Analysis of Classical and Quantum Resources for the Quantum Linear Systems Algorithm

The quantum algorithm by Harrow, Hassidim, and Lloyd solves a system of N linear equations and achieves exponential speedup over classical algorithms under certain conditions. The advantage to the algorithm is that log(N) rather than N registers are required. Given an N x N matrix A and vectors x and b, the quantum algorithm seeks to find x such that Ax = b. By representing vector b as a superposition of quantum states vertical bar b>, quantum phase estimation is used to find the corresponding eigenvalues of A. Applying the inverse Fourier transform, we solve for vertical bar x> such that vertical bar x> = A(-1)vertical bar b>. We model the algorithm using a quantum circuit diagram, with data qubits encoded using the Steane code for fault tolerant quantum phase estimation. Fresh ancilla for error correction are provided using an oracular pipelined ancilla architecture. We then analyze the classical and quantum resources needed for implementation. The significance of this case study is to examine how classical and quantum resources interact in implementing this algorithm. The issues raised in this analysis, such as fault tolerant phase estimation using pipelined ancilla, garbage collection, and the preparation of I/O registers to this architecture, will be explored in more detail in future research.