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arXiv:2406.08320 · Journal of Physics A Mathematical and Theoretical

Geometric representations of braid and Yang–Baxter gates

Kun Zhang, Kun Hao, Kwangmin Yu, V. E. Korepin, Wen‐Li Yang

Published Oct 10, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ad85b2

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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.
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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.
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 \emph{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.
Abstract Quantum computers provide a promising method to study the dynamics of many‐body systems beyond classical simulation. On the other hand, the analytical methods developed and results obtained from the integrable systems provide deep insights on the many‐body system. Quantum simulation of the integrable system not only provides a valid benchmark for quantum computers but is also the first step in studying integrable‐breaking systems. The building block for the simulation of an integrable system is the Yang–Baxter gate. It is vital to know how to optimally realize the Yang–Baxter gates on quantum computers. Based on the geometric picture of the Yang–Baxter gates, the optimal realizations of two types of Yang–Baxter gates with a minimal number of controlled NOT (CNOT) or gates are presented. It is also shown how to systematically realize the Yang–Baxter gates via the pulse control. The different realizations on IBM quantum computers are tested and compared. It is found that the pulse realizations of the Yang–Baxter gates always have a higher gate fidelity compared to the optimal CNOT or realizations. On the basis of the above optimal realizations, the simulation of the Yang–Baxter equation on quantum computers is demonstrated. These results provide a guideline and standard for further experimental studies based on the Yang–Baxter gate.
We study the operator entanglement of local operators in one-dimensional brickwork circuits whose two-site gate satisfies the braid relation; throughout this work, we call such a gate a Yang--Baxter gate. We establish upper bounds for several structured, overlapping classes of Yang--Baxter gates. We show that the operator Schmidt rank remains uniformly bounded in time for all qubit Yang--Baxter gates and, in arbitrary local dimension, for permutation gates obtained from non-degenerate Yang--Baxter maps. We also show that it grows at most polynomially for involutive dual-unitary Yang--Baxter gates and for arbitrary phase dressings of permutation gates obtained from non-degenerate Yang--Baxter maps. These results imply, respectively, constant and logarithmic upper bounds on the operator entanglement. Conversely, we construct a seven-state involutive Yang--Baxter gate without dual unitarity and a one-site operator whose exact operator Schmidt rank grows exponentially, although the corresponding operator entropies remain undetermined. Entanglement growth in the general Yang--Baxter case remains open. All proofs and selected examples were constructed by ChatGPT 5.6 Sol.