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University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Physics and Research Papers in Physics and Astronomy Astronomy

2014 Parafermion stabilizer codes Utkan Güngördü University of Nebraska-Lincoln, [email protected]

Rabindra Nepal University of Nebraska-Lincoln, [email protected]

Alexey Kovalev University of Nebraska - Lncoln, [email protected]

Follow this and additional works at: http://digitalcommons.unl.edu/physicsfacpub Part of the Quantum Physics Commons

Güngördü, Utkan; Nepal, Rabindra; and Kovalev, Alexey, "Parafermion stabilizer codes" (2014). Faculty Publications, Department of Physics and Astronomy. 131. http://digitalcommons.unl.edu/physicsfacpub/131

This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Physics and Astronomy by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. PHYSICAL REVIEW A 90, 042326 (2014)

Parafermion stabilizer codes

Utkan Gung¨ ord¨ u,¨ Rabindra Nepal, and Alexey A. Kovalev Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA (Received 23 September 2014; published 21 October 2014)

We define and study parafermion stabilizer codes, which can be viewed as generalizations of Kitaev’s one- dimensional (1D) model of unpaired Majorana . Parafermion stabilizer codes can protect against low- weight errors acting on a small subset of parafermion modes in analogy to qudit stabilizer codes. Examples of several smallest parafermion stabilizer codes are given. A locality-preserving embedding of qudit operators into parafermion operators is established that allows one to map known qudit stabilizer codes to parafermion codes. We also present a local 2D parafermion construction that combines topological protection of Kitaev’s with additional protection relying on parity conservation.

DOI: 10.1103/PhysRevA.90.042326 PACS number(s): 03.67.Pp, 05.30.Pr, 71.10.Pm

I. INTRODUCTION prime or a prime power [42,48,49], while generalizations to composite D are also possible [50]. Topologically protected systems are potentially useful for Parafermion codes can also be interpreted in terms of realizations of fault-tolerant elements in a quantum com- termwise commuting Hamiltonians of interacting parafermion puter [1,2]. The zero-temperature stability of such systems zero modes, thus generalizing Kitaev’s one-dimensional (1D) leads to exponential suppression of decoherence induced by model of unpaired Majorana fermions to the D>2 case local environmental perturbations. On the other hand, the and to arbitrary interactions preserving the commutativity manipulation of the degenerate ground state can be achieved of terms in the Hamiltonian. Of particular interest are the by braiding operations with non-Abelian [3,4]. Hamiltonians corresponding to geometrically local interac- The Kitaev chain provides an enlightening example of how tions on a d-dimensional lattice. Thus, one can ask similar interactions can result in non-Abelian [5]. Net- questions to those posed in Ref. [51] in relation to Majorana works of one-dimensional realizations of such quasiparticles codes, i.e., what is the role of superselection rules in the can be employed for realizations of quantum gates via braiding finite-temperature stability of defined by operations [6,7]. However, only a nonuniversal set of quantum interacting parafermion modes. Such superselection rules are gates can be realized with Majorana zero modes. A generaliza- characteristic of fermionic systems when only interactions tion of the Kitaev chain model has been proposed recently in with bosonic environments are present. On the other hand, the which quasiparticles obey parafermion Z algebra as opposed D superselection rule prohibiting parity-violating error operators to Z algebra for Majorana zero modes [8]. Many recent 2 is not likely to always hold, for instance, when the environment publications address possible realizations of parafermion zero supports gapless fermionic modes that can couple to the modes [9–29]. The braiding properties of parafermion systems system [52,53]. Parafermion stabilizer codes can help in such have some advantages over the Majorana modes, while still situations by providing protection associated with the code remaining nonuniversal [10,11,16]. However, parafermion distance of parity-violating logical operators. systems can be used for obtaining quasiparticles that permit The paper is organized as follows. In Sec. II, we intro- universal quantum computations [19]. duce notations and provide background on the theory of The presence of finite temperature introduces inevitable qudit stabilizer codes. Here we also discuss the Jordan- errors and in principle requires continuous error correc- Wigner transformation, which leads us to the introduction tion [30]. “Self-correcting” quantum memories are stable of parafermion operators. In Sec. III,wegiveaformal at finite temperatures [31,32]; however, they cannot be definition of parafermion stabilizer codes and establish their realized in two dimensions with local interactions [33,34]. basic properties. We also discuss the commutativity condition Parafermion stabilizer codes considered here can protect on stabilizer generators, define the code distance, and prove against low-weight fermionic errors, i.e., errors that act on basic results on the dimension of the code space. In Sec. IV,we a small subset of parafermion modes. The measurement present several examples of the smallest parafermion stabilizer and manipulation schemes required for code implementa- codes. In Sec. V, we construct mappings between qudit sta- tions have been formulated for Majorana zero modes [35– bilizer codes and parafermion stabilizer codes. By employing 37] and should in principle generalize to parafermion zero such mappings, we are able to construct parafermion toric modes [11]. code with an adjustable degree of protection against the In this paper, we address the possibility of active error parity-violating errors. Finally, we give our conclusions in correction in systems containing a set of parafermion modes as Sec. VI. opposed to typical systems containing or qudits. Earlier works on usually addressed the II. BACKGROUND case with a Hilbert space dimension D = 2[30,38–40]. Error correction on qudits with D>2 has also been considered, A. Qudits and qudit stabilizer codes have been introduced [41–47]. The Qudits are D-dimensional generalizations of qubits, and formalism is usually applied to situations in which D is are generally implemented using D-level physical systems.

