Generalized Yang–baxter Equations and Braiding Quantum Gates

Chen, Rebecca S.

doi:10.1142/s0218216512500873

Cited by 25 publications

(23 citation statements)

References 6 publications

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“…A (2, 3, 2) gYB-operator is given in [9] and a (2, 3, 1) gYB-operator is in [3]. Three families of variations of the latter are obtained in [2] and we will call those as (2, 3, 1) gYB-operator of type I, II, and III in the following Sec. 4.1.…”

Section: Examplesmentioning

confidence: 99%

“…On the lexicographically ordered basis of V ⊗3 , three families of (2, 3, 1) gYB-operators are represented as 8 × 8 unitary matrices for 0 ≤ θ ≤ π as follows [2]: …”

Section: (2 3 1) Egyb-operatorsmentioning

confidence: 99%

“…We discuss a (2, 3, 2)-type gYB-operator given in [9] as one of the main examples. Another example of gYB-operator, (2, 3, 1)-type, appeared in [3] and three families of its variations were discussed in [2]. In this paper, we enhance these gYB-operators and discuss corresponding isotopy invariants of oriented links.…”

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confidence: 99%

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Invariants of Links From the Generalized Yang–baxter Equation

Hong

2013

J. Knot Theory Ramifications

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Enhanced Yang-Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang-Baxter operators (gYB-operators). At present two examples of gYB-operators are known and recently three types of variations for one of these were discovered. We present the definition of enhanced generalized Yang-Baxter operators and show that all known examples of gYB-operators can be enhanced to give corresponding invariants of oriented links. Most of these invariants are specializations of the polynomial invariant P . Invariants from gYB-operators are multiplicative after a normalization.We define enhanced generalized Yang-Baxter operators (briefly, EgYBoperators). These operators are new, but very few nontrivial examples of gYB-operators are known. In fact, so far only unitary solutions have been considered. We discuss a (2, 3, 2)-type gYB-operator given in [9] as one of the main examples. Another example of gYB-operator, (2, 3, 1)-type, appeared in [3] and three families of its variations were discussed in [2]. In this paper, we enhance these gYB-operators and discuss corresponding isotopy invariants of oriented links.Note that from any ribbon category we have a link invariant for each object. If there is a (generalized) localization in the sense of [3,8], then some enhancement should exist and it is reasonable to expect that the corresponding invariant recovers the one defined directly from the category. In [3], it is shown that SU(3) 3 theory has no 4 × 4 ordinary localization which corresponds to the usual YB-operator, but does have an 8 × 8 generalized localization which corresponds to gYB-operator. No 4 × 4 unitary YB-operator can yield the invariant discussed in [7] (and identified in [6]), and it seems unlikely that any ordinary YB-operator can. This is one of the motivations for this work. On the other hand, it is possible to find a 4 × 4 operator that together with associativities can produce this invariant. So it is possible that the gYB-operator can somehow absorb the nontrivial associativities.After online version of this work appeared, a method to obtain gYB-operators from certain objects in ribbon fusion categories was suggested [5]. Following this line a family of (2, 3, 1) gYB-operators were constructed from the unitary modular categories SO(N ) 2 and corresponding invariants of links were studied [4]. Furthermore, it is proved that the invariants are the same as those obtained directly from the categories.Here are the contents of this paper. In Sec. 2, we recall the notion of gYBoperator and introduce the EgYB-operators. In the definition of EgYB-operator, the orthogonality condition is weaker than the corresponding condition in the definition of EYB-operator. In Sec. 3, we define an invariant of oriented links associated with each EgYB-operator in a similar way as in [10]. The invariant is projectively multiplicative on a disjoint union of links, that is, multiplicative up to a (uniform) constant factor. In Sec. 4, some examples of EgYB-operators and corresponding lin...

