Polyadic braid operators and higher braiding gates

Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.

The modern development of the quantum computing technique implies various extensions of its foundational concepts [1][2][3]. One of the main problems in the physical realization of quantum computers is presence of errors, which implies that it is desirable that quantum computations be provided with error correction, or that ways be found to make the states more stable, which leads to the concept of topological quantum computation (for reviews, see, e.g., [4][5][6], and references therein). In the Turaev approach [7], link invariants can be obtained from the solutions of the constant Yang-Baxter equation (the braid equation). It was realized that the topological entanglement of knots and links is deeply connected with quantum entanglement [8,9]. Indeed, if the solutions to the constant Yang-Baxter equation [10] (Yang-Baxter operators/maps [11,12]) are interpreted as a special class of quantum gate, namely braiding quantum gates [13,14], then the inclusion of non-entangling gates does not change the relevant topological invariants [15,16]. For further properties and applications of braiding quantum gates, see [17][18][19][20].
In this paper we obtain and study the solutions to the higher arity (polyadic) braid equations introduced in [21,22], as a polyadic generalization of the constant Yang-Baxter equation (which is considerably different from the generalized Yang-Baxter equation of [23][24][25][26]). We introduce special classes of matrices (star and circle types), to which most of the solutions belong, and find that the so-called magic matrices [18,27,28] belong to the star class. We investigate their general non-trivial group properties and polyadic generalizations. We then consider the invertible and non-invertible matrix solutions to the higher braid equations as the corresponding higher braiding gates acting on multi-qubit states. It is important that multi-qubit entanglement can speed up quantum key distribution [29] and accelerate various algorithms [30]. As an example, we have made detailed computations for the ternary braiding gates as solutions to the ternary braid equations [21,22]. A particular solution to the n-ary braid equation is also presented. It is shown, that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete relations for that are obtained, which can allow us to build non-entangling networks.

II. YANG-BAXTER OPERATORS
Recall here [9,13] the standard construction of the special kind of gates we will consider, the braiding gates, in terms of solutions to the constant Yang-Baxter equation [10] (called also algebraic Yang-Baxter equation [31]), or the (binary) braid equation [21].

A. Yang-Baxter maps and braid group
First we consider a general abstract construction of the (binary) braid equation. Let V be a vector space over a field K and the mapping C V 2 : V ⊗ V → V ⊗ V, where ⊗ = ⊗ K is the tensor product over K. A linear operator (braid operator) C V 2 is called a Yang-Baxter operator (denoted by R in [13] and by B in [10]) or Yang-Baxter map [12] (denoted by F in [11]), if it satisfies the braid equation [32][33][34] where id V : V → V, is the identity operator in V. The connection of C V 2 with the R-matrix R is given by C V 2 = τ • R, where τ is the flip operation [10,11,32]. Let us introduce the operators A 1,2 : It follows from (2.1) that If C V 2 is invertible, then C −1 V 2 is also the Yang-Baxter map with A −1 1 and A −1 2 . Therefore, the operators A i represent the braid group B 3 = {e, σ 1 , σ 2 | σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 } by the mapping π 3 as (2.4) The representation π m of the braid group with m strands B m = {e, σ 1 , . . . , σ m−1 σ i σ i+1 σ i = σ i+1 σ i σ i+1 , i = 1, . . . , m − 1, can be obtained using operators A i (m) :V ⊗m →V ⊗m analogous to (2.2) by the mapping π m : B m → EndV ⊗m in the following way π m (σ i ) = A i (m) , π m (e) = A 0 (m) . (2.7) In this notation (2.2) is A i = A i (2), i = 1, 2, and therefore (2.3) represents B 3 by (2.4).

B. Constant matrix solutions to the Yang-Baxter equation
Consider next a concrete version of the vector space V which is used in the quantum computation, a d-dimensional euclidean vector space V d over complex numbers C with a basis {e i }, i = 1, . . . , d. A linear operator V d → V d is given by a complex d × d matrix, the identity operator id V becomes the identity d × d matrix I d , and the Yang-Baxter map C V 2 is a d 2 × d 2 matrix C d 2 (denoted by R in [31]) satisfying the matrix algebraic Yang-Baxter equation , (2.8) being an equality between two matrices of size d 3 × d 3 . We use the unified notations which can be straightforwardly generalized for higher braid operators. In components . (2.10) The system (2.10) is highly overdetermined, because the matrix C d 2 contains d 4 unknown entries, while there are d 6 cubic polynomial equations for them. So for d = 2 we have 64 equations for 16 unknowns, while for d = 3 there are 729 equations for the 81 unknown entries of C d 2 . The unitarity of C d 2 imposes a further d 2 quadratic equations, and so for d = 2 we have in total 68 equations for 16 unknowns. This makes the direct discovery of solutions for the matrix Yang-Baxter equation (2.10) very cumbersome. Nevertheless, using a conjugation classes method, the unitary solutions and their classification for d = 2 were presented in [31].
In the standard matrix form (2.9) can be presented by introducing the 4-dimensional vector spaceṼ 4 = V ⊗ V with the natural basisẽk = {e 1 ⊗ e 1 , e 1 ⊗ e 2 , e 2 ⊗ e 1 , e 2 ⊗ e 2 }, wherek = 1, . . . , 8 is a cumulative index. The linear operatorC 4 :Ṽ 4 →Ṽ 4 corresponding to (2.9) is given by 4 × 4 matrixcĩj asC 4 •ẽĩ = 4 j=1cĩj ·ẽj. The operators (2.2) become two 8 × 8 matrices A 1,2 asÃ 1 =c ⊗ K I 2 ,Ã 2 = I 2 ⊗ Kc , (2.11) where ⊗ K is the Kronecker product of matrices and I 2 is the 2 × 2 identity matrix. In this notation (which is universal and also used for higher braid equations) the operator binary braid equations (3.7) become a single matrix equatioñ which we call the matrix binary braid equation (and also the constant Yang-Baxter equation [31]). In component form (2.12) is a highly overdetermined system of 64 cubic equations for 16 unknowns, the entries ofc. The matrix equation (2.12) has the following "gauge invariance", which allows a classification of Yang-Baxter maps [35]. Introduce an invertible operator Q : V → V in the two-dimensional vector space V ≡ V d=2 . In the basis {e 1 , e 2 } its 2 × 2 matrix q is given by Q • e i = 2 j=1 q ij · e j . In the natural 4-dimensional basisẽk the tensor product of operators Q ⊗ Q is presented by the Kronecker product of matricesq 4 = q ⊗ K q. If the 4 × 4 matrixc is a fixed solution to the Yang-Baxter equation (2.12), then the family of solutionsc (q) corresponding to the invertible 2 × 2 matrix q is the conjugation ofc byq 4 such that which follows from conjugating (2.12) by q ⊗ K q and using (2.11). If we include the obvious invariance of (2.12) with respect to an overall factor t ∈ C, the general family of solutions becomes (cf. the Yang-Baxter equation [35]) c (q, t) = tq 4cq −1 (2.14) It follows from (2.13) that the matrix q ∈ GL (2, C) is defined up to a complex non-zero factor. In this case we can put q = a 1 c d , (2.15) and The matrixq ⋆ 4q4 (where ⋆ represents Hermitian conjugation) is diagonal (this case is important in a further classification similar to the binary one [31]), when the condition holds, and so the matrix q takes the special form (depending on 2 complex parameters) We call two solutionsc 1 andc 2 of the constant Yang-Baxter equation and we will not distinguish between them. The q-conjugation in the form (2.19) does not require the invertibility of the matrix q, and therefore the solutions of different ranks (or invertible and not invertible) can be q-conjugated (for the invertible case, see [35][36][37]). The matrix equation (2.12) does not imply the invertibility of solutions, i.e. matricesc being of full rank (in the binary Yang-Baxter case of rank 4 and d = 2). Therefore, below we introduce in a unified way invertible and non-invertible solutions to the matrix Yang-Baxter equation (2.10) for any rank of the corresponding matrices.

