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Persistent homology can read structural change in a financial time series that a price chart alone does not show. We embed market data into point clouds, estimate Betti numbers with classical solvers and with quantum phase estimation, and show that spikes in β₀ line up with crash regimes — then train a simple LSTM on those topological features.
The quantum Fourier transform is the same classical Fourier transform, applied to amplitudes. On a periodic amplitude comb it piles constructive interference at multiples of N/r and cancels everywhere else — which is why period finding, and therefore Shor, can read a hidden period from a single measurement.