Braiding quantum gates from partition algebras

Unitary braiding operators can be used as robust entangling quantum gates. We introduce a solution-generating technique to solve the $(d,m,l)$-generalized Yang-Baxter equation, for $m/2\leq l \leq m$, which allows to systematically construct such braiding operators. This is achieved by using partition algebras, a generalization of the Temperley-Lieb algebra encountered in statistical mechanics. We obtain families of unitary and non-unitary braiding operators that generate the full braid group. Explicit examples are given for a 2-, 3-, and 4-qubit system, including the classification of the entangled states generated by these operators based on Stochastic Local Operations and Classical Communication.


Introduction
The fragile nature of quantum entanglement is a central issue in quantum computing, which can in principle be alleviated by the use of topology. Drawing inspiration from the Aravind hypothesis [1][2][3], it has been proposed that braiding operators -operators that obey braiding relations and create knots from unlinked strands -could be thought of as quantum entanglers, i.e. gates that create entanglement from product states [4][5][6][7][8]. These initial studies about the relation between entangling gates and knots were then pushed forward in [9][10][11][12][13], paving the way to the proposal of topological quantum circuits with gates given by braiding operators [14,15].
It is expected that a physical realization of these braiding operators/entangling gates could be obtained using anyons.
One way to get braiding operators is by solving the parameter-independent Yang-Baxter equation (YBE), which we briefly review. 1 The YBE is an operator equation for an invertible matrix, R : V ⊗ V → V ⊗ V , given by (R ⊗ I) (I ⊗ R) (R ⊗ I) = (I ⊗ R) (R ⊗ I) (I ⊗ R) , (1.1) where V is a d-dimensional complex vector space and I is the identity operator on V . We use the terms Yang-Baxter operator and R-matrix interchangeably for the operator R. Solutions to (1.1) for some cases are presented in [17,18].
The R-matrix can be seen as a generalization of the permutation operator that swaps two vector spaces. This point of view is useful if one notices that the R-matrices can be used to construct representations of the Artin braid group B n on n-strands, with generators σ i satisfying for i = 1, . . . , n − 1. The first relation above is called the braid relation, whereas the second relation is the far-commutativity condition. Representations for σ i can be constructed out of the R-matrices that solve (1.1) as follows ρ(σ i ) = I ⊗i−1 ⊗ R i,i+1 ⊗ I ⊗n−i−1 .
( 1.4) Notice that this representation satisfies far-commutativity trivially. This implies that every Rmatrix that solves (1.1) can be used to construct a representation of the braid group, denoted a braiding operator.
The distinction between R-matrices and braiding operators become essential when introducing a natural generalization of the YBE [10,11] which involves two extra parameters, m and l. The linear invertible operator R : V ⊗m → V ⊗m now acts on m copies of the d-dimensional vector space V with l identity operators, and obeys R ⊗ I ⊗l I ⊗l ⊗ R R ⊗ I ⊗l = I ⊗l ⊗ R R ⊗ I ⊗l I ⊗l ⊗ R , (1.5) prompting the notation (d, m, l)-gYBE, as used in [19]. We dub this generalized R-operator as either the generalized Yang-Baxter operator or the generalized R-matrix. This generalization is important for quantum information processes that involve more than two qubits.
Unlike the R-matrix that solves the YBE in (1.1), not all generalized R-matrices that solve the (d, m, l)-gYBE in (1.5) provide a representation of the braid group, as they do not always satisfy the far-commutativity condition in (1.3). However, for the cases when 2l ≥ m (assuming l < m) far-commutativity is trivially satisfied, just as in the case of the Yang-Baxter operators. This is seen through the representations of the braid group given in terms of the generalized R-matrices by ρ(σ i ) = I ⊗l ⊗i−1 ⊗ R i,··· ,i+m−1 ⊗ I ⊗l ⊗n−i−m+1 . (1.6) We will then be interested in finding the generalized Yang-Baxter operators that satisfies the (d, m, l)-gYBE when 2l ≥ m, thus automatically leading to representations of the braid group. 2 The (d, m, l)-gYBE in (1.5) involves d 2m+2l cubic polynomial relations for d 2m unknowns (the entries of the generalized R-matrix) and is in general hard to solve. In this work we use algebraic methods to solve for the R-matrices and generalized R-matrices using partition algebras [21]. We obtain families of both unitary and non-unitary operators, with the former finding use as quantum gates in quantum computing and the latter being useful to investigate topological aspects of the gYBE that include the study of knot invariants. We focus on the quantum entangling aspects of these operators.
The paper is structured as follows. Partition algebras are reviewed in Sec. 2 where we obtain and discuss in detail R-matrices for the 2-, 3-, and 4-qubit cases. We also study the SLOCC classes of entangled states generated by these matrices. The structure of these generalized R-matrices allows to find an algorithm to systematically generate solutions of the (d, m, l)-gYBE. There are three kinds of generalized R-matrices known in the 3-qubit case [10,19,12]. In Sec. 5 we show that the 3-qubit solutions obtained in this paper are inequivalent to the known solutions. We conclude with some open questions and an outlook in Sec. 6.
2 A different approach to representations of the braid group in terms of twisted tensor products is discussed in [20].