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Section: Examplesmentioning

confidence: 99%

“…On the lexicographically ordered basis of V ⊗3 , three families of (2, 3, 1) gYB-operators are represented as 8 × 8 unitary matrices for 0 ≤ θ ≤ π as follows [2]: …”

Section: (2 3 1) Egyb-operatorsmentioning

confidence: 99%

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Invariants of Links From the Generalized Yang–baxter Equation

Hong

2013

J. Knot Theory Ramifications

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“…The R-matrices related to involutive, non-degenerate settheoretic solutions of the QYBE give cocycles into abelian groups [20], therefore they can be given as an input in the construction of universal R-matrices and twists for Hopf algebras, for example as in Theorem 4.2, [18].Most of the R-matrices constructed in our paper are unitary. A unitary R-matrix leads to a unitary representation of the Braid group, and the resulting unitary matrices associated to braids can be used to process quantum information [38,15,21]. In connection with the topological quantum computation, it was conjectured in [24,42] that a single unitary R-matrix can generate only finite representations of Braid groups, and in [24] it was confirmed in several important classes of R-matrices.…”

mentioning

confidence: 98%

“…Most of the R-matrices constructed in our paper are unitary. A unitary R-matrix leads to a unitary representation of the Braid group, and the resulting unitary matrices associated to braids can be used to process quantum information [38,15,21]. In connection with the topological quantum computation, it was conjectured in [24,42] that a single unitary R-matrix can generate only finite representations of Braid groups, and in [24] it was confirmed in several important classes of R-matrices.…”

mentioning

confidence: 98%

Set-theoretic solutions of the Yang–Baxter equation and new classes of R-matrices

Smoktunowicz

2018

Linear Algebra and its Applications

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We describe several methods of constructing R-matrices that are dependent upon many parameters, for example unitary R-matrices and R-matrices whose entries are functions. As an application, we construct examples of R-matrices with prescribed singular values. We characterise some classes of indecomposable set-theoretic solutions of the quantum Yang-Baxter equation (QYBE) and construct R-matrices related to such solutions. In particular, we establish a correspondence between one-generator braces and indecomposable, non-degenerate involutive set-theoretic solutions of the QYBE, showing that such solutions are abundant. We show that R-matrices related to involutive, non-degenerate solutions of the QYBE have special form. We also investigate some linear algebra questions related to R-matrices.constructed using set-theoretic solutions of the quantum Yang-Baxter equation, and have only one nonzero element in each column. This type of matrices appear frequently in the literature, for example in [17], where Baxterisation of some R-matrices of this type was obtained, and in [29], where they appear as universal gates for the quantum computation related to the circuit model (see Theorem 1,[29]). In [38,24,21] they appear in connection with Braid groups and topological quantum computation (see also [7,35,36,31]). They also appear as combinatorial R-matrices in the theory of crystal bases, geometric crystals and box-bell systems. The R-matrices related to involutive, non-degenerate settheoretic solutions of the QYBE give cocycles into abelian groups [20], therefore they can be given as an input in the construction of universal R-matrices and twists for Hopf algebras, for example as in Theorem 4.2, [18].Most of the R-matrices constructed in our paper are unitary. A unitary R-matrix leads to a unitary representation of the Braid group, and the resulting unitary matrices associated to braids can be used to process quantum information [38,15,21]. In connection with the topological quantum computation, it was conjectured in [24,42] that a single unitary R-matrix can generate only finite representations of Braid groups, and in [24] it was confirmed in several important classes of R-matrices. In [40], Rowell made the following comment: "From the quantum information point of view, a unitary R-matrix can be used to directly simulate topological quantum computers on the quantum circuit model. More recently people have begun to study what extra gates one needs to supplement braiding with in order to achieve universality (see e.g. [16]). If a single R-matrix can only generate a (nearly) finite group, can an additional small gate lead to a universal gate set?". This question provides the inspiration for our construction of R-matrices with many parameters. All of our examples are locally monomial BVS (we recall the definition in Section 2).Recall that locally monomial braided vector spaces (BVS) were introduced by Galindo and Rowell in [24], and localisation was introduced by Rowell and Wang in Definition 2.3 in [42]. A related notion of br...

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Modified Newton Integration Neural Algorithm for Solving Time-Varying Yang-Baxter-Like Matrix Equation

Huang

et al. 2022

Neural Process Lett

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Generalized Yang–baxter Equations and Braiding Quantum Gates

Cited by 25 publications

References 6 publications

Invariants of Links From the Generalized Yang–baxter Equation

Invariants of Links From the Generalized Yang–baxter Equation

Set-theoretic solutions of the Yang–Baxter equation and new classes of R-matrices

Modified Newton Integration Neural Algorithm for Solving Time-Varying Yang-Baxter-Like Matrix Equation

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