On Computational Complexity of Clifford Algebra

After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced in a previous paper.

💡 Research Summary

The paper begins with a concise review of the computational landscape of Clifford algebras Cl(p,q). A Clifford algebra generated by n = p+q orthogonal vectors γ_i has a basis of 2ⁿ elements, and the product of two arbitrary multivectors requires expanding each in this basis and applying the anti‑commutation rule γ_iγ_j + γ_jγ_i = 2η_{ij}. Consequently, the naïve multiplication cost grows exponentially, typically expressed as O(2^{2n}). The standard way to analyse this cost is to invoke the well‑known matrix isomorphism Cl(p,q) ≅ Mat_{2^{⌊n/2⌋}}(ℝ) or Mat_{2^{⌊n/2⌋}}(ℂ). While this representation yields the same asymptotic bound, it introduces substantial overhead: one must allocate large matrices, manage floating‑point arithmetic, and handle the conversion between algebraic and matrix forms.

The core contribution of the work is a new basis tailored specifically for even‑dimensional algebras Cl(2m). Starting from the original vector basis γ_i (1 ≤ i ≤ 2m), the authors define m new generators e_i = γ_iγ_{i+m}. These generators satisfy a remarkably simple anti‑commutation relation e_i e_j + e_j e_i = 0 for i ≠ j, and each squares to −1 or +1 depending on the signature. Any multivector can then be written as a linear combination of products e_I = ∏{i∈I} e_i, where I is a subset of {1,…,m} and e∅ is the identity. The multiplication rule in this basis collapses to a pure set‑theoretic operation: \