1050-2947/2014/90(4)/042326(7) 042326-1 ©2014 American Physical Society GUNG¨ ORD¨ U,¨ NEPAL, AND KOVALEV PHYSICAL REVIEW A 90, 042326 (2014)

One of the well-known generating sets for qudit operations is The robustness of a quantum code can be measured by how constructed by the generators of the finite discrete Weyl group far two encoded states are apart, which is quantified through WD that obey the defining relations [54,55] the notion of distance. The weight of the logical operators implies the separation of the encoded states. Therefore, the XD = ZD = 1,ZX= ωXZ. (1) distance of a is defined as This group is sometimes referred to as the discrete Heisenberg d = min |Li |. (4) group [42], and the generators are sometimes referred to Li ∈C(S)\S as generalized Pauli matrices [50]. By diagonalizing one The longer the code distance is, the better protection the code of these operators, say Z, one obtains the D-dimensional provides. A code of distance d can detect any error of weight up representation to d − 1, and correct up to d/2 . A quantum error-correcting D−1 D−1 code that encodes n physical qudits into k logical qudits with = | + | = j | | X j 1 j ,Z ω j j , (2) distance d is denoted as [[n,k,d]]D. j=0 j=0 where ω = e2πi/D and the addition j + 1isinmodD. Above C. Parafermion operators and throughout the paper, 1 denotes the identity operator with Parafermion operators can be obtained by the Jordan- proper dimensions. Products of X and Z span the Lie algebra Wigner transformation of the D-state operators { }∈W⊗n su(D), hence their linear combinations can generate universal Xj ,Zj D as SU(D) operations. Operations on multiple qudits are tensor j−1 products of the single-qudit operators, hence operators acting = on distinct qudits commute. We will denote an X operator γ2j−1 Xk Zj , k=1 actingonthejth site as Xj , which is equivalent to an X operator at the jth slot of the tensor product padded with j−1 = (d−1)/2 identity operators: Xj = 1 ⊗···⊗X ⊗···⊗1 (and similar γ2j ω Xk Zj Xj , (5) for Zj ). k=1 which is a mapping of n local spin operators into 2n B. Stabilizer codes for qudits nonlocal parafermion operators, therefore the total number Stabilizer codes are an important class of quantum error- of parafermion modes is always even. Parafermion operators correcting codes [30,56], which, under appropriate mapping, γj obey the following relations: can be also thought of as additive classical codes [57]. γ d = 1,γγ = ωγ γ (j

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A parafermion operator ωλγ α ∈ PF(D,2n) will preserve PF(D,2n). We list the defining properties of parafermion parity iff stabilizer codes as follows: (i) Elements of SPF are parity-preserving operators. 2n j (ii) SPF is an Abelian group not containing ω 1, where αi = 0modD. (8) j ∈ ZD and j = 0. = i 1 Whether these conditions hold for a given parafermion m n = mn n m One can generalize Eq. (6) to obtain γi γj ω γj γi stabilizer code or not can be verified using Eqs. (8) and (9), for i

042326-3 GUNG¨ ORD¨ U,¨ NEPAL, AND KOVALEV PHYSICAL REVIEW A 90, 042326 (2014) is a projection operator satisfying P 2 = P = P †. Clearly, for operators. Even though this code does not provide protection | ∈ | =| W against parity-violating errors, in the absence of such errors any ψj CSPF , P ψj ψj holds. Thus the subspace ⊆ W the code protection can be described by the diameter of even that P projects onto includes CSPF ,orCSPF . Next we show that this relation holds the other way around. logical operators, i.e., lcon = 2n. Let |φ be an arbitrary element of W (thus P |φ = |φ) and Sk be an arbitrary element of SPF. Since SkP = P for all k,we obtain Sk(P |φ) = P |φ, meaning all |φ ∈ W is stabilized B. Minimal parafermion stabilizer codes S W ⊆ W = by PF or CSPF , leading us to the conclusion that Quantum error-correcting schemes come at the expense of

