Extraspecial Two-Groups, Generalized Yang-Baxter Equations and Braiding Quantum Gates

Abstract

In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an odd ) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.

Generating W states with braiding operators

“…It seems quite hard to obtain unitary braiding operators for W states and we are not aware of any example in the literature. This is in sharp contrast to the case of GHZ states, for which various unitary representations have been found in [10,11,16]. It would be interesting to understand this difference better and to see whether it has any relevance in distinguishing between topological aspects of gYBOs and quantum entanglement.…”

“…Braiding operators that are unitary enjoy special significance, as they also serve as quantum gates, but non-unitary operators have also been considered. Central to the task of finding braiding operators is the systematic construction of solutions to the (d, m, l)-generalized Yang-Baxter Equation (gYBE) [10,11]…”

“…In these notes, we obtain W states from the braiding operators constructed in [16] using partition algebras, something that was left out of that reference, and also present a novel construction in terms of extraspecial 2-groups [10] adapted to a four-qubit setting. It turns out that these methods only provide non-unitary braiding operators.…”

Local invariants of braiding quantum gates -- associated link polynomials and entangling power

“…One of possible future directions would be to analyze robustness of entanglement [37] for braiding quantum gates and to understand how topological properties coming from the braid contribute to the robustness of the quantum entanglement. In addition, it would be crucial to check these features for multi-qubit braid operators that can be constructed using the generalized Yang-Baxter equation [35,36] for which several solutions have been found [38][39][40][41][42].…”

Quantum entanglement, supersymmetry, and the generalized Yang-Baxter equation

“…A natural question that arises is whether generic entangled states in multi-qubit systems can also be produced from solutions to the YBE, the so-called R-matrices. Besides the already mentioned Bell matrix, this was shown to be the case for the GHZ states in [22,23]. As the Rmatrices producing the GHZ states must act on three qubits simultaneously, this necessitates the introduction of the generalized Yang-Baxter equation (gYBE), which accommodates Rmatrices with support on more than two qubits.…”