C. Partial identity and unitarity
To be as close as possible to the invertible case, we introduce "non-invertible analogs" of identity and unitarity. Let M be a diagonal n × n matrix of rank r ≤ n, and therefore with n − r zeroes on the diagonal. If the other diagonal elements are units, such a diagonal M can be reduced by row operations to a block matrix, being a direct sum of the identity matrix I r×r and the zero matrix Z (n−r)×(n−r) . We call such a diagonal matrix a block r-partial identity I and without the block reduction-a shuffle r-partial identity I (shuf f le) n (r) (these are connected by conjugation). We will use the term partial identity and I n (r) to denote any matrix of this form. Obviously, with the full rank r = n we have I n (n) ≡ I n , where I n is the identity n × n matrix. As with the invertible case and identities, the partial identities (of the corresponding form) are trivial solutions of the Yang-Baxter equation.
If a matrix M = M (r) of size n × n and rank r satisfies the following r-partial unitarity condition where M (r) ⋆ is the conjugate-transposed matrix and I (1) n (r), I n (r) are partial identities (of any kind, they can be different), then M (r) is called a r-partial unitary matrix. In the case, when I (1) n (r) = I (2) n (r), the matrix M (r) is called normal. If M (r) ⋆ = M (r), then it is called r-partial self-adjoint. In the case of full rank r = n, the conditions (2.20)-(2.21) become ordinary unitarity, and M (n) becomes an unitary (and normal) matrix, while a r-partial self-adjoint matrix becomes a selfadjoint matrix or Hermitian matrix.
As an example, we consider a 4 × 4 matrix of rank 3 which satisfies the 3-partial unitarity conditions (2.20)-(2.21) with two different 3-partial identities on the r.h.s.
D. Permutation and parameter-permutation 4-vertex Yang-Baxter maps The system (2.12) with respect to all 16 variables is too cumbersome for direct solution. The classification of all solutions can only be accomplished in special cases, e.g. for matrices over finite fields [35] or for fewer than 16 vertices. Here we will start from 4-vertex permutation and parameter-permutation matrix solutions and investigate their group structure. It was shown [13,31] that the special 8-vertex solutions to the Yang-Baxter equation are most important for further applications including braiding gates. We will therefore study the 8-vertex solutions in the most general way: over C and in various configurations, invertible and not invertible, and also consider their group structure.
First, we introduce the permutation Yang-Baxter maps which are presented by the permutation matrices (binary matrices with a single 1 in each row and column), i.e. 4-vertex solutions. In total, there are 64 permutation matrices of size 4 × 4, while only 4 of them have the full rank 4 and simultaneously satisfy the Yang-Baxter equation (2.12). These are the following trc = 2, detc = −1, eigenvalues: {1} [2] , {−1} [2] , (2.28) (2.29) Here and next we list eigenvalues to understand which matrices are conjugated, and after that, if and only if the conjugation matrix is of the form (2.16), then such solutions to the Yang-Baxter equation (2.12) coincide. The traces are important in the construction of corresponding link invariants [7] and local invariants [39,40], and determinants are connected with the concurrence [41,42]. Note that the first matrix in (2.28) is the SWAP quantum gate [1].
To understand symmetry properties of (2.28)-(2.29), we introduce the so called reverse matrix J ≡ J n of size n × n by For any n × n matrix M ≡ M n the matrix JM is the matrix M reflected vertically, and the product M J is M reflected horizontally. In addition to the standard symmetric matrix satisfying M = M T (T is the transposition), one can introduce Thus, a persymmetric matrix is symmetric with respect to the minor diagonal, while a 90 • -symmetric matrix is symmetric under 90 • -rotations. A bisymmetric matrix is symmetric and persymmetric simultaneously. In this notation, the first family of the permutation solutions (2.28) are bisymmetric, but not 90 • -symmetric, while the second family of the solutions (2.29) are, oppositely, 90 • -symmetric, but not symmetric and not persymmetric (which explains their notation).
In the next step, we define the corresponding parameter-permutation solutions replacing the units in (2.28) by parameters. We found the following four 4-vertex solutions to the Yang-Baxter equation (2.12) (2.34) The first pair of solutions (2.33) correspond to the bi-symmetric permutation matrices (2.28), and we call them star-like solutions, while the second two solutions (2.34) correspond to the 90 • -symmetric matrices (2.28) which are called circle-like solutions.
The first (second) star-like solution in (2.33) with y = z (x = t) becomes symmetric (persymmetric), while on the other hand with x = t (y = z) it becomes persymmetric (symmetric). They become bisymmetric parameter-permutation solutions if all the parameters are equal x = y = z = t. The circle-like solutions (2.34) are 90 • -symmetric when x = y.
Using q-conjugation (2.14) one can next get families of solutions depending from the entries of q, the additional complex parameters in (2.15). E. Group structure of 4-vertex and 8-vertex matrices Let us analyze the group structure of 4-vertex matrices (2.33)-(2.34) with respect to matrix multiplication, i.e. which kinds of subgroups in GL (4, C) they can form. For this we introduce four 4-vertex 4 × 4 matrices over C: two star-like matrices and two circle-like matrices . (2.36) Denoting the corresponding sets by N star1,2 = {N star1,2 } and N circ1,2 = {N circ1,2 }, these do not intersect and are closed with respect to the following multiplications (2.40) Note that there are no closed binary multiplications among the sets of 4-vertex matrices (2.35)-(2.36).
To give a proper group interpretation of (2.37)-(2.40), we introduce a k-ary (polyadic) general linear semigroup , where M f ull = {M n×n } is the set of n × n matrices over C and µ [k] is an ordinary product of k matrices. The full semigroup GLS [k] (n, C) is derived in the sense that its product can be obtained by repeating the binary products which are (binary) closed at each step. However, n × n matrices of special shape can form k-ary subsemigroups of GLS [k] (n, C) which can be closed with respect to the product of at minimum k matrices, but not of 2 matrices, and we call such semigroups k-non-derived. Moreover, we have for the sets N star1,2 and N circ1,2 (2.41) A simple example of a 3-nonderived subsemigroup of the full semigroup GLS [k] (n, C) is the set of antidiagonal matrices M adiag = {M adiag } (having nonzero elements on the minor diagonal only): the product µ [3] of 3 matrices from M adiag is closed, and therefore M adiag is a subsemigroup S [3] adiag = M adiag | µ [3] of the full ternary general linear semigroup GLS [3] (n, C) with the multiplication µ [3] as the ordinary triple matrix product.