CSPF . The dimension of the code space is then given as tr(P ). introducing additional qudits in order to protect information S α = Since PF is an Abelian group and the trace condition tr(γ ) encoded into quantum states. The ratio of the number of α α 0 when γ = 1 and tr(1) = Dn for γ ,1 ∈ PF(D,2n), holds, encoded qudits k (whose state can be restored after deco- we arrive at the result herence) to the number of underlying physical qudits n is = 1 n called encoding rate r k/n. The relative distance is defined tr(P ) =|CS |= D . (18) PF as δ = d/n. Codes with higher encoding rate r and relative |SPF| distance are preferable, and it is known that both δ and r can l lk | S |= Corollary. When D is a prime power p , C PF p and be finite for a particular code family [58]. In this section, we |S |= r = − PF p with r l(n k) (we refer the reader to [42]for discuss the minimal stabilizer codes encoding the k = 1 qudit a detailed derivation). and try to find codes with the best encoding rate r for the In later sections, we will also use a matrix form of the = α α minimal nontrivial distance d 3 for prime D. S = = 1 l stabilizer code PF S1,...,Sl γ ,...,γ , whose Using an exhaustive search, we find that for D = 3the α rows are given by i , that is, smallest nontrivial code requires eight parafermion modes and ⎛ ⎞ results in an [[8,1,3]]3 parafermion stabilizer code: α1 ⎜ ⎟ S = ⎝ . ⎠. (19) PF . S = † † † † † † PF γ1 γ2γ4 γ6,γ2 γ3γ5 γ7,γ3 γ4γ6 γ8 , αl L S = † † † The same construction is also extended for the logical ( PF) γ1 γ2γ3γ7,γ2 γ3 γ6 . (23) operators, yielding the matrix LPF. Since SPF is an Abelian T = The logical operators generate W , encoding eight group, due to Eq. (9), we have SPFSPF 0modD.The 3 logical operator matrix LPF, on the other hand, obeys the parafermion modes into a single logical . T = T = Realizations of D = 6 parafermion zero modes have relations LPFSPF 0 and LPFLPF 0in modD. been proposed recently [11], making this case particularly = IV. EXAMPLES OF PARAFERMION STABILIZER CODES interesting. Because D 6 is not a prime or prime power, the original construction for qudit stabilizer codes [42]isnot A. Three-state quantum clock model directly applicable. We will instead “double” the D = 3 code We present a simple example of a parafermion code starting given above by squaring all the generators. However, this is from a three-state quantum clock model Hamiltonian (for h = a mapping onto a larger space and we need to take care of 0): the additional operators that commute with the new stabilizer generators. The full set of generators for D = 6 thus becomes n−1 † † H =−J (Z Z + + Z Z ). (20) 3 j j 1 j+1 j S = 3 3 3 3 3 3 3 3 = PF γ1 γ2 ,γ3 γ4 ,γ5 γ6 ,γ7 γ8 j 1 † † 2 † † 2 † † 2 By employing the Jordan-Wigner transformation, this Hamil- (γ1 γ2γ4 γ6) ,(γ2 γ3γ5 γ7) ,(γ3 γ4γ6 γ8) , tonian can be rewritten in terms of parafermion operators in † † † L S =γ γ γ γ 2, γ γ γ 2. the following form: ( PF) ( 1 2 3 7) ( 2 3 6) (24) n−1 † † Since these logical operators behave like X2 and Z2 for D = 6 H = iJ (γ γ + − γ γ ), (21) 2j 2j 1 2j+1 2j qudits, the code above essentially encodes a qutrit using 2n = j=1 8 parafermion zero modes. We also note that this code may which is known as the Fendley [8] generalization of the Kitaev not have the best encoding rate for D = 6. chain model. For D = 2, Eq. (20) reduces to the familiar Ising However, the minimal number of modes depends on D.For = model with h 0. the case of D = 7, there exists a [[6,1,3]]7 code that requires We form the corresponding stabilizer group taking individ- only six modes, ual terms of the Hamiltonian for each value of j as † † † † S = 5 5 iγ γ , − iγ γ ,...,iγ γ − , − iγ γ − . (22) PF γ1γ2γ5 ,γ1γ4 γ6 , 2 3 3 2 2n−2 2n 1 2n−1 2n 2 (25) L S = 3 6 2 5 Logical operators of the code can be chosen as Z¯ = γ1 and ( PF) γ1 γ2 γ6,γ1 γ2 γ3 . X¯ = γ2n. Then the distance of the code is d = 1. But these logical operators are not parity-preserving, we can combine This indicates that there is a minimal D for which the encoding † † them as γ1 γ2n and γ1γ2n to obtain parity-preserving logical rate is optimal [59].

