Thesis: The Power of Quantum Fourier Sampling by William Jason Fefferman (2014)

Fefferman, 2014

Category: Quantum Computing

Overall Rating

2.0/5 (14/35 pts)

This paper presents a specific theoretical pathway to demonstrating quantum sampling advantage based on novel mathematical structures (ESPs and the Squashed QFT).
However, this path is contingent on proving challenging, unproven complexity conjectures and requires the efficient realization of a specific quantum unitary, which is posed as an open problem.
These dependencies significantly limit the paper's direct actionable value for current research, despite the mathematical novelty of its constructs.

this thesis introduces specific concepts like "Efficiently Specifiable Polynomials" (ESP) and the "Squashed QFT" as a means to generate sampling distributions.
These specific structures, particularly the "Squashed QFT" as a transform linking assignments to symmetric polynomial evaluations at integer points, might contain unexplored mathematical properties or algorithmic insights that haven't been fully leveraged beyond their original context of proving quantum hardness relative to classical complexity classes.
An unconventional research direction could explore using the Squashed QFT... as a structured feature transformation layer in classical neural networks or kernel methods.
Experimental progress in building quantum computers since 2014, particularly those targeting sampling-based demonstrations of quantum advantage, makes the experimental realization of the types of sampling distributions discussed here (or close variants) more feasible now than at the time of publication.

its central theoretical leverage point – the "Squashed QFT"... hinges on a major open question: whether this specific unitary can be realized by an efficient quantum circuit (Ch 8).
the connection between approximate classical sampling hardness and #P-hardness relies on strong, specific anti-concentration and hardness conjectures (Conjectures 1, 2, 3) that the authors themselves note "seem out of reach" (Ch 8).
The absence of a concrete path to proving these conjectures or building the required quantum resource weakens its foundational impact...
the field's focus for demonstrating quantum supremacy has largely shifted to Random Circuit Sampling (RCS)... Re-treading this specific path might be redundant compared to pushing the boundaries of more established or experimentally-driven paradigms.

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