Focus beyond Quadratic Speedups for Error-Corrected Quantum Advantage

We discuss conditions under which it would be possible for a modest fault-tolerant quantum computer to realize a runtime advantage by executing a quantum algorithm with only a small polynomial speedup over the best classical alternative. The challenge is that the computation must finish within a reasonable amount of time while being difficult enough that the small quantum scaling advantage would compensate for the large constant factor overheads associated with error-correction. We compute several examples of such runtimes using state-of-the-art surface code constructions for superconducting qubits under a variety of assumptions. We conclude that quadratic speedups will not enable quantum advantage on early generations of such fault-tolerant devices unless there is a significant improvement in how we would realize quantum error-correction. While this conclusion persists even if we were to increase the rate of logical gates in the surface code by more than an order of magnitude, we also repeat this analysis for speedups by other polynomial degrees and find that quartic speedups look significantly more practical.