Two excellent questions. Regarding claims of factoring small numbers: First note that error correction becomes exponentially more important for longer computat

Two excellent questions. Regarding claims of factoring small numbers: First note that error correction becomes exponentially more important for longer computations. Second, see this https://arxiv.org/abs/1301.7007 https://arxiv.org/abs/1301.7007 Regarding building a tower of ever less noisy qubits: it turns out there is something called a threshold theorem which requires a certain minimal level of noise before concatenating error correction schemes makes things better rather than worse. https://en.m.wikipedia.org/wiki/Quantum_threshold_theorem https://en.m.wikipedia.org/wiki/Quantum_threshold_theorem Reaching this threshold for physical qubits in the laboratory is perhaps the primary goal of experimental quantum computing.