In the theory of polyadic groups [43] an analog of the binary inverse M −1 is given by the querelement, which is denoted bȳ M and in the matrix k-ary case is defined by whereM can be on any place. If each element of the k-ary semigroup GLS [k] (n, C) (or its subsemigroup) has its querelement M , then this semigroup is a k-ary general linear group GL [k] (n, C).
In the set of n × n matrices the binary (ordinary) product is defined (even it is not closed), and for invertible matrices we formally determine the standard inverse M −1 , but for arity k ≥ 4 it does not coincide with the querelementM , because, as follows from (2.42) and cancellativity in C thatM (2.43) The k-ary (polyadic) identity I n is called left (right) polyadic identity. For instance, in the subsemigroup (in GLS [k] (n, C)) of antidiagonal matrices S [3] adiag the ternary identity I [3] n can be chosen as the n × n reverse matrix (2.30) having units on the minor diagonal, while the ordinary n × n unit matrix I n is not in S [3] adiag . It follows from (2.44), that for matrices over C the (left, right) polyadic identity I [k] n is (2. 45) which means that for the ordinary matrix product I n is a reflection of (k − 1) degree), while both sides cannot belong to a subsemigroup S [k] of GLS [k] (n, C) under consideration (as in S [3] adiag ). As the solutions of (2.45) are not unique, there can be many k-ary identities in a k-ary matrix semigroup. We denote the set of k-ary identities by I In the case of S [3] adiag the ternary identity I [3] n can be chosen as any of the n × n reverse matrices (2.30) with unit complex numbers e iαj , j = 1, . . . , n on the minor diagonal, where α j satisfy additional conditions depending on the semigroup. In the concrete case of S [3] adiag the conditions, giving (2.45), are (k − 1) α j = 1 + 2πr j , r j ∈ Z, j = 1, . . . , n. In the framework of the above definitions, we can interpret the closed products (2.37)-(2.38) as the multiplications µ [3] of the ternary semigroups S [3] star1,2 (4, C) = N star1,2 | µ [3] . The corresponding querelements are given bȳ The ternary semigroups having querelements for each element (i.e. the additional operation ( ) defined by (2.46)) are the ternary groups G [3] star1,2 (4, C) = N star1,2 | µ [3] , ( ) which are two (non-intersecting because N star1 ∩N star2 = ∅) subgroups of the ternary general linear group GL [3] (4, C). The ternary identities in G [3] star1,2 (4, C) are the following different continuous sets I [3] star1,2 = I [3] star1,2 , where In the particular case α j = 0, j = 1, 2, 3, 4, the ternary identities (2.47)-(2.48) coincide with the bisymmetric permutation matrices (2.28).
First, we have the triple relations "inside" star and circle matrices (2.57) We observe the following module structures on the left column above (elements of the corresponding module are in brackets, and we informally denote modules by their sets): 1) from (2.52)-(2.54), the set N star2 is a middle, right and left module over N star1 ; 2) from (2.55)-(2.57), the set N star1 is a middle, right and left module over N star2 ; 3) from (2.60)-(2.63), the sets N circ1,2 are a right and left module over N star1,2 ; 4) from (2.64)-(2.65), the sets N star1,2 are a middle ternary module over N circ1,2 ; If M star and M circ are invertible (the determinants in (2.76)-(2.77) are non-vanishing), then and in terms of sets we can write M star = N star1 ∪ N star2 and M circ = N circ1 ∪ N circ2 , while N star1 ∩ N star2 = ∅ and N circ1 ∩ N circ2 = ∅ (see (2.41)). Note that, if M star and M circ are treated as elements of an algebra, then (2.80)-(2.82) are reminiscent of the Cartan decomposition (see, e.g., [46]), but we will consider them from a more general viewpoint, which will treat such structures as semigroups, ternary groups and modules. Thus the set M 8vertex = M star ∪ M circ is closed, and because of the associativity of matrix multiplication, M 8vertex forms a non-commutative semigroup which we call a 8-vertex matrix semigroup S 8vertex (4, C), which contains the zero matrix Z ∈ S 8vertex (4, C) and is a subsemigroup of the (binary) general linear semigroup GLS (4, C). It follows from (2.80), that M star is its subsemigroup S star 8vertex (4, C). Moreover, the invertible elements of S 8vertex (4, C) form a 8-vertex matrix group G 8vertex (4, C), because its identity is a unit 4 × 4 matrix I 4 ∈ M star , and so M star is a subgroup G star 8vertex (4, C) of G 8vertex (4, C) and a subgroup of the (binary) general linear group GL (4, C). The structure of S 8vertex (4, C) (2.80) is similar to that of block-diagonal and block-antidiagonal matrices (of the necessary sizes). So the 8-vertex (binary) matrix semigroup S 8vertex (4, C) in which the parameters satisfy (2.79) is a 8-vertex (binary) matrix group G 8vertex (4, C), having a subgroup G star 8vertex (4, C) = M star | ·, I 4 , where (·) is an ordinary matrix product, and I 4 is its identity. The group structure of the circle matrices M circ (2.77) follows from M circ | ν [3] is a ternary (3-nonderived) semigroup with the zero Z ∈ M circ which is a subsemigroup of the ternary (derived) general linear semigroup GLS [3] (4, C). Instead of the inverse, for each invertible element M circ ∈ M circ \ Z we introduce the unique querelementM circ [43] by (2.42), and since the ternary product is the triple ordinary product, we haveM circ = M −1 circ from (2.43). Thus, if the conditions of invertibility (2.79) hold valid, then the ternary semigroup S circ(3) [3] , () which does not contain the ordinary (binary) identity, since Nevertheless, the ternary group of circle matrices G circ [3] 8vertex (4, C) has the following set I [3] circ = I [3] circ of left-right 6-vertex and 8-vertex ternary identities (see (2.44)-(2.45)) which (without additional conditions) depend upon the free parameters a, b, c, d ∈ C, b, c = 0, and I circ ∈ M circ . In the binary sense, the matrices from (2.84) are mutually similar, but as ternary identities they are different.
If we consider the second operation for matrices (as elements of a general matrix ring), the binary matrix addition (+), the structure of M 8vertex = M star ∪ M circ becomes more exotic: the set M star is a (2, 2)-ring R star [2,2] 8vertex = M star | +, · with the binary addition (+) and binary multiplication (·) from the semigroup S star [3] with the binary matrix addition (+), the ternary matrix multiplication ν [3] and the zero Z.
Moreover, because of the distributivity and associativity of binary matrix multiplication, the relations (2.81) mean that the set M circ (being an abelian group under binary addition) can be treated as a left and right binary module M circ 8vertex over the ring R star (2,2) 8vertex with an operation ( * ): the module action M circ * M star = M circ , M star * M circ = M circ (coinciding with the ordinary matrix product (2.81)). The left and right modules are compatible, since the associativity of ordinary matrix multiplication gives the compatibility condition 8vertex , and therefore M circ (as an abelian group under the binary addition (+) and the module action ( * )) is a R The last relation (2.82) shows another interpretation of M circ as a formal "square root" of M star (as sets).

