difference that gives the ψ holes additional 0 00 … p; p ;p ; that have been combined into a single hole. properties, namely, the localized parafermion modes at The basis state jgi of the qubit stored in this hole is their endpoints, that the e and m holes do not possess. ωg … associated with the eigenstate of EpEp0 Ep00 We wish The topological degeneracy and protection, however, is a to shrink this hole so that p is no longer a part of it. property shared by all three. 1 The state will then have a þ eigenvalue for Ep, and the An e-type hole and an m-type hole together correspond qudit state jgi will be associated with the ωg eigenstate to a two-qudit space. However, let us consider the subspace of Ep0 Ep00 …. spanned by states jg; gi. These are such that the e-type hole

041040-18 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015)

end result is that the ψ g originally stored in the defect line now resides in the e-type hole as an eg and the m-type hole as an mg, corresponding to the state jg; gi of their individual qudits. As for the shrinking of holes, this process does not have sufficient support to distinguish between different basis states. The process therefore does not decohere any superpositions of these states, nor does it assign any relative phases. To recombine the two holes into a single defect line, the process is simply reversed. This will be straightforward if the two holes are in a state of the form jg; gi since the state of the defect line will simply become jgi. However, some processes, such as mistakes during error correction, could result in holes whose states are not of this form. This will introduce frustration that will not allow all of the involved triangular and hexagonal plaquettes to return to their ground state after recombination. FIG. 14. Red circles denote defects as in Eq. (13), with a pair of As an example, consider a state of the form jg; hi. This parafermions (purple) emerging at the ends of the defect line. corresponds to an e m † g and an h and could arise from Blue circles correspond to terms of the form Xj þ Xj , while green † an initial state jg; gi if an mh-g were added in error to the circles correspond to terms of the form Zj þ Z . These grow the e j m-type hole, or from jh; hi with an eg-h error on the and m parts of the defect line, respectively. Once both the blue e-type hole. and green defect lines have been added, the red defect line can be The anyons eg and an mh can combine either to a ψ g removed. The ψ g particle initially stored in the red defect line has and an m − or a ψ and an e . The former will be then been split into its e and m components. h g h g-h g g energetically favorable because of the weaker strength of the Mp terms. The adiabatic process will therefore result in ψ g being stored on the defect line and an mh-g anyon carries an eg anyon whenever the m-type hole carries an mg. A single qudit can be stored in this subspace present as an excitation on one of the triangular plaquettes that was once part of the holes. Syndrome measurement Since ψ g ¼ eg × mg, two holes as described above hold a will then detect the mg h. However, since the value g-h gives net ψ g. Similar fusion can also be applied to holes, as we - will now show. Specifically, an e-type and an m-type hole information only about the error that occurred and not can be combined into a ψ-type hole, and a ψ-type hole can the value of g or h, this does not extract any information be split into an e-type and an m-type one. about the stored qudit. These processes are, in fact, a simple generalization of the hole extension and shrinking processes described above. Suppose we have a defect line along double plaquettes P; P0;P00; … This stores a qudit whose basis [1] A. Yu. Kitaev, Fault-Tolerant Quantum Computation by g states jgi are ω eigenstates of WPWP0 WP00 … Let us now Anyons, Ann. Phys. (Amsterdam) 303, 2 (2003). extend this line. However, rather than adding another [2] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. double plaquette, we instead add a triangular plaquette Sarma, Non-Abelian Anyons and Topological Quantum p. This can be done by adiabatically introducing the strong Computation, Rev. Mod. Phys. 80, 1083 (2008). † [3] J. K. Pachos, Introduction to Topological Quantum term Xj þ Xj to the Hamiltonian on a site shared by p and Computation (Cambridge University Press, Cambridge, the triangular part of P. This term does not commute with England, 2012). either Ep or WP. However, it does commute with their [4] J. R. Wootton, J. Burri, S. Iblisdir, and D. Loss, Error product. The final state will then have the qudit basis states Correction for Non-Abelian Topological Quantum Compu- defined by the operator EpWPWP0 WP00 …. Further such tation, Phys. Rev. X 4, 011051 (2014). processes can be used to extend the e-type part of the defect [5] C. G. Brell, S. Burton, G. Dauphinais, S. T. Flammia, and line. Corresponding processes on the hexagonal plaquettes D. Poulin, Thermalization, Error Correction, and Memory Lifetime for Ising Anyon Systems, Phys. Rev. X 4, 031058 can be used to grow an m-type part of the defect line. (2014). An illustration is given in Fig. 14. [6] F. L. Pedrocchi and D. P. DiVincenzo, Majorana Braiding When both e-type and m-type parts have been added, the with Thermal Noise, Phys. Rev. Lett. 115, 120402 (2015). original ψ-type defect line can be removed. This is done [7] M. A. Levin and X.-G. Wen, String-Net Condensation: A simply by removing the parity-operator terms along its Physical Mechanism for Topological Phases, Phys. Rev. B length and allowing the SP terms to again dominate. The 71, 045110 (2005).

