Braiding quantum gates from partition algebras

Pramod Padmanabhan1, Fumihiko Sugino2, and Diego Trancanelli3,4

1Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon, South Korea
2Center for Theoretical Physics of the Universe, Institute for Basic Science, Daejeon, South Korea
3Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, via Campi 213/A, 41125 Modena, Italy
4INFN Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy

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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.

The quantum world is filled with counter-intuitive phenomena, with entanglement – the “spooky action at a distance” in Einstein’s words – lying at the heart of the quantum weird- ness. In spite of this, entanglement is a resource in quantum computing, where entangled qubits execute quantum algorithms that are expected to surpass classical methods. How- ever entanglement is fragile and susceptible to decoherence and thus requires protection from disturbances.
A clever and robust way to achieve such protection is by using physical implementations enjoying topological properties: entangling gates should be realized through the action of braiding operators, with the entanglement between quantum states being similar to knotted links. For example, two qubits can be either entangled in an EPR state or disentangled in a product state, precisely as two links can be either knotted or separated, as seen in an Hopf link.
Just as it is impossible to undo links tied into a knot without cutting them, quantum states which got entangled together in a topological manner should be resilient to disturbances from the environment.
In this work, we employ techniques from partition algebras as a way to systematically construct braid operators to entangle multiple qubits in a topological manner. The method we present produces all the known classes of entangled states in a two- or three-qubit system. In addition, we explicitly see that it generates various entangled states in a four-qubit system. Further investigation of this method is expected to shed more light on the relation between topological and quantum entanglement. Its application to quantum field theories will also be an intriguing direction for future research.

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Cited by

[1] Pramod Padmanabhan, Fumihiko Sugino, and Diego Trancanelli, "Local invariants of braiding quantum gates—associated link polynomials and entangling power", arXiv:2010.00270, Journal of Physics A: Mathematical and Theoretical 54 13, 135301 (2021).

[2] Pramod Padmanabhan, Fumihiko Sugino, and Diego Trancanelli, "Generating W states with braiding operators", arXiv:2007.05660.

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