A Graphical Calculus for Quantum Computing with Multiple Qudits using
Generalized Clifford Algebras
- URL: http://arxiv.org/abs/2103.16081v2
- Date: Wed, 15 Mar 2023 02:02:58 GMT
- Title: A Graphical Calculus for Quantum Computing with Multiple Qudits using
Generalized Clifford Algebras
- Authors: Robert Lin
- Abstract summary: We show that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators.
We derive several new identities for the braid elements, which are key to our proofs.
In terms of quantum computation, we show that it is feasible to envision implementing the braid operators for quantum computation.
- Score: 0.0
- License: http://creativecommons.org/licenses/by-nc-nd/4.0/
- 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.
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