Fast Black-Box Quantum State Preparation

Johannes Bausch

Google DeepMind
CQIF, DAMTP, University of Cambridge, UK

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Abstract

Quantum state preparation is an important ingredient for other higher-level quantum algorithms, such as Hamiltonian simulation, or for loading distributions into a quantum device to be used e.g. in the context of optimization tasks such as machine learning. Starting with a generic "black box" method devised by Grover in 2000, which employs amplitude amplification to load coefficients calculated by an oracle, there has been a long series of results and improvements with various additional conditions on the amplitudes to be loaded, culminating in Sanders et al.'s work which avoids almost all arithmetic during the preparation stage.
In this work, we construct an optimized black box state loading scheme with which various important sets of coefficients can be loaded significantly faster than in $O(\sqrt N)$ rounds of amplitude amplification, up to only $O(1)$ many. We achieve this with two variants of our algorithm. The first employs a modification of the oracle from Sanders et al., which requires fewer ancillas ($\log_2 g$ vs $g+2$ in the bit precision $g$), and fewer non-Clifford operations per amplitude amplification round within the context of our algorithm. The second utilizes the same oracle, but at slightly increased cost in terms of ancillas ($g+\log_2g$) and non-Clifford operations per amplification round. As the number of amplitude amplification rounds enters as multiplicative factor, our black box state loading scheme yields an up to exponential speedup as compared to prior methods. This speedup translates beyond the black box case.

Data loading is a crucial step for many algorithms, classical or quantum. A generic formulation of this task comprises two components, a "black box" subroutine that one can query and ask about parts of the data (for instance a specific pixel in an image), and the host routine that decides how to query the data, and takes the information it receives to prepare a state encoding the data to be loaded.
In this work, we improve upon the host routine, by significantly reducing the number of necessary queries to the black box, yielding an up to exponential speedup – naturally depending on the data to be loaded, but the results hold for a wide range of realistic datasets or distributions of interest. We further devise a specific black box subroutine, tailored to work particularly well with our host data loading scheme which further reduces the required qubit and gate overhead.

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