Quantum state preparation via piecewise QSVT

Oliver O'Brien; Christoph Sünderhauf

doi:10.22331/q-2025-07-03-1786

Quantum state preparation via piecewise QSVT

Oliver O'Brien^1,2 and Christoph Sünderhauf¹

¹Riverlane, Cambridge, United Kingdom
²Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom

Published:	2025-07-03, volume 9, page 1786
Editor:	Daniel Malz
Eprint:	arXiv:2409.07332v2
Doi:	https://doi.org/10.22331/q-2025-07-03-1786
Citation:	Quantum 9, 1786 (2025).

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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.

Featured image: High level overview of algorithm: Target state is approximated by a piecewise polynomial, then piecewise QSVT is used prepare a state with amplitudes uniformly sampled from this piecewise polynomial.

Popular summary

Loading classical data into a quantum computer is a major bottleneck that limits the practical usefulness of many quantum algorithms. This challenge affects a broad range of fields, including quantum chemistry, computational fluid dynamics, and quantum finance. Sometimes running a quantum algorithm on the data takes significantly less time than preparing the data in the first place!

While some classes of data can be compressed and loaded efficiently, many real-world datasets contain sharp discontinuities or singularities that make this much more difficult. In this work, we introduce a new method for efficiently loading such challenging data into quantum systems, enabling faster access to a broader range of inputs.

We achieve this by extending a powerful framework known as Quantum Singular Value Transformation (QSVT). QSVT allows polynomial transformations of matrices that are encoded into quantum circuits—a process known as block encoding. We develop a piecewise variant of QSVT that applies different polynomial transformations to different regions of the input, allowing us to better match the structure of complex data. To support this, we introduce a new block encoding that enables efficient sampling from piecewise polynomial transformations.

To demonstrate the impact of our method, we show how it can efficiently prepare a classical B-spline window function—valued for its excellent spectral properties. This function enhances the performance of Quantum Phase Estimation (QPE), a key subroutine in many quantum algorithms. Using our technique, this enhancement can be achieved with 50 times fewer resources than the best previous methods.

Our work provides a general framework for preparing a wider range of quantum states based on structured classical data, especially those with localized features or sharp transitions. We expect it to be broadly useful in early fault-tolerant quantum computing, where minimizing circuit costs is essential.

► BibTeX data

@article{OBrien2025quantumstate,
  doi = {10.22331/q-2025-07-03-1786},
  url = {https://doi.org/10.22331/q-2025-07-03-1786},
  title = {Quantum state preparation via piecewise {QSVT}},
  author = {O'Brien, Oliver and S{\"{u}}nderhauf, Christoph},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1786},
  month = jul,
  year = {2025}
}

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