041040-19 ADRIAN HUTTER, JAMES R. WOOTTON, and DANIEL LOSS PHYS. REV. X 5, 041040 (2015)

[8] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. [27] F. Zhang and C. L. Kane, Time-Reversal-Invariant Z4 Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell Fractional Josephson Effect, Phys. Rev. Lett. 113, et al., Superconducting Quantum Circuits at the Surface 036401 (2014). Code Threshold for Fault Tolerance, Nature (London) 508, [28] Y. Oreg, E. Sela, and A. Stern, Fractional Helical Liquids in 500 (2014). Quantum Wires, Phys. Rev. B 89, 115402 (2014). [9] C. Monroe and J. Kim, Scaling the Ion Trap Quantum [29] J. Klinovaja and D. Loss, Time-Reversal Invariant Paraf- Processor, Science 339, 1164 (2013). ermions in Interacting Rashba Nanowires, Phys. Rev. B 90, [10] Ch. Kloeffel and D. Loss, Prospects for Spin-Based 045118 (2014). Quantum Computing, Annu. Rev. Condens. Matter Phys. [30] J. Klinovaja, A. Yacoby, and D. Loss, Kramers Pairs of 4, 51 (2013). Majorana Fermions and Parafermions in Fractional [11] A. Negretti, P. Treutlein, and T. Calarco, Quantum Comput- Topological Insulators, Phys. Rev. B 90, 155447 ing Implementations with Neutral Particles, Quantum Inf. (2014). Process. 10, 721 (2011). [31] M. Barkeshli, Y. Oreg, and X.-L. Qi, Experimental Proposal [12] A. Yu. Kitaev, Anyons in an Exactly Solved Model and to Detect Topological Ground State Degeneracy, Beyond, Ann. Phys. (Amsterdam) 321, 2 (2006). arXiv:1401.3750. [13] H. Yao and S. A. Kivelson, Exact Chiral Spin Liquid with [32] C. P. Orth, R. P. Tiwari, T. Meng, and T. L. Schmidt, Non- Non-Abelian Anyons, Phys. Rev. Lett. 99, 247203 (2007). Abelian Parafermions in Time-Reversal Invariant Interact- [14] H. Bombin, Topological Order with a Twist: Ising Anyons ing Helical Systems, Phys. Rev. B 91, 081406 (2015). from an Abelian Model, Phys. Rev. Lett. 105, 030403 [33] J. Klinovaja and D. Loss, Fractional Charge and Spin (2010). States in Topological Insulator Constrictions, Phys. Rev. B [15] O. Petrova, P. Mellado, and O. Tchernyshyov, Unpaired 92, 121410(R) (2015). Majorana Modes on Dislocations and String Defects in [34] J. Alicea and P. Fendley, Topological Phases with Paraf- Kitaev’s Honeycomb Model, Phys. Rev. B 90, 134404 ermions: Theory and Blueprints, arXiv:1504.02476. (2014). [35] M. Barkeshli, C.-M. Jian, and X.-L. Qi, Twist Defects and [16] J. Alicea, New Directions in the Pursuit of Majorana Projective Non-Abelian Braiding Statistics, Phys. Rev. B Fermions in Solid State Systems, Rep. Prog. Phys. 75, 87, 045130 (2013). 076501 (2012). [36] M. Barkeshli and X.-L. Qi, Topological Nematic States and [17] S. Bravyi, Universal Quantum Computation with the ν ¼ Non-Abelian Lattice Dislocations, Phys. Rev. X 2, 031013 5=2 Fractional Quantum Hall State, Phys. Rev. A 73, (2012). 042313 (2006). [37] Y.-Z. You and X.