Deterministic Bethe state preparation

David Raveh; Rafael I. Nepomechie

doi:10.22331/q-2024-10-24-1510

Deterministic Bethe state preparation

Department of Physics, PO Box 248046, University of Miami, Coral Gables, FL 33124 USA

Published:	2024-10-24, volume 8, page 1510
Eprint:	arXiv:2403.03283v3
Doi:	https://doi.org/10.22331/q-2024-10-24-1510
Citation:	Quantum 8, 1510 (2024).

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Abstract

We present an explicit quantum circuit that prepares an arbitrary $U(1)$-eigenstate on a quantum computer, including the exact eigenstates of the spin-$1/2 XXZ$ quantum spin chain with either open or closed boundary conditions. The algorithm is deterministic, does not require ancillary qubits, and does not require QR decompositions. The circuit prepares such an $L$-qubit state with $M$ down-spins using $\binom{L}{M}-1$ multi-controlled rotation gates and $2M(L-M)$ CNOT-gates.

Featured image: The complete circuit diagram for a Bethe state of the open XXZ chain with L=4 and M=2.

“Bethe state preparation”, Simons Center, May 2024: http://scgp.stonybrook.edu/video_portal/video.php?id=6518

“Preparing exact eigenstates on a quantum computer”, Australian Institute of Physics Theoretical Seminar, July 2024:
https://www.aip.org.au/Theoretical-Physics-(TPG)/13382858

Popular summary

The spin-1/2 XXZ quantum spin chain is a paradigmatic model of theoretical physics, describing a one-dimensional magnet: a “chain’’ of L quantum spins (qubits) with nearest-neighbor interactions that preserve the number M of down-spins (qubits in the state 1). In this paper, we present an algorithm for preparing on a quantum computer exact eigenstates of this model, with either periodic or open boundary conditions. The only needed input is the set of M so-called Bethe roots that completely characterize the desired eigenstate. (This model has the remarkable feature of being “integrable’’, which implies that its eigenstates can be so characterized. The Bethe roots are in general complex numbers, which in principle can be obtained by solving a system of M simultaneous equations, called Bethe equations.) The algorithm is explicit, deterministic, and does not require ancillary qubits. However, both the classical and quantum complexity of the algorithm scale exponentially with M. Simple examples are presented, and Qiskit code that implements this algorithm is provided. Based on this algorithm, we present in a separate work [1] an algorithm for obtaining Bethe roots on a quantum computer.

[1] D. Raveh and R.I. Nepomechie, “Estimating Bethe roots with VQE”, J. Phys. A 57, 355303 (2024), arXiv:2404.18244

► BibTeX data

@article{Raveh2024deterministicbethe,
  doi = {10.22331/q-2024-10-24-1510},
  url = {https://doi.org/10.22331/q-2024-10-24-1510},
  title = {Deterministic {B}ethe state preparation},
  author = {Raveh, David and Nepomechie, Rafael I.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1510},
  month = oct,
  year = {2024}
}

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Cited by

[1] Yuxuan Guo, Sheng Yang, and Xue-Jia Yu, "Quantum Strong-To-Weak Spontaneous Symmetry Breaking in Decohered One-Dimensional Critical States", PRX Quantum 6 4, 040311 (2025).

[2] D. Faílde, A. Gómez, and J. Fernández-Rossier, "Preparation of the single-spinon wave function on a quantum computer", Physical Review Research 8 3, 033152 (2026).

[3] Roberto Ruiz, Alejandro Sopena, Esperanza López, German Sierra, and Balázs Pozsgay, "Bethe Ansatz, quantum circuits, and the F-basis", SciPost Physics 18 6, 187 (2025).

[4] Xiaowei Huang, Fei Shi, Lijun Zhang, and Lvzhou Li, "Quantum states supported by matroids", Physical Review A 113 6, 062452 (2026).

[5] Maximilian Lutz, Lorenzo Piroli, Georgios Styliaris, and J. Ignacio Cirac, "Adiabatic quantum state preparation in integrable models", Quantum 10, 2032 (2026).

[6] R. Stagraczyński and T. Lulek, "Symbolic evaluation of transfer matrices for the XXX model", Computer Physics Communications 324, 110135 (2026).

[7] Leandro Hayato Ymai, Karin Wittmann Wilsmann, Arlei Prestes Tonel, Angela Foerster, and Jon Links, "High-fidelity transfer of entangled states on a quantum turntable", Newton 1 9, 100226 (2025).

[8] Subhayan Sahu and Guifre Vidal, "Fractal decompositions and tensor network representations of Bethe wavefunctions", SciPost Physics Core 8 4, 067 (2025).

[9] Roberto Ruiz, Alejandro Sopena, Balázs Pozsgay, and Esperanza López, "Efficient Eigenstate Preparation in an Integrable Model with Hilbert Space Fragmentation", PRX Quantum 6 3, 030316 (2025).

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[12] David Raveh and Rafael I Nepomechie, "Estimating Bethe roots with VQE", Journal of Physics A: Mathematical and Theoretical 57 35, 355303 (2024).

[13] Hyeonjun Yeo, Ha Eum Kim, IlKwon Sohn, and Kabgyun Jeong, "Reducing circuit depth in quantum state preparation for quantum simulation using measurements and feedforward", Physical Review Applied 23 5, 054066 (2025).

[14] Nabi Zare Harofteh and Rafael I. Nepomechie, "Spin‐sU(1)‐Eigenstate Preparation", Annalen der Physik 538 7, e70239 (2026).

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