Maximum expectation of observables with restricted purity states

Vikesh Siddhu; John Aaron Smolin

doi:10.22331/q-2024-08-13-1437

Maximum expectation of observables with restricted purity states

Vikesh Siddhu^1,2 and John Aaron Smolin¹

¹IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA
²IBM Quantum, IBM Research India

Published:	2024-08-13, volume 8, page 1437
Eprint:	arXiv:2311.07680v2
Doi:	https://doi.org/10.22331/q-2024-08-13-1437
Citation:	Quantum 8, 1437 (2024).

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Abstract

Assessment of practical quantum information processing (QIP) remains partial without understanding limits imposed by noise. Unfortunately, mere description of noise grows exponentially with system size, becoming cumbersome even for modest sized systems of imminent practical interest. We fulfill the need for estimates on performing noisy quantum state preparation, verification, and observation. To do the estimation we propose fast numerical algorithms to maximize the expectation value of any $d$-dimensional observable over states of bounded purity. This bound on purity factors in noise in a measurable way. Our fastest algorithm takes $O(d)$ steps if the eigendecomposition of the observable is known, otherwise takes $O(d^3)$ steps at worst. The algorithms also solve maximum likelihood estimation for quantum state tomography with convex and even non-convex purity constraints. Numerics show performance of our key sub-routine (it finds in linear time a probability vector with bounded norm that most overlaps with a fixed vector) can be several orders of magnitude faster than a common state-of-the-art convex optimization solver. Our work fosters a practical way forward to asses limitations on QIP imposed by quantum noise. Along the way, we also give a simple but fundamental insight, noisy systems (equivalently noisy Hamiltonians) always give higher ground-state energy than their noiseless counterparts.

► BibTeX data

@article{Siddhu2024maximumexpectation,
  doi = {10.22331/q-2024-08-13-1437},
  url = {https://doi.org/10.22331/q-2024-08-13-1437},
  title = {Maximum expectation of observables with restricted purity states},
  author = {Siddhu, Vikesh and Smolin, John Aaron},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1437},
  month = aug,
  year = {2024}
}

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