-G. Wen, Projective Non-Abelian Statistics [18] M. Freedman, C. Nayak, and K. Walker, Towards Universal of Dislocation Defects in a ZN Rotor Model, Phys. Rev. B Topological Quantum Computation in the ν ¼ 5=2 Frac- 86, 161107(R) (2012). tional Quantum Hall State, Phys. Rev. B 73, 245307 (2006). [38] Y.-Z. You, C.-M. Jian, and X.-G. Wen, Synthetic Non- [19] N. H. Lindner, E. Berg, G. Refael, and A. Stern, Fraction- Abelian Statistics by Abelian Anyon Condensation, Phys. alizing Majorana Fermions: Non-Abelian Statistics on the Rev. B 87, 045106 (2013). Edges of Abelian Quantum Hall States, Phys. Rev. X 2, [39] A. Hutter, D. Loss, and J. R. Wootton, Improved HDRG 041002 (2012). Decoders for Qudit and Non-Abelian Quantum Error [20] M. Cheng, Superconducting Proximity Effect on the Edge of Correction, New J. Phys. 17, 035017 (2015). Fractional Topological Insulators, Phys. Rev. B 86, 195126 [40] J. R. Wootton and A. Hutter, Active Error Correction for (2012). Abelian and Non-Abelian Anyons, arXiv:1506.00524. [21] D. J. Clarke, J. Alicea, and K. Shtengel, Exotic Non-Abelian [41] S. Burton, C. G. Brell, and S. T. Flammia, Classical Sim- Anyons from Conventional Fractional Quantum Hall States, ulation of Quantum Error Correction in a Fibonacci Anyon Nat. Commun. 4, 1348 (2013). Code, arXiv:1506.03815. [22] A. Vaezi, Fractional Topological Superconductors with [42] A. Hutter and J. R. Wootton, Quantum Computing with Fractionalized Majorana Fermions, Phys. Rev. B 87, Parafermions, arXiv:1511.02704. 035132 (2013). [43] E. Fradkin and L. P. Kadanoff, Disorder Variables and [23] M. Burrello, B. van Heck, and E. Cobanera, Topological Para-fermions in Two-Dimensional Statistical Mechanics, Phases in Two-Dimensional Arrays of Parafermionic Zero Nucl. Phys. B170, 1 (1980). Modes, Phys. Rev. B 87, 195422 (2013). [44] J. Kempe, A. Yu. Kitaev, and O. Regev, The Complexity of [24] R. S. K. Mong, D. J. Clarke, J. Alicea, N. H. Lindner, P. the Local Hamiltonian Problem, SIAM J. Comput. 35, 1070 Fendley, C. Nayak, Y. Oreg, A. Stern, E. Berg, K. Shtengel, (2006). and M. P. A. Fisher, Universal Topological Quantum Com- [45] S. P. Jordan and E. Farhi, Perturbative Gadgets at Arbitrary putation from a Superconductor/Abelian Quantum Hall Orders, Phys. Rev. A 77, 062329 (2008). Heterostructure, Phys. Rev. X 4, 011036 (2014). [46] R. Oliveira and B. M. Terhal, The Complexity of Quantum [25] M. Barkeshli and X.-L. Qi, Synthetic Topological Qubits in Spin Systems on a Two-Dimensional Square Lattice, Conventional Bilayer Quantum Hall Systems, Phys. Rev. X Quantum Inf. Comput. 8, 900 (2008). 4, 041035 (2014). [47] J. K. Pachos and M. B. Plenio, Three-Spin Interactions in [26] J. Klinovaja and D. Loss, Parafermions in an Interacting Optical Lattices and Criticality in Cluster Hamiltonians, Nanowire Bundle, Phys. Rev. Lett. 112, 246403 (2014). Phys. Rev. Lett. 93, 056402 (2004).