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V. MAPPINGS BETWEEN QUDITS AND PARAFERMION It turns out that we can do a similar mapping in the opposite MODES direction, albeit without preserving the locality of stabilizer generators. A. Mappings to and from parafermion codes Lemma. Any parafermion stabilizer code with parame- There is an established literature on stabilizer codes for ters [[2n,k,d]]D and stabilizer group SPF can generate a  qudits when D is prime or a prime power [60,61]. Recently, [[2n,2k,d ]]D qudit CSS code. some properties of qudit stabilizer codes for the nonprime Proof. Consider the check matrix case have been discussed in [50]. An isomorphism between multi-qudit and multi-parafermion mode operators will let us SPF 0 SCSS = . (27) construct parafermion stabilizer codes based on qudit codes. 0 SPF In this section, we establish such an isomorphism by mapping four parafermion modes to a single qudit. For a parafermion code, k = n − rank(SPF), whereas for the Remark. Let X˜ and Z˜ (j = 1,...,k) denote the generat-  j j CSS code, k = 2n − 2 × rank(SPF) = 2k ( is a full-rank W⊗k  ing operators of D embedded into PF(D,2n), encoding k matrix). Hence SCSS is the check matrix of a [[2n,2k,d ]]D qudits into 2n parafermion modes. Such an embedding has the CSS code. The corresponding logical operator matrices LPF following properties: and LPF behave like X- and Z-type logical qudit operators. (i) Logical qudit operators {X˜ j ,Z˜ j } obey Eq. (1), that is, We note that this construction is a proper generalization W⊗k ⊆ they generate the embedded Weyl group D PF(D,2n). of the doubling lemma described in [51], which maps a (ii) Logical qudit operators for different sites commute code to a weakly self-dual CSS code. ([X˜ i ,X˜ j ] = [Z˜ i ,Z˜ j ] = [X˜ i ,Z˜ j ] = 0 when i = j). Unfortunately, for D>2 this mapping becomes nonlocal, W⊗k (iii) The embedding of D into the larger group PF(D,2n) i.e., a local qudit operator will generally map to a nonlocal { ˜ (i)} parafermion operator. may require additional parafermion operators Qj that commute with the original qudit stabilizer group S or its corresponding logical operators L(S ). Such operators must PF B. Parafermion toric code with parity-violating be included in the parafermion stabilizer group S and hence PF logical operators must preserve parity [an example is given in Eq. (26) below]. In turns out that the minimum number of parafermion In this section, we construct a parafermion analog of modes required for such an embedding is four, that is, four Kitaev’s toric code [1] for qudits [60]. The toric code is a parafermion modes will map to a single qudit. This mapping stabilizer code defined on an a × b lattice on the surface of leads to the following lemma. a torus. A portion of the lattice is depicted in Fig. 1, where Lemma. Every [[n,k,d]]D stabilizer code can be mapped each dot represents a single qudit (hence, there are 2ab qudits onto a [[4n,k,2d]]D parafermion stabilizer code, encoding four overall). 2l + parafermion modes into a single qudit. Let D = p , where p is a prime number and l ∈ Z .The Proof. Let us define the operators operators

† † pl −1 pl −1 ˜ = ˜ = Z˜ + = γ γ + , X˜ + = γ γ + , Zj+1 γ1+4j γ2+4j , Xj+1 γ1+4j γ3+4j , j 1 1+4j 2 4j j 1 1+4j 3 4j (26) (28) † † † † ˜ = Q˜ + = γ γ γ + γ + Qj+1 γ1+4j γ2+4j γ3+4j γ4+4j . j 1 1+4j 2+4j 3 4j 4 4j