Partition algebras
We review the notion of partition algebras following [22]. We present just the bare minimum needed in this work, pointing the reader to that reference for more details. The elements of the partition algebras, denoted CA k , are the partitions of two copies of a set: {1, 2, · · · , k} and {1 , 2 , · · · , k }. The generators of CA k are 3) whose action can be represented diagrammatically, see Fig. 1. An example of composition of these generators is shown in Fig. 2. Using these diagrams one can easily verify that the Figure 2: Example of composition in CA k . The nodes 1, · · · i − 1, i + 3, · · · , k on which the generators act trivially are suppressed.
generators satisfy the following relations Note that p i and p i+ 1 2 generate planar diagrams. Non-planarity is introduced by the permutation group generators s i . The mixed relations are and To emphasize that s i swaps the elements on the vector spaces at sites i and i+1, one could also write it as s i,i+1 , but we will stick to the notation s i to avoid cluttering. The second relations in (2.8)-(2.9) can be understood as the fundamental property of the permutation operator: The index swapping is obvious in the first relation and it becomes obvious also in the second one, when one notices that p i+ 1 2 has non-trivial support on sites i and i + 1 whereas p i+ 3 2 has non-trivial support on sites i + 1 and i + 2. To make this more transparent, one can change notation by identifying p i+ 1 2 with p i,i+1 and p i+ 3 2 with p i+1,i+2 , prompting the definition whose diagrammatic representation is shown in Fig. 3 and can be worked out using the composition laws in Fig. 2. The figure suggests one can generalize the p i+ 1 to the cases with support on two arbitrary sites, i and i + j, as satisfying the relations 14)

Representations
We employ hermitian representations for the generators, which come in two kinds: the generators of the planar diagrams can be rescaled either asymmetrically (Qubit representation) or symmetrically (Temperley-Lieb representation).
Qubit representation In this representation one of the relations satisfied by p i,i+1 is modified to corresponding to the scaling of p i,i+1 .
In this paper we deal with qubits and hence the case of d = 2. The qudit realizations can be similarly obtained through an appropriate generalization. Qubit representations are given by where 1 is the identity operator acting on the relevant Hilbert space and X i , Y i , Z i are the usual Pauli matrices acting on the qubit space on site i, written in the basis {|0 , |1 } where Z is diagonal. Another representation which is unitarily equivalent to the above is given by (2.20) Temperley-Lieb representation Now both generators of the planar diagrams are rescaled by the same factor, with the rest of the relations of the planar part of the partition algebra unchanged. The planar part of CA k can be realized using the generators of the Temperley-Lieb algebra with doubled dimensions, e 1 , · · · , e 2k−1 ∈ TL 2k , which satisfy the relations in (2.5)-(2.6), see [22]. Notice the doubling of the number of sites in this realization, as shown in Fig. 4.  R-matrices on twice the number of sites, i.e. if one obtains the generalized R-matrix that solves the (d, m, l)-gYBE, then this representation will yield another solution that solves the (d, 2m, 2l)-gYBE.

