Towards Quantum Advantage on Noisy Quantum Computers

Quantum computers offer the potential of achieving significant speedup for certain computational problems. Yet, many existing quantum algorithms with notable asymptotic speedups require a degree of fault tolerance that is currently unavailable. The quantum algorithm for topological data analysis (TDA) by Lloyd et al. is believed to be one such algorithm. TDA is a powerful technique for extracting complex and valuable shape-related summaries of high-dimensional data. However, the computational demands of classical TDA algorithms are exorbitant, and become impractical for high-order characteristics. In this paper, we present NISQ-TDA, the first fully implemented end-to-end quantum machine learning algorithm needing only a short circuit-depth, that is applicable to non-handcrafted high-dimensional classical data, and with provable asymptotic speedup for certain classes of problems. The algorithm neither suffers from the data-loading problem nor does it need to store the input data on the quantum computer explicitly. Our approach includes three key innovations: an efficient realization of the full boundary operator; a quantum rejection sampling and projection approach to restrict a quantum state to the simplices of the desired order in the given complex; and a stochastic rank estimation method to estimate the topological features in the form of approximate Betti numbers. We present theoretical results that establish additive error guarantees, along with computational cost and circuit-depth complexities for normalized output estimates, up to the error tolerance. The algorithm was successfully executed on quantum computing devices, as well as on noisy quantum simulators, applied to small datasets. Preliminary empirical results suggest that the algorithm is robust to noise. Finally, we provide target depths and noise level estimates to realize near-term, non-fault-tolerant quantum advantage.