F. Star 8-vertex and circle 8-vertex Yang-Baxter maps
Let us consider the star 8-vertex solutionsc to the Yang-Baxter equation (2.12), having the shape (2.76), in the most general setting, over C and for different ranks (i.e. including noninvertible ones). In components they are determined by Solutions from, e.g. [31,35], etc., should satisfy this overdetermined system of 24 cubic equations for 8 variables.
We search for the 8-vertex constant solutions to the Yang-Baxter equation over C without additional conditions, unitarity, etc. (which will be considered in the next sections). We also will need the matrix functions tr and det which are related to link invariants, as well as eigenvalues which help to find similar matrices and q-conjugated solutions to braid equations. Take into account that the Yang-Baxter maps are determined up to a general complex factor t ∈ C (2.14). For eigenvalues (which are determined up to the same factor t) we use the notation: {eigenvalue} [algebraic multiplicity] .
We found the following 8-vertex solutions, classified by rank and number of parameters.
Note that only the first and the last cases are genuine 8-vertex Yang-Baxter maps, because the three-parameter matrices (2.87) are q-conjugated with the 4-vertex parameter-permutation solutions (2.33). Indeed, where b ∈ C is a free parameter. If b = y z two matrices q in (2.90) are similar, and we have the unique q-conjugation (2.89). Another solution in (2.87) is q-conjugated to the second 4-vertex parameter-permutation solutions (2.33) such that where q's are pairwise similar in (2.92), and therefore we have 2 different q-conjugations.
The circle 8-vertex solutionsc to the Yang-Baxter equation (2.12) of the shape (2.77) are determined by the following system of 32 cubic equations for 8 unknowns over C (2.95) We found the 8-vertex solutions, classified by rank and number of parameters.
• Rank = 4 (invertible circle Yang-Baxter map) are quadratic in parameters There are no circle 8-vertex solutions of rank 3. The corresponding families of solutions can be derived from the above by using the q-conjugation (2.14).
A particular case of the 8-vertex circle solution (2.96) was considered in [49].
G. Triangle invertible 9and 10-vertex solutions There are some higher vertex solutions to the Yang-Baxter equations which are not in the above star/circle classification. They are determined by the following system of 15 cubic equations for 9 unknowns over C We found the following 9-vertex Yang-Baxter maps 99) trc = 2x, detc = −x 4 , x = 0, eigenvalues: {x} [3] , −x. (2.100) The third matrix in (2.99) is conjugated with the 4-vertex parameter-permutation solutions (2.33) of the form (which has the same the same eigenvalues (2.100))c The matrix (2.102) cannot be presented as the Kronecker product q ⊗ K q (2.16), and so the third matrix in (2.99) The 4-parameter 9-vertex solution is We also found 5-parameter, 9-vertex solution of the form Finally, we found the following 3-parameter 10-vertex solutioñ This solution is conjugated with the 4-vertex parameter-permutation solutions (2.33) of the form (which has the same the same eigenvalues as (2.111))c by the conjugated matrix  .12) can be obtained from the above ones by using the q-conjugation (2.14).