Equivalence classes of R-matrices and SLOCC classes
In quantum information theory the idea of SLOCC was introduced to classify entangled states.
It states that two quantum states are equivalent when there exists an Invertible Local Operator (ILO) that maps one state into the other: where the states |ψ 1 and |ψ 2 live in the Hilbert space ⊗ n i=1 H i [23]. A i is an ILO acting only at the site i. This equivalence relation appeals to the intuition of entangled states, as one expects local operations to not disturb the non-local entanglement property of the state.
One can also define an equivalence class of the R-matrices satisfying the parameterindependent YBE. To identify this class, one observes that if R is a solution to the (d, m, l)-gYBE, then so are αR (with α a constant), R −1 , and (A 1 ⊗ · · · ⊗ A m ) R A −1 1 ⊗ · · · ⊗ A −1 m , where A 1 , · · · , A m is an ILO that also appears in the definition of the SLOCC classes of quantum states. We can now prove the following theorem: Theorem Two entangling R-matrices R 1 and R 2 , which are equivalent under ILO, produce entangled states of the same SLOCC class.
Proof R 1 produces an entangled state |E acting on the product state |P , By assumption, one can express R 1 = AR 2 A −1 , where A is an ILO, so that This means R 2 A −1 |P = A −1 |E , by definition, both A −1 |P and A −1 |E are in the same SLOCC classes as |P and |E , respectively, hence proving the assertion.
This theorem naturally implies that if two R-matrices produce states of two different SLOCC classes, then they cannot be related by an ILO. However, the converse of the theorem is not always true: if two entangled states |E 1 and |E 2 belonging to the same SLOCC class are generated by two entangling R-matrices R 1 and R 2 , respectively, then they need not be related by an ILO. One has in fact As |E 2 = A|E 1 and |P 2 = L|P 1 , where A and L are two ILOs, one obtains Note that the ILOs A and L need not be the same. For unitary R-matrices, this relation holds on all the product states that span the Hilbert space, so that one can identify We shall use the definitions and this result to determine the classes of R-matrices.

2-qubit SLOCC classes
There are two SLOCC classes in the 2-qubit case, the Bell state class and the product state class.

3-qubit SLOCC classes
There are six SLOCC classes in the 3-qubit case [23]. Two tripartite entangled classes, GHZ state class and W state class, also denoted as ABC to symbolize the three parties of the state. Three partially entangled state classes, A−BC, AC −B, AB −C and finally the product state class, A − B − C.

4-qubit SLOCC classes
In the 4-qubit case, it was discussed in [23] that there are infinitely many SLOCC classes. Later, it was shown in [24] that there are nine families in the sense of nine different ways of entangling 4-qubits. On the other hand, it was reported in [25,26] that the number of the families is eight instead of nine for genuinely entangled states. 3 Furthermore, it was discovered in [27] that the nine families in [24] further split into 49 SLOCC entanglement classes by looking at SLOCC invariant quantities.

Generalized R-matrices
The generators of the permutation group s i solve the (d, 2, 1)-gYBE. In fact, the transposition operators s i,i+l solve the (d, m, l)-gYBE, assuming l ≤ m, with non-trivial support on the sites i and i + l. The ansatze with non-trivial support on all the m sites used in this paper are modifications of these transposition operators, with generators given by the planar part of the partition algebra. In the language of quantum gates, these ansatze are generalized SWAP gates.
In the following we discuss the 2-qubit and 3-qubit cases in detail before writing down the answers for the 4-qubit case and outlining an algorithm for an arbitrary multi-qubit generalized R-matrix.