It is straightforward to show that X˜ ,Z˜ generate the define a mapping of four parafermion modes onto a single j j S =˜ W⊗k ⊆ ˜ ˜ = qudit via the one-qudit stabilizer group PF Qj and its embedded Weyl group D PF(D,2n) (that is, Zi Xj L S =˜ ˜ ˜ ˜ ˜ D = ˜ D = corresponding logical operators ( PF) Xj ,Zj . ωXj Zi δij and Xj Zj 1) and are parity-preserving. We can treat L(SPF) =X˜ j ,Z˜ j as the logical operators of a stabilizer group SPF =Q˜ j . This makes the purpose of the l A additional fourth mode (which does not appear in the logical ˜ p −1 s X = γ1+4jγ3+4j operators) clear: without it, the stabilizer group would include X˜ As a non-parity-preserving operator. Finally, since every Weyl X˜ −1 operator is mapped to a parafermion operator with two modes, X˜ −1 the distance of the new code is 2d. Bp This mapping allows us to construct families of parafermion Bp stabilizer codes from known families of qudit stabilizer codes. Z˜ In particular, one can map the qudit toric codes [60] (and their ˜ pl−1 Z˜−1 Z = γ1+4jγ2+4j generalizations [62,63]) to the corresponding parafermion Z˜−1 code. The advantage of this mapping is that a local stabilizer generator in a d-dimensional lattice will map to a local FIG. 1. (Color online) A portion of the lattice place on a torus, parafermion operator. The disadvantage is that all logical where each dot represents four parafermion modes (the index j  0 operators preserve parity, thus there is no additional protec- uniquely denotes the lattice point). On the right, parafermion star and l tion associated with the presence of parity-violating logical plaquette operators As and Bp are given in detail [p is a prime power; operators. further details are given in Eq. (28)].

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topological protection of Kitaev’s toric code with additional protection relying on suppression of parity-violating errors.

VI. CONCLUSION

X We have introduced stabilizer codes in which parafermion 1 zero modes represent the constructing blocks as opposed to Z2 qudit stabilizer codes. Our work generalizes earlier construc- tions based on Majorana zero modes [51]. While it is possible X2 Z1 in general to start with a stabilizer code for qudits and use it with parafermion zero modes through the mapping given in W⊗n ⊂ Eq. (26), which utilizes the embedding D PF(D,4n), FIG. 2. (Color online) Loops corresponding to the logical opera- we find that there are more efficient codes in PF(D,2n) tors of the toric code. requiring fewer parafermion modes, as we have exemplified in Sec. IV B. These results also show that the parafermions Consider the operators defined on a star-shaped and can achieve a better encoding rate than Majorana fermions. plaquette-shaped portion of the lattice: We have also shown that by using a similar embedding with a qudit toric code, it is possible to construct a code = ˜ aj = ˜ bj As Xj ,Bp Zj , (29) protecting parafermion modes against parity-violating errors j∈star(s) j∈plaquette(p) where the degree of protection (i.e., distance) can be adjusted. where aj and bj are ±1, specified on the right side of Fig. 1. A similar construction has been introduced for color codes In general, As and Bp either do not overlap or overlap at using Majorana zero modes [51]. two sites. One can easily verify that the construction given in Parafermion stabilizer codes can be used for construct- Eq. (29) ensures that the commutator [As ,Bp] vanishes in both ing Hamiltonians in which commuting terms correspond to cases. We also note that both As and Bp are parity-conserving stabilizer generators. Parafermion stabilizer codes thus lead operators. The set of all As and Bp forms a stabilizer group. to a multitude of models generalizing Kitaev’s 1D chain of Due to the fact that the lattice is defined on the surface of a unpaired Majorana zero modes to higher dimensions (D>2) torus, the lattice is periodic in both dimensions, leading to the and to arbitrary interactions defined by the choice of stabilizer result generators. An important question arising here is related = = to finite-temperature stability of topological order in such As 1, Bp 1. (30) systems. In general, a 2D lattice with local interactions cannot s p lead to stable topological order at finite temperature. Thus, This implies |S|=2(ab − 1), and using Eq. (16), we find that it could be plausible to assume that by requiring some of k = 2. The logical operators Xl,Zl (l = 1,2) are horizontal the logical operators to be parity-violating operators, one and vertical loops along the lattice, as given in Fig. 2. Since can add additional protection to topological order where this these loops go all the way through the torus, they commute additional protection relies on superselection rules. Whether with the stabilizer generators As and Bp at all sites. such constructions can lead to the absence of parity-conserving We note that the parity (charge) associated with operators stringlike logical operators (e.g., stringlike logical operators is pl = 0modD [64]. Hence, the parity of the horizontal are absent in Haah’s code [65]) is an open problem. (vertical) logical operators of the parafermion toric code is a × pl (b × pl)modD. By tuning a and b, we can ensure ACKNOWLEDGMENTS that one of the logical operators will violate parity (that is, pl divides a but does not divide b). The choice of the smallest b We are grateful to L. Pryadko, K. Shtengel, and S. Bravyi would correspond to the absence of parity-violating errors. In for multiple helpful discussions. This work was supported general, b can be tuned depending on the probability of parity- in part by the NSF under Grants No. Phy-1415600 and No. violating errors. Therefore, this code construction combines NSF-EPSCoR 1004094.

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