Hypersphere product code[1]

Cousins

• Quantum locally testable code (QLTC)— The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance of order \(\Theta(\sqrt{n})\). Applying Hastings’ weight-reduction construction yields QLDPC families with distance \(\Theta^*(\sqrt{n})\) and inverse-polylogarithmic soundness [2]. Application of generalized distance balancing [3] to hypersphere product codes using an asymptotically good classical code of length \(t\) yields \(O( 1/(\log(n)^2 t^2) )\) soundness and order \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as order \(\Theta(t^2)\) [4].
• Distance-balanced code— Application of generalized distance balancing [3] to hypersphere product codes using an asymptotically good classical code of length \(t\) yields \(O( 1/(\log(n)^2 t^2) )\) soundness and order \(\Theta(\sqrt{n}t)\) distance while maintaining locality scaling and at the expense of a dimension scaling as order \(\Theta(t^2)\) [4].