III. POLYADIC BRAID OPERATORS AND HIGHER BRAID EQUATIONS
The polyadic version of the braid equation (2.1) was introduced in [21]. Here we define higher analog of the Yang-Baxter operator and develop its connection with higher braid groups and quantum computations.
Let us consider a vector space V over a field K. A polyadic (n-ary) braid operator C V n is defined as the mapping [21] C V n : The polyadic analog of the braid equation (2.1) was introduced in [21] using the associative quiver technique [45]. Let us introduce n operators

2)
i.e. p is a place of C V n instead of one id V in id ⊗n V . A system of (n − 1) polyadic (n-ary) braid equations is defined by (3.6) Example 1. In the lowest non-binary case n = 3, we have the ternary braid operator Note that the higher braid equations presented above differ from the generalized Yang-Baxter equations of [23,24,50]. The higher braid operators (3.1) satisfying the higher braid equations (3.4)-(3.6) can represent the higher braid group [22] using (2.6) and (3.3). By analogy with (2.6) we introduce m operators by The representation π [n] m of the higher braid group B [n+1] m (of (n + 1)-degree in the notation of [22]) (having m − 1 generators σ i and identity e) is given by In this way, the generators σ i of the higher braid group B [n+1] m satisfy the relations • n higher braid relations
In the case m = 4 and n = 3 the higher braid group B [4] 4 is represented by (3.7) and generated by 3 generators σ 1 , σ 2 , σ 3 , which satisfy 2 braid relations only (without far commutativity) According to (3.16)-(3.17), the far commutativity relations appear when the number of elements of the higher braid groups satisfy such that all conditions (3.18) should hold. Thus, to have the far commutativity relations in the ordinary (binary) braid group (2.5) we need 3 generators and B 4 , while for n = 3 we need at least 7 generators σ i and B [4] 8 (see Example 7.12 in [22]). In the concrete realization of V as a d-dimensional euclidean vector space V d over the complex numbers C and basis {e i }, i = 1, . . . , d, the polyadic (n-ary) braid operator C V n becomes a matrix C d n of size d n × d n which satisfies n − 1 higher braid equations (3.4)-(3.6) in matrix form. In the components, the matrix braid operator is

IV. SOLUTIONS TO THE TERNARY BRAID EQUATIONS
Here we consider some special solutions to the minimal ternary version (n = 3) of the polyadic braid equation (3.4)-(3.6), the ternary braid equation (3.7).

A. Constant matrix solutions
Let us consider the following two-dimensional vector space V ≡ V d=2 (which is important for quantum computations) and the component matrix realization (3.21) of the ternary braiding operator where ⊗ K is the Kronecker product of matrices and I 2 is the 2 × 2 identity matrix. In this notation the operator ternary braid equations (3.7) become the matrix equations (cf. (3.4)-(3.6)) with n = 3) which we call the total matrix ternary braid equations. Some weaker versions of ternary braiding are described by the partial braid equations where, obviously, two of them are independent. It follows from (3.4)-(3.6) that the weaker versions of braiding are possible for n ≥ 3, only, so for higher than binary braiding (the Yang-Baxter equation (2.8)). Thus, comparing (4.3) and (3.19) we conclude that (for each invertible matrixc in (4.2) satisfying (4.3)) the isomorphism π [4] 4 : σ i →Ã i , i = 1, 2, 3 gives a representation of the braid group B [4] 4 by 32 × 32 matrices over C. Now we can generate families of solutions corresponding to (4.2)-(4.3) in the following way. Consider an invertible operator Q : V → V in the two-dimensional vector space V ≡ V d=2 . In the basis {e 1 , e 2 } its 2 × 2 matrix q is given by Q • e i = 2 j=1 q ij · e j . In the natural 8-dimensional basisẽk the tensor product of operators Q ⊗ Q ⊗ Q is presented by the Kronecker product of matricesq 8 = q ⊗ K q ⊗ K q. Let the 8 × 8 matrixc be a fixed solution to the ternary braid matrix equations (4.3). Then the family of solutionsc (q) corresponding to the invertible 2 × 2 matrix q is the conjugation ofc byq 8 so that This also follows directly from the conjugation of the braid equations (4.3)-(4.6) by q ⊗ K q ⊗ K q ⊗ K q ⊗ K q and (4.2). If we include the obvious invariance of the braid equations with the respect of an overall factor t ∈ C, the general family of solutions becomes (cf. the Yang-Baxter equation [35]) and then the manifest form ofq 8 isq a 3 a 2 b a 2 b ab 2 a 2 b ab 2 ab 2 b 3 a 2 c a 2 d abc abd abc abd b 2 c b 2 d a 2 c abc a 2 d abd abc b 2 c abd b 2 d ac 2 acd acd ad 2 bc 2 bcd bcd bd 2 a 2 c abc abc b 2 c a 2 d abd abd b 2 d ac 2 acd bc 2 bcd acd ad 2 bcd bd 2 ac 2 bc 2 acd bcd acd bcd ad 2 bd 2 It is important that not every conjugation matrix has this very special form (4.10), and that therefore, in general, conjugated matrices are different solutions of the ternary braid equations (4.3). The matrixq ⋆ 8q 8 (⋆ being the Hermitian conjugation) is diagonal (this case is important for further classification similar to the binary one [31]), when the conditions ab * + cd * = 0 (4.11) hold, and so the matrix q has the special form (depending of 3 complex parameters, for d = 0) We can present the families (4.7) for different ranks, because the conjugation by an invertible matrix does not change rank. To avoid demanding (4.11), due to the cumbersome calculations involved, we restrict ourselves to a triangle matrix for q (4.9). In general, there are 8 × 8 = 64 unknowns (elements of the matrixc), and each partial braid equation (4.4)-(4.6) gives 32 × 32 = 1024 conditions (of power 4) for the elements ofc, while the total braid equations (4.3) give twice as many conditions 1024 × 2 = 2048 (cf. the binary case: 64 cubic equations for 16 unknowns (2.8)). This means that even in the ternary case the higher braid system of equations is hugely overdetermined, and finding even the simplest solutions is a non-trivial task.

