A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras
Robert Lin
TL;DR
This work introduces a self-contained graphical calculus for multi-qudit computations based on generalized Clifford algebras, anchored by a fixed ground state and representation-theoretic axioms. It provides an entirely algebraic route to the Yang-Baxter equation and braid-group representations, resolving open questions for even qudit dimensions and yielding 2-local, near-Clifford braid gates suitable for quantum computation. Key contributions include an intertwining framework with a charge-grading, a complete operator-level analysis of braid elements, and a vector-identity–driven reduction toolkit that extends the calculus to multi-qudit states. The approach offers a rigorous, diagrammatic alternative to category-theoretic formalisms and demonstrates practical pathways to implementing braid-based operations on qudit systems.
Abstract
In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem of Cobanera and Ortiz on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor $ζ$, which is an appropriate square root of a primitive root of unity.