2-qubits
On two sites i and i + 1, there are various choices of the generators p i , p i+1 , p i,i+1 and s i to construct the R-matrices. We consider the different possibilities separately.
Using s i , p i and p i+1 Consider the following ansatz for the Yang-Baxter operator with support on sites i and i + 1, with constants α, β, γ ∈ C. This operator satisfies the (d, 2, 1)-gYBE for all α, β, γ, as seen by evaluating the two sides of the YBE where repeated use of the permutation operator given in (2.10) has been applied. In a similar manner one can compute the right hand side R i+1 R i R i+1 , which turns out to be equal to (4.2).
Using p 2 i = p i , p i p i+1 = p i+1 p i and s 2 i = 1, we can show that the inverse is given by This expression is needed to check for which values of the parameters the R-matrix is unitary.
It is also easy to check that the operators in (4.1) satisfy far-commutativity for braid has trivial common support with the operator in (4.1) for all j > 1.
In general, these solutions are non-unitary and generate the infinite-dimensional braid group, i.e. the image of braid group representations built using these R-matrices is infinite. This is seen by computing the powers of R i , where the parameters are defined recursively as after identifying α 1 , β 1 and γ 1 with α, β and γ in (4.1).
By equating (4.5) and R † i , the conditions .
give a family of unitary solutions that generate the infinite-dimensional braid group just as the non-unitary case. (4.7) are explicitly solved by with θ, ϕ and φ angles between 0 and 2π. The recursive definitions of the parameters of R n i (4.6) show that R n i = 1 for any finite n when θ, ϕ and φ are generic. This can also be seen in the eigenvalues {1, ±e There are eight real unitary solutions, four of which are shown in Table 1 in the qubit representations of (2.18). (The remaining four generate the same SLOCC classes as these four.) denotes the multiplicity of the eigenvalue a. In the last column, n|R n = 1 means the lowest positive integer n satisfying R n = 1.
In the qubit representation, the (2, 2, 1)-Yang-Baxter operator takes the explicit form The unitary operators can act as quantum gates, however not all may lead to a universal set of gates. According to a theorem by Brylinski [28] for a 2-qubit space, a gate helps building a universal set if and only if it is entangling. We can use this criterion to check which of the operators in Table 1 are entangling and can potentially lead to a universal set.
A quantum gate is entangling if there is a vector |v 1 ⊗|v 2 ∈ C 2 ⊗C 2 that gets mapped to an entangled state by the quantum gate. With this definition the gates corresponding to (0, 0, −2) and (0, −2, 2) are entangling. This assertion can be checked by seeing that these gates map the most general product state in C 2 ⊗ C 2 , given by an entangled state. For example, the operator corresponding to (0, 0, −2) maps this product Using s i and p i,i+1 The operator satisfies the (d, 2, 1)-gYBE for all values of α ∈ C, as can be checked by computing The inverse, when d = 2, is given by The image of the braid group representation built using the (d, 2, 1)-R-matrix in (4.10) is infinite, as seen through its powers: with the parameters, when d = 2, defined recursively as α n = α 1 + α n−1 + 2α 1 α n−1 , after identifying α 1 with α in (4.10).
This is unitary for α * = − α 1+2α , i.e. α = 1 2 (e iθ − 1) for arbitrary angle θ, which gives real solutions α = 0, −1. From the above recursion formula, unitary solutions with generic θ generate the infinite-dimensional braid group (R n i = 1 for any finite n). For the representation of the partition algebra generators in (2.18) one obtains (4.13) The case α = 0 is just the permutation operator s i , which is not an entangler by previous considerations. That leaves us with α = −1, which is not an entangler either.

