Abstract
In this perspective, 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 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.
Highlights
One of the most important goals of the field of quantum computing is to eventually build a fault-tolerant quantum computer
Toffoli distillation rates increase by 2 orders magnitude it would still be challenging to obtain quantum advantage with a quadratic speedup but we cannot categorically rule it out for all algorithms
We investigate simple conditions that must be satisfied to realize a quantum advantage through polynomial speedups on a small fault-tolerant quantum computer
Summary
One of the most important goals of the field of quantum computing is to eventually build a fault-tolerant quantum computer. The central issue is that quantum error correction and the device operation time introduce significant constant factor slowdowns to the algorithm runtime (see Fig. 1) These large overheads present many challenges for the practical realization of useful fault-tolerant devices. This is borne out through numerous studies on the cost of error-correcting applications with an exponential scaling advantage in areas such as quantum chemistry [14,15,16], quantum simulation of lattice models [17,18], and prime factoring [19] In this perspective we discuss when it would be practical for a modest fault-tolerant quantum computer to realize a quantum advantage with quantum algorithms giving only a small polynomial speedup over their classical competition. We hope this perspective will encourage the field to critically examine the prospects for quantum advantage with errorcorrected quadratic speedups and either produce examples where it is feasible or focus more effort on algorithms with larger speedups
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