Braiding quantum gates from partition algebras

Abstract

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 ≤ l ≤ 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.

Generating W states with braiding operators

“…Braiding operators were systematically constructed in [16] out of generators of partition algebras. We do not go over the construction here, but refer the reader to [16,22] for details.…”

“…Braiding operators were systematically constructed in [16] out of generators of partition algebras. We do not go over the construction here, but refer the reader to [16,22] for details. We use two types of 3-qubit braiding operators from [16] for the purpose of obtaining W states.…”

“…Several solutions to the gYBE in (1.1) have been found in [14,15] and more recently using partition algebras in [16]. Among the quantum states obtained from these operators, one finds maximally entangled states such as the two-qubit Bell states, of typical form 1 other classes of states such as partially entangled states and product states can also be obtained from these braiding operators.…”

“…One has also to guarantee that the far-commutativity condition σ i σ j = σ j σ i for |i − j| > 1 is satisfied, which happens automatically for m 2 ≤ l < m [16]. 1 Here we shall focus our attention on this particular case.…”

“…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.…”