3-qubits
The number of possible operators on three sites i, i + 1 and i p i+1,i+2 and p i,i+2 , along with the permutation generators s i and s i+1 . In order to obtain valid representations of the braid group we obtain solutions to the (d, 3, 2)-gYBE.
Using s i , p i , p i+1 and p i+2 As ansatz we propose the natural generalization of (4.1) from the 2-qubit case: where s i,i+2 = s i s i+1 s i and the parameters are complex. This operator does not satisfy the (d, 3, 2)-gYBE for all values of the parameters. One can however use the identities in (2.8) to The inverse is given by where .
The image of the braid group representation constructed out of (4.14) with parameters satisfying (4.15) is infinite, as seen by computing the powers of the generalized R-matrix with the parameters defined recursively as and after identifying α 3 and γ (1) with α 1 , α 3 , β 1 , β 2 , β 3 and γ in (4.14).
There are 16 real unitary points for the parameters in (4.15), of which we discuss eight.  Tables 2, 3 and 4, respectively. The generalized R-matrices in Table 2 generate the following entangled states in the GHZ SLOCC class which are equivalent to the standard GHZ state, |000 +|111 , by the application of appropriate ILOs, such as, for example, The generalized R-matrices in Table 3 generate the partially entangled states AC − B given by respectively. The product state SLOCC class is reported instead on Table 4.   Using s i , p i,i+1 and p i+1,i+2 The ansatz with s i,i+2 = s i s i+1 s i , satisfies the (d, 3, 2)-gYBE when γ = − α+β 2 . This is seen after simplifying the expressions in the (d, 3, 2)-gYBE using the swapping property of the permutation operator in (2.10), as done before.
The generalized R-matrices in the first two rows of Table 5 are not entanglers, as they generate just the product state SLOCC class. The generalized R-matrices of the second two rows, on the other hand, are both entanglers generating the GHZ SLOCC class. In particular the states they generate by acting on |000 are which are equivalent to the standard GHZ state |000 + |111 by appropriate ILOs, such as, for example,

4-qubits
As one increases the number of qubits, the analytic computation for the generalized R-matrices gets more tedious. To illustrate the feasibility of the method, we write down the answers for the 4-qubit case as well. We use the operators s i , p i , p i+1 , p i+2 and p i+3 to build the generalized R-matrices. The generalized R-matrices that satisfy the (d, 4, 2)-gYBE and the (d, 4, 3)-gYBE also satisfy far-commutativity yielding braid group representations. These are the cases we focus on.
As in the (2, 4, 2) case, there are 64 real unitary generalized R-matrices and we write down one unitary solution from each set of eigenvalue. When the parameters are complex we obtain a family of unitary solutions that generate the full braid group.

4-qubits from 2-qubits via Temperley-Lieb generators
Up to this point we used the qubit representations of (2.20) , but this should be recognized as the (d, 4, 2)-gYBE instead of (d, 4, 1)-gYBE. Note that R i has a nontrivial support on the sites 2i − 1 to 2i + 3, whereas R i+1 has that on the sites 2i + 1 to 2i + 5. A shift of the index i of the R-matrix by one corresponds to a shift of the sites by two. The far-commutativity relation is If the Temperley-Lieb generators are expressed by hermitian matrices, we can see that the solution is unitary for with ∆ ≡ Q + Q −1 , and θ, ϕ and φ angles taking any value. For real α, β and γ, there are 8 unitary points: As a representation of the Temperley-Lieb generators for qubits at the sites i and i + 1, we use the following matrix [29]  In what follows, we discuss the corresponding entangled states for the non-trivial cases of (4.49). We obtain the R-matrices which are not equivalent with those in the (d, 4, 2)-cases in the previous subsection.
The solution satisfies R 2 i = 1, and generates entangled states as For the other states, R i does not generate entanglement. We can see that each of the four states in (4.51) is SLOCC equivalent to with some coefficient λ. This falls into what is called G abcd in [24] with a = 1 + λ, b = c = 0, d = 1 − λ, or into what is called the class of span{0 k Ψ, 0 k Ψ} in [26]. For example, the first state in (4.51) is mapped to (4.52) by successively operating the following two ILOs: (4.53) The matrix R i has eigenvalues 1 (9) and −1 (7) , which is inequivalent to any of (4.35)-(4.38).
Case III: − 2 ∆ , 0, 0 This case provides essentially the same entanglement as the case II, since the two R-matrices are connected by swapping sites as (4.54) The solution satisfies R 4 i = 1 (but R 2 i , R 3 i = 1), and gives entangled states as map the first state in (4.55) to (4.52). The eigenvalues of is inequivalent to any of (4.35)-(4.38).
This case is essentially the same as the case IV, because The solution satisfies R 2 i = 1, and generates entangled states as  (9) and −1 (7) , which is inequivalent to any of (4.35)-(4.38).
Since the solution in this case is factorized as it is easy to see that R 2 i = 1, and the entangled states obtained fall into a class of the direct product of the two Bell states or the direct product of the Bell state and a separable 2-qubit state. The eigenvalues of R i are 1 (10) and −1 (6) , which is inequivalent to any of (4.35)-(4.38).
Although various inequivalent solutions would be obtained by choosing other representations of the Temperley-Lieb algebra, we leave this issue as a future subject.

