A triple homological product of good quantum LDPC codes achieves distance N^(2/3) with transversal CCZ gates and prepares N^(1/3) magic states in a single round.
Collapsibility and vanishing of top homology in random simplicial complexes
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abstract
Let Y be a random d-dimensional subcomplex of the (n-1)-dimensional simplex S obtained by starting with the full (d-1)-dimensional skeleton of S and then adding each d-simplex independently with probability p=c/n. We compute an explicit constant gamma_d=Theta(log d) so that for c < gamma_d such a random simplicial complex either collapses to a (d-1)-dimensional subcomplex or it contains the boundary of a (d+1)-simplex. We conjecture this bound to be sharp. In addition we show that there exists a constant gamma_d< c_d <d+1 such that for any c>c_d and a fixed field F, asymptotically almost surely H_d(Y;F) \neq 0.
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Transversal non-Clifford gates on qLDPC codes breaking the $\sqrt{N}$ distance barrier and quantum-inspired geometry with $\mathbb{Z}_2$ systolic freedom
A triple homological product of good quantum LDPC codes achieves distance N^(2/3) with transversal CCZ gates and prepares N^(1/3) magic states in a single round.