Quantum state preparation via piecewise QSVT

TL;DR

This paper introduces a piecewise Quantum Singular Value Transformation method for efficient quantum state preparation, enabling new classes of states and improving quantum phase estimation with significantly reduced resource costs.

Contribution

It presents a novel piecewise QSVT technique and a new piecewise linear diagonal block encoding for preparing states with piecewise polynomial amplitudes, enhancing quantum algorithm efficiency.

Findings

Efficiently prepares states approximated by piecewise polynomials.

Reduces Toffoli gate count by 50 times compared to Kaiser window.

Replicates exponential tail probability reduction in QPE using B-spline window.

Abstract

Efficient state preparation is essential for implementing efficient quantum algorithms. Whilst several techniques for low-cost state preparation exist, this work facilitates further classes of states, whose amplitudes are well approximated by piecewise polynomials. We show how such states can be efficiently prepared using a piecewise Quantum Singular Value Transformation along with a new piecewise linear diagonal block encoding. We illustrate this with the explicit examples of $x^\alpha|x\rangle$ and $\log x|x\rangle$. Further, our technique reduces the cost of window boosted Quantum Phase Estimation by efficiently preparing the B-spline window state. We demonstrate this window state requires 50 times fewer Toffolis to prepare than the state-of-the-art Kaiser window state, and we show that the B-spline window replicates the Kaiser window's exponential reduction in tail probability for QPE.