Algorithm for multi-qubit generalized R-matrices
The ansatze in (4.1), (4.14), (4.31), and (4.39) act as a guide for constructing the generalized R-matrices in the multi-qubit case, while using the available operators: s j j = i, · · · , i + m − 2 and p j j = i, · · · , i + m − 1. Throughout we assume that 2l ≥ m in order to ensure farcommutativity in addition to obeying the (d, m, l)-gYBE.
The generalized R-matrix that satisfies the (d, m, l)-gYBE and is made up of products of the p i operators is  where s i,i+l = s i+l−1 · · · s i+1 s i s i+1 · · · s i+l−1 swaps the sites i and i + l. For r = 1, r−1 j=1 p i+k j and r−2 j=1 p i+k j should be regarded as 1 and 0, respectively. The coefficients can be determined by requiring this to satisfy R i R i+l R i = R i+l R i R i+l . This computation can be done analytically, but it is tedious. At the moment we do not have a general analytic expression for the generalized R-matrix, but this algorithm works as studied in the cases of three and four qubits. As seen in those cases, we expect to obtain a family of both unitary and non-unitary generalized R-matrices. The image of the braid group representations using the non-unitary matrices and the complex unitary matrices will be infinite, whereas the image of the braid group representations of the real unitary matrices is expected to be finite. respectively. However, these solve the (2, 3, 1)-gYBEs and hence cannot be compared to the solutions in this paper, which solve instead the (2, 3, 2)-gYBE.
Comparison with the Kitaev-Wang solutions The Kitaev-Wang solutions in [12] and their qubit realizations in [13] solve the (d, 3, 1)-gYBE, whereas the methods presented in this paper generate the generalized Yang-Baxter operators that solve the (d, 3, 2)-gYBE. Thus we cannot compare the two generalized R-matrices.

Outlook
In this paper we have used partition algebras to introduce a solution-generating technique to obtain parameter-independent generalized R-matrices. This is quite remarkable, since solving the gYBE is a notoriously difficult task. This is especially true for the parameter-independent gYBE, for which very few solutions are known in the literature. In some recent work [16], we have focused on the parameter-dependent case, using supersymmetry algebras instead of partition algebras. In that case, the relation between R-matrices and braid group representations was not very clear, so that the present work should be considered as an improvement of our previous analysis.
Improved as it may be, we need however to remark that the method based on partition algebras has certain limitations. The main issue is that not all SLOCC classes of entangled states seem to be obtainable from the generalized R-matrices via partition algebras. In particular, we do not obtain the W-state SLOCC class of a 3-qubit system. We suspect that this absence of the W-state class is not peculiar to 3-qubit systems, and it might extend to the multi-qubit case as well. It would be interesting to check whether this is true, although it would represent a very laborious computation.
Of course, the physical realization of braiding operators is of the utmost interest. A natural next step would then be to try to identify the anyons corresponding to these representations and study their computational power. This could possibly help identifying the unitary modular tensor categories that describe these anyons [30].
On a complementary direction, one could construct new integrable spin chains upon Baxterizing the 2-qubit R-matrices. In particular, using the Temperley-Lieb representations of the 2-qubit R-matrices one could obtain new 4-site interaction spin chains that are integrable.