Accelerating Classical and Quantum Tensor PCA
Matthew B. Hastings
TL;DR
This work studies accelerating both classical and quantum spectral algorithms for tensor PCA with a spiked Gaussian tensor. It introduces a quadratic-speedup framework that preserves the quartic separation between quantum and classical runtimes by projecting onto a reduced eigenspace via decorrelated input states and approximate projectors, and it further advances with a multi-step quantum procedure achieving up to a sixth-power speedup. The results include provable detection guarantees for an ideal input and a probabilistic recovery argument for the spiked case, contingent on density-of-states assumptions. The paper also discusses limitations: a tighter bound on the spectral norm may remove provable polynomial speedups, while DOS considerations suggest possible applicability to a broader class of planted problems.
Abstract
Spectral methods are a leading approach for tensor PCA with a ``spiked" Gaussian tensor. The methods use the spectrum of a linear operator in a vector space with exponentially high dimension and in Ref. 1 it was shown that quantum algorithms could then lead to an exponential space saving as well as a quartic speedup over classical. Here we show how to accelerate both classical and quantum algorithms quadratically, while maintaining the same quartic separation between them. That is, our classical algorithm here is quadratically faster than the original classical algorithm, while the quantum algorithm is eigth-power faster than the original classical algorithm. We then give a further modification of the quantum algorithm, increasing its speedup over the modified classical algorithm to the sixth power. We only prove these speedups for detection, rather than recovery, but we give a strong plausibility argument that our algorithm achieves recovery also. Note added: After this paper was prepared, A. Schmidhuber pointed out to me Ref. 3. This improves the best existing bounds on the spectral norm of a certain random operator. Because the norm of this operator enters into the runtime, with this improvement on the norm, we no longer have a provable polynomial speedup. Our results are phrased in terms of certain properties of the spectrum of this operator (not merely the largest eigenvalue but also the density of states). So, if these properties still hold, the speedup still holds. Rather than modify the paper, I have left it unchanged but added a section at the end discussing the needed property of density of states and considering for which problems (there are several problems for which this kind of quartic quantum speedup has been used and the techniques here will likely be applicable to several of them) the property is likely to hold.