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Entanglement scaling in matrix product state representation of smooth functions and their shallow quantum circuit approximations

Vladyslav Bohun1,*, Illia Lukin1,2,*, Mykola Luhanko1,3, Georgios Korpas4,5,6, Philippe J. S. De Brouwer7, Mykola Maksymenko1, and Maciej Koch-Janusz1,8,†

  • *These authors contributed equally to this work.
  • †Contact author: maciej@haiqu.ai
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Phys. Rev. Research 8, 023062 – Published 20 April, 2026

DOI: https://doi.org/10.1103/yyvr-dtsb

Abstract

Encoding classical data in a quantum state is a key prerequisite of many quantum algorithms. Recently, matrix product state (MPS) methods emerged as the most promising approach for constructing shallow quantum circuits approximating input functions, including probability distributions, with only linear number of gates. We derive rigorous asymptotic expansions for the decay of entanglement across bonds in the MPS representation depending on the smoothness of the input function, real or complex. We also consider the dependence of the entanglement on localization properties and function support. Based on these analytical results, we construct an improved MPS-based algorithm yielding shallow and accurate encoding quantum circuits. By using tensor cross interpolation, we are able to construct utility-scale quantum circuits in a compute- and memory-efficient way. We validate our methods by loading heavy-tailed distributions, including Lévy, important in finance, but they apply to any smooth function inputs. We test the performance of the resulting quantum circuits by executing and sampling from them on IBM quantum devices, for up to 156 qubits.

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Phys. Rev. Research 8, 023062– Published 20 April, 2026
  • Received 1 April 2025
  • Accepted 2 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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