We construct explicit families of asymptotically good quantum locally testable codes over qubits. More precseily, we construct explicit quantum LDPC CSS codes over qubits having constant rate, constant relative distance and constant weight local testers with constant soundness.
We construct quantum locally testable codes (LTCs) with constant rate, distance, soundness and locality under a variant of a product expansion conjecture of Bafna and Vyas about Reed-Solomon codes. In particular, we use the high-dimensional expansion framework of Dinur, Lin and Vidick for constructing quantum LTCs, instantiated with the non-Abelian cubical complexes of Rungtanapirom, Stix and Vdovina. Our code is obtained by equipping the complex with carefully chosen Reed-Solomon local codes whose symmetries are compatible with those of the complex.
In this paper we study classical and quantum LDPC codes with constant rate obtained using tensor product operation over noncommutative group algebra for chain complexes. It is shown that these families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture. The existence of asymptotically good families of locally testable codes with constant parameters of locality and soundness is also shown, which gives the solution of the known conjecture in the field of classical locally testable codes.
Recent work by Wills constructed codes with constant rate and transversal T. Extending this line of work, we construct asymptotically good CSS codes with \textbf{fully addressable} transversal T, permitting selective control over which logical qubits receive a T gate. We also give Pauli-LDPC codes with transversal T and growing distance at vanishing rate; and a protected-subsystem construction with fully addressable transversal T, constant protected rate, distance Ω(n), and sparse (non-Pauli) checks. More generally, we obtain asymptotically good binary codes with full transversal addressability for every fixed finite-order one-qubit gate, retaining CSS realizations for dyadic phase gates such as T and using generally nonadditive codes for rotations such as Rz(π/7). The remaining challenge is to construct asymptotically good Pauli-LDPC codes with fully addressable transversal non-Clifford one-qubit gates.
In this paper, we prove a lower bound on the soundness of quantum locally testable codes under the distance balancing construction of Evra et al. Our technical contribution is that the soundness of the quantum code after this procedure is at least its soundness before the procedure divided by the length of the classical code used to perform distance balancing. This allows us to use any classical code when distance balancing, where previously only the repetition code had been considered for these codes. By using a good classical low-density parity check (LDPC) code, we are able to grow the dimension of the hypersphere product codes and the hemicubic codes while maintaining their distance and locality, but at the expense of soundness. From this, and also by distance balancing a chain complex of Cross et al., we obtain quantum locally testable codes of new parameters.