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arXiv:2303.02964 · Physica Scripta

The yang-baxter equation, quantum computing and quantum entanglement

Fabienne Chouraqui

Published Sep 23, 2024• 3 citations
arXivHTMLPDF

Abstract

Abstract We present a method to construct infinite families of entangling (and primitive) 2-qudit gates, and amongst them entangling (and primitive) 2-qudit gates which satisfy the Yang-Baxter equation. We show that, given 2-qudit gates c and d, if c or d is entangling, then their Tracy-Singh product c ⊠ d is also entangling and we can provide decomposable states which become entangled after the application of c ⊠ d.

DOI
10.1088/1402-4896/ad7e57

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Abstract We present a method to construct infinite families of entangling (and primitive) 2-qudit gates, and amongst them entangling (and primitive) 2-qudit gates which satisfy the Yang-Baxter equation. We show that, given 2-qudit gates c and d, if c or d is entangling, then their Tracy-Singh product c ⊠ d is also entangling and we can provide decomposable states which become entangled after the application of c ⊠ d.
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 ′ .
We construct solutions of the Yang–Baxter equation in any dimension d≥2d\ge 2d≥2 by directly generalizing the previously found solutions for d=2d=2d=2 . We equip those solutions with unitarity and entangling properties. Being unitary, they can be turned into 222 -qudit quantum logic gates for qudit-based systems. The entangling property enables each of those solutions, together with all 111 -qudit gates, to form a universal set of quantum logic gates.
Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any Yang-Baxter solution can be interpreted as a two-qudit quantum gate. Here we show that if this gate is non-entangling, then the resulting invariant of knots is trivial. We thus obtain a general connection between topological entanglement and quantum entanglement, as suggested by Kauffman et al.
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.
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.
Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding operators as quantum entanglers, and is part of a larger speculated connection between topological and quantum entanglement. We push the analysis of this connection forward, by showing that supersymmetry algebras can be used to construct large families of solutions of the spectral parameter-dependent generalized Yang-Baxter equation. We present a number of explicit examples and outline a general algorithm for arbitrary numbers of qubits. The operators we obtain produce, in turn, all the entangled states in a multi-qubit system classified by the Stochastic Local Operations and Classical Communication protocol introduced in quantum information theory.