  Program implementation and simulation are essential for research in the field
of quantum algorithms. However, complex and large-scale quantum algorithms can
pose challenges for existing quantum programming languages and simulators.
Here, we present a scalable program implementation of the quantum walk on a
sparse matrix and the quantum linear solver based on the quantum walk. Our
implementation is based on a practical scenario in which the sparse matrix is
stored in the compressed-sparse-column format in quantum random access memory.
All necessary modules are implemented unitarily and are ensured to be
decomposed at the quantum gate level, including implementing a quantum binary
search and a modification of the original algorithm. The program is validated
using a highly efficient quantum circuit simulator which is based on the
register level and sparse state representation. With only a single core, we
simulate the quantum walk on a 16384-dimensional matrix with 582 qubits in 1.1
minutes per step, as well as a quantum linear solver up to 1024 dimensions and
212245 steps in 70 hours. Our work narrows the gap between the simulation of a
quantum algorithm and its classical counterparts, where the asymptotic
complexity of our quantum linear solver simulation approximates a classical
linear solver. These program implementation and simulation techniques have the
potential to expand the boundary of numerical research for large-scale quantum
algorithms, with implications for the development of error-correction-era
quantum computing solutions.
