A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras

Abstract: In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, using the algebraic framework developed in a previous 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. In terms of physics, we connect these braid identities to physics by showing the presence of a conserved charge. 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. We show how to explicitly compute various vector states in an efficient manner using algebraic methods. 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 $\zeta$, which is an appropriate square root of a primitive root of unity.