Quantum state preparation for multivariate functions

Matthias Rosenkranz; Eric Brunner; Gabriel Marin-Sanchez; Nathan Fitzpatrick; Silas Dilkes; Yao Tang; Yuta Kikuchi; Marcello Benedetti

doi:10.22331/q-2025-04-09-1703

Quantum state preparation for multivariate functions

Matthias Rosenkranz¹, Eric Brunner¹, Gabriel Marin-Sanchez¹, Nathan Fitzpatrick², Silas Dilkes², Yao Tang², Yuta Kikuchi^3,4, and Marcello Benedetti¹

¹Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom
²Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom
³Quantinuum K.K., Otemachi Financial City Grand Cube 3F, 1-9-2 Otemachi, Chiyoda-ku, Tokyo, Japan
⁴Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan

Published:	2025-04-09, volume 9, page 1703
Editor:	Tongyang Li
Eprint:	arXiv:2405.21058v2
Doi:	https://doi.org/10.22331/q-2025-04-09-1703
Citation:	Quantum 9, 1703 (2025).

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Abstract

A fundamental step of any quantum algorithm is the preparation of qubit registers in a suitable initial state. Often qubit registers represent a discretization of continuous variables and the initial state is defined by a multivariate function. We develop protocols for preparing quantum states whose amplitudes encode multivariate functions by linearly combining block-encodings of Fourier and Chebyshev basis functions. Without relying on arithmetic circuits, quantum Fourier transforms, or multivariate quantum signal processing, our algorithms are simpler and more effective than previous proposals. We analyze requirements both asymptotically and pragmatically in terms of near/medium-term resources. Numerically, we prepare bivariate Student's t-distributions, 2D Ricker wavelets and electron wavefunctions in a 3D Coulomb potential, which are initial states with potential applications in finance, physics and chemistry simulations. Finally, we prepare bivariate Gaussian distributions on the Quantinuum H2-1 trapped-ion quantum processor using 24 qubits and up to 237 two-qubit gates.

Featured image: Quantum circuit for state preparation in one dimension. Circuit $U$ is the block-encoding of a Fourier or Chebyshev basis function. Each bit in the coefficient register drives the application of powers of $U$. The algorithm succeeds if we measure the zero state on all $a + b$ ancilla qubits. The circuit for the multidimensional case has the same structure with one coefficient register, one main register, and one block-encoding register (for Chebyshev only) per dimension.

► BibTeX data

@article{Rosenkranz2025quantumstate,
  doi = {10.22331/q-2025-04-09-1703},
  url = {https://doi.org/10.22331/q-2025-04-09-1703},
  title = {Quantum state preparation for multivariate functions},
  author = {Rosenkranz, Matthias and Brunner, Eric and Marin-Sanchez, Gabriel and Fitzpatrick, Nathan and Dilkes, Silas and Tang, Yao and Kikuchi, Yuta and Benedetti, Marcello},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1703},
  month = apr,
  year = {2025}
}

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