B. Permutation and parameter-permutation 8-vertex solutions
First we consider the case whenc is a binary (or logical) matrix consisting of {0, 1} only, and, moreover, it is a permutation matrix (see Subsection II D). In the latter casec can be considered as a matrix over the field F 2 (Galois field GF (2)). In total, there are 8! = 40 320 permutation matrices of the size 8 × 8. All of them are invertible of full rank 8, because they are obtained from the identity matrix by permutation of rows and columns.
We have found the following four invertible 8-vertex permutation matrix solutions to the ternary braid equations (4.3)   , trc = 4, detc = 1, eigenvalues: {1} [4] , {−1} [4] . (4.14) The first two solutions (4.13) are given by bisymmetric permutation matrices (see (2.31)), and we call them 8-vertex bisymm1 and bisymm2 respectively. The second two solutions (4.14) are symmetric matrices only (we call them 8-vertex symm1 and symm2), but one matrix is a reflection of the other with respect to the minor diagonal (making them mutually persymmetric). No 90 • -symmetric (see (2.32)) solution for the ternary braid equations (4.3) was found. The bisymmetric and symmetric matrices have the same eigenvalues, and are therefore pairwise conjugate, but not q-conjugate, because the conjugation matrices do not have the form (4.10). Thus they are 4 different permutation solutions to the ternary braid equations (4.3). Note that the bisymm1 solution (4.13) coincides with the three-qubit swap operator introduced in [18].
All the permutation solutions are reflections (or involutions)c 2 = I 8 having detc = +1, eigenvalues {1, −1}, and are semimagic squares (the sums in rows and columns are 1, but not the sums in both diagonals). The 8-vertex permutation matrix solutions do not form a binary or ternary group, because they are not closed with respect to multiplication.
By analogy with (2.33)-(2.34), we obtain the 8-vertex parameter-permutation solutions from (4.13)-(4.14) by replacing units with parameters and then solving the ternary braid equations (4.3). Each type of the permutation solutions bisymm1, 2 and symm1, 2 from (4.13)-(4.14) will give a corresponding series of parameter-permutation solutions over C. The ternary braid maps are determined up to a general complex factor (see (2.14) for the Yang-Baxter maps and (4.8)), and therefore we can present all the parameter-permutation solutions in polynomial form.
The products (4.57) mean that both M ′ star and M ′ circ are separately closed with respect to binary matrix multiplication (·), and therefore S star 16vert = M ′ star | · and S circ 16vert = M ′ circ | · are semigroups. We denote their intersection by S diag 8vert = S star 16vert ∩ S circ 16vert which is a semigroup of diagonal 8-vertex matrices. In case, the invertibility conditions  54), the subgroups G star 16vert and G circ 16vert (as well as the semigroups S star 16vert and S circ 16vert ) are isomorphic by the obvious isomorphism where U ′ is in (4.54). The "interaction" between M ′ star and M ′ circ also differs from the binary case (2.81), because where M ′ quad is a set of 32-vertex so called quad-matrices of the form Because of (4.60), the set M ′ quad is closed with respect to matrix multiplication, and therefore (for invertible matrices M ′ quad ) the group G quad 32vert = M ′ quad | ·, ( ) −1 , I 8 is a subgroup of GL (8, C). So, in trying to find higher 32-vertex solutions (having at most half as many unknown variables as a general 8 × 8 matrix) to the ternary braid equations (4.3) it is worthwhile to search within the class of quad-matrices (4.61). Thus, the group structure of the above 16-vertex 8 × 8 matrices (4.57)-(4.60) is considerably different to that of 8-vertex 4 × 4 matrices (2.76)-(2.77)as the former contains two isomorphic binary subgroups G star 16vert and G circ 16vert of GL (8, C) (cf. (2.80)-(2.82) and (4.57)).
The sets M ′ star , M ′ circ and M ′ quad are closed with respect to matrix addition as well, and therefore (because of the distributivity of C) they are the matrix rings R star 16vert , R circ 16vert and R quad 32vert , respectively. In the invertible case (4.53) and det M ′ quad = 0, these become matrix fields.

E. Pauli matrix presentation of the star and circle 16-vertex constant matrices
The main peculiarity of the 16-vertex 8 × 8 matrices (4.57)-(4.60) is the fact that they can be expressed as special tensor (Kronecker) products of the Pauli matrices (see, also, [18,27]). Indeed, let where ρ i are Pauli matrices (we have already used the letter "σ" for the braid group generators (2.5)) In particular, for the 8-vertex permutation solutions (4.13)-(4.14) of the ternary braid equations (4.3) we havẽ Similarly, one can obtain the Pauli matrix presentation for the general star and circle 16-vertex matrices (4.49) and (4.50) which will contain linear combinations of the 16 parameters as coefficients before the Σ's.
G. Higher 2 n -vertex constant solutions to n-ary braid equations Next we considered the 4-ary constant braid equations (3.4)-(3.6) and found the following 32-vertex star solutioñ (4.81) We may compare ( (4.82) Informally we call such solutions the "Minkowski" star solutions, since their legs have the "Minkowski signature". Thus, we assume that in the general case for the n-ary braid equation there exist 2 n+1 -vertex 2 n × 2 n matrix "Minkowski" star invertible solutions of the above formc This allows us to use the general solution (4.83) as n-ary braiding quantum gates with an arbitrary number of qubits.

