Schrödinger as a Quantum Programmer: Estimating Entanglement via Steering

Aby Philip; Soorya Rethinasamy; Vincent Russo; Mark M. Wilde

doi:10.22331/q-2024-06-11-1366

Schrödinger as a Quantum Programmer: Estimating Entanglement via Steering

Aby Philip¹, Soorya Rethinasamy¹, Vincent Russo², and Mark M. Wilde^3,1

¹School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14850, USA
²Unitary Fund
³School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA

Published:	2024-06-11, volume 8, page 1366
Eprint:	arXiv:2303.07911v4
Doi:	https://doi.org/10.22331/q-2024-06-11-1366
Citation:	Quantum 8, 1366 (2024).

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Abstract

Quantifying entanglement is an important task by which the resourcefulness of a quantum state can be measured. Here, we develop a quantum algorithm that tests for and quantifies the separability of a general bipartite state by using the quantum steering effect, the latter initially discovered by Schrödinger. Our separability test consists of a distributed quantum computation involving two parties: a computationally limited client, who prepares a purification of the state of interest, and a computationally unbounded server, who tries to steer the reduced systems to a probabilistic ensemble of pure product states. To design a practical algorithm, we replace the role of the server with a combination of parameterized unitary circuits and classical optimization techniques to perform the necessary computation. The result is a variational quantum steering algorithm (VQSA), a modified separability test that is implementable on quantum computers that are available today. We then simulate our VQSA on noisy quantum simulators and find favorable convergence properties on the examples tested. We also develop semidefinite programs, executable on classical computers, that benchmark the results obtained from our VQSA. Thus, our findings provide a meaningful connection between steering, entanglement, quantum algorithms, and quantum computational complexity theory. They also demonstrate the value of a parameterized mid-circuit measurement in a VQSA.

Featured image: Quantum part of the variational quantum steering algorithm for estimating the fidelity of separability $F_s(\rho_{AB})$. The unitary circuit $U^\rho$ prepares the state $\psi_{RAB}$, which is a purification of $\rho_{AB}$. The parameterized circuit $W_R(\Theta)$ acts on $R$ to evolve $\psi_{RAB}$ to another purification of $\rho_{AB}$. The following measurement, labeled "steering measurement," steers the systems $AB$ to be in a pure state $\psi_{AB}^{x}$ if the measurement outcome $x$ occurs. Conditioned on the outcome $x$, the final parameterized circuit $U^{x}_{A}(\Theta^x)$ and the subsequent measurement accepts with a maximum probability of $F_s(\rho_{AB})$.

Popular summary

Entanglement is a unique feature of quantum mechanics, initially brought to light by Einstein, Podolsky, and Rosen. Determining whether a general quantum state is entangled or not, known as the separability problem, is of fundamental interest and relevant to various fields of physics: condensed matter, quantum gravity, quantum optics, and quantum information science. Motivated by the rapid development of quantum computers, we provide a variational quantum steering algorithm (VQSA) to address the separability problem. The quantum steering effect, discovered by Schrödinger, is central to the design of our algorithm, which is implementable on quantum computers that are available today. We provide benchmarks and simulations of our algorithm and establish a meaningful connection between steering, entanglement, quantum algorithms, and quantum computational complexity theory.

► BibTeX data

@article{Philip2024schrodingeras,
  doi = {10.22331/q-2024-06-11-1366},
  url = {https://doi.org/10.22331/q-2024-06-11-1366},
  title = {Schr{\"{o}}dinger as a {Q}uantum {P}rogrammer: {E}stimating {E}ntanglement via {S}teering},
  author = {Philip, Aby and Rethinasamy, Soorya and Russo, Vincent and Wilde, Mark M.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1366},
  month = jun,
  year = {2024}
}

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Cited by

[1] L. Zambrano, A. D. Muñoz-Moller, M. Muñoz, L. Pereira, and A. Delgado, "Avoiding barren plateaus in the variational determination of geometric entanglement", Quantum Science and Technology 9 2, 025016 (2024).

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