Quantum computation has shown advantages in several problems over the corresponding classical algorithms. The noisy intermediate-size quantum devices with dozens of qubits make it more promising in the past decade, as the quantum advantage was demonstrated experimentally. Before entering the era of scalable quantum computation, one has to resolve the errors in a quantum many-body system, which is inevitable due to the environment interaction during quantum control processes. The goal of quantum error correction is to reduce such errors and increase decoherence time.

The most successful candidates of quantum error correction codes are topological codes, especially surface codes, which were discovered by Alexei Kitaev. The ordered phase of the 2D Ising model on a torus ensures that toric codes are fault-tolerant below a critical error probability. Other than the FT threshold, the topological codes feature efficient encoding and decoding, local measurements, but suffer from asymptotically zero code rate.

To encode more logical qubits with finite resources, one can loosen the condition on locality and extend to a broader class of quantum low-density parity-check (LDPC)codes. There are many known algebraic constructions for such codes, but only a precious few of them have finite code rates and meet the fault-tolerant condition: the stabilizer weight is bounded and the distance scales at least logarithmically with the code size. In this thesis, I construct the higher-dimensional quantum hypergraph product (HQHP) codes, which generalize quantum hypergraph product (QHP) codes and toric codes in all dimensions. The HQHP codes projected into a single space gives subsystem product codes, which can then be gauge fixed to concatenated codes or homological product codes. Those include some common CSS codes, like Shor’s codes, Bacon Shor’s codes, and subsystem QHP codes. The HQHP codes can be mapped to the tensor product of chain complexes, which provides an algebraic framework to construct quantum LDPC codes with finite code rates, square root distances, FT thresholds and single-shot properties with redundant checks. Meanwhile, their rich connection to other codes are very instructive and may lead to further optimizations. Regarding the remained procedures towards fault-tolerant universal quantum computation through quantum LDPC codes, I will discuss the fault-tolerant condition for each code, including distance, stabilizer weight, decoding, fault-tolerant gates and measurement protocols.