V. INVERTIBLE AND NONINVERTIBLE QUANTUM GATES
Informally, quantum computing consists of preparation (setting up an initial quantum state), evolution (by a quantum circuit) and measurement (projection onto the final state). Mathematically (in the computational basis) the initial state is a vector in a Hilbert space (multi-qubit state), the evolution is governed by successive (quantum circuit) invertible linear transformations (unitary matrices called quantum gates) and the measurement is made by non-invertible projection matrices to leave only one final quantum (multi-qubit) state. So, quantum computing is non-invertible overall, and we may consider non-invertible transformations at each step. It was then realized that one can "invite" the Yang-Baxter operators (solutions of the constant Yang-Baxter equation) into quantum gates , providing a means of entangling otherwise non-entangled states. This insight uncovered a deep connection between quantum and topological computation (see for details, e.g. [9,13]).
Here we propose extending the above picture in two directions. First, we can treat higher braided operators as higher braiding gates. Second, we will analyze the possible role of non-invertible linear transformations (described by the partial unitary matrices introduced in (2.20)-(2.21)), which can be interpreted as a property of some hypothetical quantum circuit (for instance, with specific "loss" of information, some kind of "dissipativity" or "vagueness"). This can be considered as an intermediate case between standard unitary computing and the measurement only computing of [51].
To establish notation recall [1], that in the computational basis (vector representation) and Dirac notation, a (pure) one-qubit state is described by a vector in two-dimensional Hilbert space where a i is a probability amplitude of |i . Sometimes, for a one-qubit state it is convenient to use the Bloch unit sphere representation (normalized up to a general unimportant and unmeasurable phase) A (pure) state of L-qubits ψ (L) is described by 2 L amplitudes, and so is a vector in 2 L -dimensional Hilbert space. If ψ (L) cannot be presented as a tensor product of L one-qubit states (5.1), it is called entangled. For instance, a two-qubit pure state ψ (2) = a 00 |00 + a 01 |01 + a 10 |10 + a 11 |11 , |a 00 | 2 + |a 01 | 2 + |a 10 | 2 + |a 11 | 2 = 1, a ij ∈ C, , i, j = 1, 2, (5 .3) is entangled, if det (a ij ) = 0, and the concurrence is the measure of entanglement 0 ≤ C (2) ≤ 1. It follows from (5.1), that the tensor product of states has vanishing concurrence C (2) (|ψ 1 ⊗ |ψ 2 ) = 0. An example of the maximally entangled (C (2) = 1) two-qubit states is the (first) Bell state ψ (2) Bell = (|00 + |11 )/ √ 2. The concurrence of the three-qubit state Thus, if the three-qubit state (5.5) is not entangled, then C (3) = 0 (for the tensor product of one-qubit states). One of the maximally entangled (C (3) = 1) three-qubit states is the GHZ state ψ (3) A quantum L-qubit gate is a linear transformation of 2 L -dimensional Hilbert space C 2 ⊗L → C 2 ⊗L which in the computational basis (5.1) is described of the 2 L × 2 L matrix U (L) such that the L-qubit state transforms as ψ ′(L) = U (L) ψ (L) . In this way, a quantum circuit is described as the successive application of elementary gates to an initial quantum state, that is the product of the corresponding matrices (for details, see, e.g., [1]). It is a standard assumption that each elementary L-qubit transformation is unitary, which implies the following strong restriction on the corresponding matrix U ≡ U (L) as where I is the 2 L × 2 L identity matrix for L-qubit state and the operation (⋆) is the conjugate-transposition. The first equality in (5.8)  applied to this simplest example of L-qubits in the 2 L -dimensional Hilbert space C 2 ⊗L (for the general case the derivation almost literally coincides), which we write in the following special form (in Dirac notation with bra-and ket-vectors) with explicitly added identities. Then (5.8) follows from (5.9) as and any unitary matrix preserves the inner product which means that unitary operators satisfying (5.8) are bounded operators (bounded matrices in our case) and invertible with the inverse U −1 = U ⋆ . Let us consider a possibility of non-invertible intermediate transformations of L-qubit states, i.e. non-invertible gates which are described by the 2 L × 2 L matrices U (r) of (possibly) less than full rank 1 ≤ r ≤ 2 L . This can be related to the production of "degenerate" states (see, e.g. [42]), "particle loss" [52][53][54], and the role of ranks in multiparticle entanglement [55,56].
In the limited cases U r = 2 L ≡ U = U (L) , and U (1) corresponds to the measurement matrix being the projection to one final vector |i f inal . In this case, for non-invertible transformations with r < 2 L instead of unitarity (5.8) we consider partial unitarity (2.20)-(2.21) as where I 1 (r) and I 2 (r) are (or may be) different partial shuffle identities having r units on the diagonal. There is an exotic limiting case, which is impossible for the identity I: we call two partial identities orthogonal, if where Z = Z 2 L ×2 L is the zero 2 L × 2 L matrix.
We propose corresponding non-invertible analogs of (5.9)-(5.11) as follows. The partial adjoint operator U (r) ⋆ in the 2 L -dimensional Hilbert space C 2 ⊗L is defined by such that (see (5.12)-(5.13)) We call the r.h.s. of (5.16) the partial inner product. So instead of (5.11) we define U (r) as the partially bounded operator Thus, if the partial identities are orthogonal (5.14), then the partial inner product vanishes identically, and the operator U (r) becomes a zero norm operator in the sense of (5.17), although (5.12)-(5.13) are not zero.
We define a general unitary semigroup as a semigroup of matrices U (r) of rank r satisfying partial regularity (5.12)-(5.13) (in the "symmetric" case I 1 (r) = I 2 (r) ≡ I (r)).
As an example, we consider two 2-qubit states (5.3) ψ (2) and ϕ (2) (with a ′ ij and |i ′ j ′ ) and the non-invertible transformation described by three-parameter 4 × 4 matrices of rank 3 (but which are not nilpotent) which directly gives the signature of the partial inner product (5.16), in our case of the Hilbert space C 2 ⊗2 . The definition of a partial adjoint operator (5.15) is satisfied with both sides being equal to a 00 a ′ 11 e iα 00 | 1 ′ 1 ′ + a 01 a ′ 01 e iβ 01 | 0 ′ 1 ′ + a 11 a ′ 10 e iγ 11 | 1 ′ 0 ′ . The partial boundedness condition (5.17) holds with the partial inner product (5.16) becoming a 01 a ′ 01 01 | 0 ′ 1 ′ + a 11 a ′ 11 11 | 1 ′ 1 ′ , thus U (3) (5.18) which is a bounded partial unitary operator. An example of a zero norm (in our sense (5.17)) operator is the two-parameter partial unitary rank 2 matrix The partial unitarity relations for U nil (2) have the form It may be seen that the partial identities I nil,1 (2) and I nil,2 (2) are now orthogonal (5.14), and the partial inner product (5.16) vanishes identically, and also the boundedness condition (5.17) holds with the r.h.s. vanishing, despite U nil (2) being a nonzero nilpotent matrix (5.22).

