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Homological connectivity of random k-dimensional complexes
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Let Delta_{n-1} denote the (n-1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of Delta_{n-1} obtained by starting with the full (k-1)-dimensional skeleton of Delta_{n-1} and then adding each k-simplex independently with probability p. Let H_{k-1}(Y;R) denote the (k-1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. Let R and k \geq 1 be fixed. It is shown that p=(k \log n)/n is a sharp threshold for the vanishing of H_{k-1}(Y;R).
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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.
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