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arXiv:2412.02345 · Quantum Information Processing

On the Realization of quantum gates coming from the Tracy-Singh product

Fabienne Chouraqui

Published Aug 11, 2025• 1 citation
arXivHTMLPDF

Abstract

Abstract The Tracy-Singh product of matrices (TS) is an efficient tool, for the construction of gates, which preserves properties such as unitary, primitive and entangling. Moreover, if c and $$c'$$ c ′ are both Yang-Baxter gates, then $$c \boxtimes c'$$ c ⊠ c ′ , their TS product, is also a Yang-Baxter gate. A natural question arises whether the TS product preserves enhanced YBE pairs and the existing connections between primitive gates and knot invariants. Another question we consider is how to implement the gates $$c \boxtimes c'$$ c ⊠ c ′ .

DOI
10.1007/s11128-025-04878-4

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Abstract The Tracy-Singh product of matrices (TS) is an efficient tool, for the construction of gates, which preserves properties such as unitary, primitive and entangling. Moreover, if c and c′c'c′ c ′ are both Yang-Baxter gates, then c⊠c′c \boxtimes c'c⊠c′ c ⊠ c ′ , their TS product, is also a Yang-Baxter gate. A natural question arises whether the TS product preserves enhanced YBE pairs and the existing connections between primitive gates and knot invariants. Another question we consider is how to implement the gates c⊠c′c \boxtimes c'c⊠c′ c ⊠ c ′ .
It has been conjectured that quantum entanglement operators can be lifted to braiding operators by the way of the topological quantum field theory axioms set forth by Witten and Atiyah. Moreover, it can be readily shown that quantum link invariants need entanglement to construct topological invariants. Given these results and the already dense mathematical framework underlying topology and quantum field theory, we propose that, through the usage of quantum algebra and bracket models, we can identify a significant area of overlap where entangling R-matrix solutions to the Yang-Baxter equation can be used to construct invariants of knots and links. Such a connection illuminates the relationship between quantum and topological entanglement, and allows us to draw novel insights about properties of entangling gates, for example the question of what it is that allows some entangling R-matrices to generate topological invariants when others do not. In conclusion, by studying the boundary between topological and quantum entanglement we construct novel topological invariants that can have significant impact on the study of quantum computing and can help make progress towards the problem of representing the Artin braid group as unitary matrices.
Abstract Brick-wall circuits composed of the Yang–Baxter gates are integrable. It becomes an important tool to study the quantum many-body system out of equilibrium. To put the Yang–Baxter gate on quantum computers, it has to be decomposed into the native gates of quantum computers. It is favorable to apply the least number of native two-qubit gates to construct the Yang–Baxter gate. We study the geometric representations of all X-type braid gates and their corresponding Yang–Baxter gates via the Yang–Baxterization. We find that the braid and Yang–Baxter gates can only exist on certain edges and faces of the two-qubit tetrahedron. We identify the parameters by which the braid and Yang–Baxter gates are the Clifford gate, the matchgate, and the dual-unitary gate. The geometric representations provide the optimal decompositions of the braid and Yang–Baxter gates in terms of other two-qubit gates. We also find that the entangling powers of the Yang–Baxter gates are determined by the spectral parameters. Our results provide the necessary conditions to construct the braid and Yang–Baxter gates on quantum computers.
A unitary operator that satisfies the constant Yang-Baxter equation immediately yields a unitary representation of the braid group B_n for every n >= 2. If we view such an operator as a quantum-computational gate, then topological braiding corresponds to a quantum circuit. A basic question is when such a representation affords universal quantum computation. In this work, we show how to classically simulate these circuits when the gate in question belongs to certain families of solutions to the Yang-Baxter equation. These include all of the qubit (i.e., d = 2) solutions, and some simple families that include solutions for arbitrary d >= 2. Our main tool is a probabilistic classical algorithm for efficient simulation of a more general class of quantum circuits. This algorithm may be of use outside the present setting.
The Yang-Baxter equation and it's various forms have applications in many fields, including statistical mechanics, knot theory, and quantum information. Unitary solutions of the braided Yang-Baxter equation are of particular interest as quantum gates for topological quantum computers. We demonstrate a simple construction for solutions in any dimension, which are both unitary and universal for quantum computation. We also fully classify a family of solutions to certain generalized Yang-Baxter equations and prove that certain instances of the equation only have solutions that are scalar multiples of the identity.
Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.