VI. BINARY BRAIDING QUANTUM GATES
Let us consider those Yang-Baxter maps which could be linear transformations of two-qubit spaces. We will pay attention to the most general . We use the exponential form of the parameters x = r x e iα , y = r y e iβ , z = r z e iγ , r x,y,z , α, β, γ ∈ R, r x,y,z ≥ 0, |α| , |β| , |γ| ≤ 2π.
With the particular choice of parameters α = β = 0 and lower signs, the solution (6.2) coincides with the 8-vertex braiding gate of [13].
Next we search for unitary solutions among the invertible circle of 8-vertex traceless solutions (2.96) to the matrix Yang-Baxter equation (2.12) with parameters in the exponential form (6.1). The unitarity conditions (5.8) give the following equations on the parameters (6.1) r = r y = r z , r 2 r 2 x + r 2 = 1, r 8 + r 6 − 2r 4 + 1 = r 2 (6.4) (6.5) The system of equations (6.4) has two real positive (or zero) solutions Thus, only the first solution leads to an 8-vertex two-parameter unitary braiding quantum gate of the form (we put γ → β in (6.1)) The second solution (6.7) gives 4-vertex two-parameter unitary braiding quantum gate (we also put γ → β in (6.1)) The non-invertible 8-vertex circle solution (2.97) to the Yang-Baxter equation (2.12) cannot be partial unitary (5.12)-(5.13) with any values of its parameters.

VII. HIGHER BRAIDING QUANTUM GATES
In general, only special linear transformations of 2 L -dimensional Hilbert space can be treated as elementary quantum gates for an L-qubit state [1]. First, in the invertible case, the transformations should be unitary (5.8), and in the hypothetical noninvertible case they can satisfy partial unitarity (5.12)-(5.13). Second, the braiding gates have to be 2 L × 2 L matrix solutions to the constant Yang-Baxter equation [13] or higher braid equations (3.4)-(3.6). Here we consider (as a lowest case higher example) the ternary braiding gates acting on 3-qubit quantum states, i.e. 8 × 8 matrix solutions to the ternary braid equations (4.3) which satisfy unitarity (5.8) or partial unitarity (5.12)- (5.13).
Note that all the permutation solutions (4.13)-(4.14) are by definition unitary, and are therefore ternary braiding gates "automatically", and we call them permutation 8-vertex ternary braiding quantum gates U 8−vertex perm . By the same reasoning the unitary version of the invertible star 8-vertex parameter-permutation solutions (4.15)- (4.22) to the ternary braid equations (4.3) will contain the complex numbers of unit magnitude as parameters.
Indeed, for the bisymmetric series (4.15)-(4.16) of star-like solutions we have 4 two real parameter unitary ternary braiding quantum gates (κ = ±1) , α, β ∈ R, |α| , |β| ≤ 2π, (7.1) which is a ternary analog of the first parameter-permutation solution to the Yang-Baxter equation from (2.33). The ternary analog of the second star solution is the following unitary version of the bisymmetric series (4.17)-(4.18) The braiding gate (7.5) is a ternary analog of (6.2), and therefore with α = 0 it can be treated as a ternary analog of the 8-vertex braiding gate considered in [13]. Note that the solution U 16−vertex 3−qubits+ (0) is transpose to the so-called generalized Bell matrix [23]. Comparing (4.49) and (7.5), we observe that the ternary braiding quantum gates (acting on 3 qubits) are those elements of the 16-vertex star semigroup G star 16vert (4.57), which satisfy unitarity (5.8). In the same way, the 32-vertex analog the 8-vertex binary braiding gate of [13] (now acting on 4 qubits) is the following constant 4-ary braiding unitary quantum gate Thus, in general, the "Minkowski" star solutions for n-ary braid equations correspond to 2 n -vertex braiding unitary quantum gates as 2 L × 2 L matrices acting on L = n qubits The braiding gate (7.7) can be treated as a polyadic (n-ary) generalization of the GHZ generator (see, e.g., [18,23]) acting on L = n qubits.

VIII. ENTANGLING BRAIDING GATES
Entangled quantum states are obtained from separable states by acting with special quantum gates on two-qubit states and multi-qubit states [41,42]. Here we consider the concrete form of braiding gates which can be entangling or not entangling. There are general considerations on these subjects for the Yang-Baxter maps [13,37,39] and generalized Yang-Baxter maps [23,25,26,50]. We present the solutions for the binary and ternary braid maps introduced above, which connect the parameters of the gate and the state.
(8.1) In general, a braiding gate is entangling if the transformed concurrence (8.1) does not vanish, and its roots give the values of the gate parameters U (α, β) for which the gate is not entangling for a given two-qubit state. In search of the solutions for the transformed concurrence C (2) s± = 0, we observe that one of the qubits has to be on the Bloch sphere equator θ 1 = π 2 (or θ 2 = π 2 ). Only in this case can the first (or second) bracket in (8.1) vanish, and we obtain 1) C (2) and α − β = 2γ 1 − π, or θ 2 = π 2 and α − β = 2γ 2 ; (8.2) 2) C s− = 0, if θ 1 = π 2 and α − β = 2γ 1 , or θ 2 = π 2 and α − β = 2γ 2 − π. In the case of the 4-vertex circle binary braiding gate (6.10) the transformed concurrence vanishes identically, and therefore this gate is not entangling for any values of its parameters.
Thus we have shown that the braiding binary and ternary quantum gates can be either entangling or not entangling, depending on how their parameters are related to the concrete quantum state on which they act. The constructions presented here could be used, e.g. in the entanglement-free protocols [57,58] and some experiments [59,60]. This can also allow us to build quantum networks without any entangling at all (non-entangling networks), when the next gate depends upon the previous state in such a way that at each step there is no entangling, as the separable, but different, final state is received from a separable initial state.