041040-20 PARAFERMIONS IN A KAGOME LATTICE OF QUBITS … PHYS. REV. X 5, 041040 (2015)

[48] H. P. Büchler, A. Micheli, and P. Zoller, Three-Body [65] A. Al-Shimary, J. R. Wootton, and J. K. Pachos, Lifetime of Interactions with Cold Polar Molecules, Nat. Phys. 3, Topological Quantum Memories in Thermal Environment, 726 (2007). New J. Phys. 15, 025027 (2013). [49] A. Micheli, G. K. Brennen, and P. Zoller, A Toolbox for [66] B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Lattice Spin Models with Polar Molecules, Nat. Phys. 2, 341 Wootton, Quantum Memories at Finite Temperature, (2006). arXiv:1411.6643. [50] C. G. Brell, S. T. Flammia, S. D. Bartlett, and A. C. Doherty, [67] S. Bravyi and A. Kitaev, Universal Quantum Computation Toric Codes and Quantum Doubles from Two-Body with Ideal Clifford Gates and Noisy Ancillas, Phys. Rev. A Hamiltonians, New J. Phys. 13, 053039 (2011). 71, 022316 (2005). [51] E. Kapit and S. H. Simon, Three- and Four-Body Inter- [68] E. T. Campbell, H. Anwar, and D. E. Browne, Magic-State actions from Two-Body Interactions in Spin Models: A Distillation in All Prime Dimensions Using Quantum Reed- Route to Abelian and Non-Abelian Fractional Chern Muller Codes, Phys. Rev. X 2, 041021 (2012). Insulators, Phys. Rev. B 88, 184409 (2013). [69] E. T. Campbell, Enhanced Fault-Tolerant Quantum Com- [52] P. Bonderson, M. Freedman, and C. Nayak, Measurement- puting in d-Level Systems, Phys. Rev. Lett. 113, 230501 Only Topological Quantum Computation, Phys. Rev. Lett. (2014). 101, 010501 (2008). [70] F. H. E. Watson, E. T. Campbell, H. Anwar, and D. E. [53] P. Bonderson, Measurement-Only Topological Quantum Browne, Qudit Colour Codes and Gauge Colour Codes Computation via Tunable Interactions, Phys. Rev. B 87, in All Spatial Dimensions, Phys. Rev. A 92, 022312 (2015). 035113 (2013). [71] E. T. Campbell (private communication). [54] J. R. Wootton, A Family of Stabilizer Codes for DðZ2Þ [72] H. Anwar, E. T. Campbell, and D. E. Browne, Qutrit Magic Anyons and Majorana Modes, J. Phys. A 48, 215302 State Distillation, New J. Phys. 14, 063006 (2012). (2015). [73] This is indeed a route we have tentatively followed. [55] A. Hutter and D. Loss, Quantum Computing with Unfortunately, the Hamiltonians necessary to generate Parafermions, arXiv:1511.02704. DðZ3Þ quantum double models in a qubit system turned [56] R. Raussendorf and J. Harrington, Fault-Tolerant Quantum out to be much more involved than Eq. (4). Computation with High Threshold in Two Dimensions, [74] J. R. Schrieffer and P. A. Wolff, Relation between the Phys. Rev. Lett. 98, 190504 (2007). Anderson and Kondo Hamiltonians, Phys. Rev. 149, 491 [57] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. (1966). Cleland, Surface Codes: Towards Practical Large-Scale [75] S. Bravyi, D. P. DiVincenzo, and D. Loss, Schrieffer-Wolff Quantum Computation, Phys. Rev. A 86, 032324 (2012). Transformation for Quantum Many-Body Systems, Ann. [58] J. R. Wootton, Quantum Memories and Error Correction, Phys. (Amsterdam) 326, 2793 (2011). J. Mod. Opt. 59, 1717 (2012). [76] Note that, for closed boundary conditions, this added [59] G. Duclos-Cianci and D. Poulin, Kitaev’s Zd-Code degeneracy does not arise for the first pair to be unpaired. Threshold Estimates, Phys. Rev. A 87, 062338 (2013). [77] S. Clark, Valence Bond Solid Formalism for d-Level [60] H. Anwar, B. J. Brown, E. T. Campbell, and D. E. Browne, One-Way Quantum Computation, J. Phys. A 39, 2701 Fast Decoders for Qudit Topological Codes, New J. Phys. (2006). 16, 063038 (2014). [78] J. M. Farinholt, An Ideal Characterization of the Clifford [61] F. H. E. Watson, H. Anwar, and D. E. Browne, A Fast Fault- Operators, J. Phys. A 47, 305303 (2014). Tolerant Decoder for Qubit and Qudit Surface Codes, Phys. [79] J. R. Wootton, V. Lahtinen, B. Boucot, and J. K. Pachos, Rev. A 92, 032309 (2015). Universal Quantum Computation with a Non-Abelian [62] V. Kolmogorov, Blossom V: A New Implementation of a Topological Memory, Ann. Phys. (Amsterdam) 326, 2307 Minimum Cost Perfect Matching Algorithm, Math. Pro- (2011). gram. Comput. 1, 43 (2009). [80] C. Cesare, A. J. Landahl, D. Bacon, S. T. Flammia, and [63] J. Edmonds, Paths, Trees, and Flowers, Can. J. Math. 17, A. Neels, Adiabatic Topological Quantum Computing, 449 (1965). Phys. Rev. A 92, 012336 (2015). [64] B. Röthlisberger, J. R. Wootton, R. M. Heath, J. K. Pachos, [81] Y.-C. Zheng and T. A. Brun, Fault-Tolerant Holonomic and D. Loss, Incoherent Dynamics in the Toric Code Quantum Computation in Surface Codes, Phys. Rev. A 91, Subject to Disorder, Phys. Rev. A 85, 022313 (2012). 022302 (2015).

041040-21