---
OA_place: repository
_id: '17222'
abstract:
- lang: eng
  text: "The quantum approximate optimization algorithm (QAOA) uses a quantum computer\r\nto
    implement a variational method with $2p$ layers of alternating unitary\r\noperators,
    optimized by a classical computer to minimize a cost function. While\r\nrigorous
    performance guarantees exist for the QAOA at small depths $p$, the\r\nbehavior
    at large depths remains less clear, though simulations suggest\r\nexponentially
    fast convergence for certain problems. In this work, we gain\r\ninsights into
    the deep QAOA using an analytic expansion of the cost function\r\naround transition
    states. Transition states are constructed in a recursive\r\nmanner: from the local
    minima of the QAOA with $p$ layers we obtain transition\r\nstates of the QAOA
    with $p+1$ layers, which are stationary points characterized\r\nby a unique direction
    of negative curvature. We construct an analytic estimate\r\nof the negative curvature
    and the corresponding direction in parameter space at\r\neach transition state.
    The expansion of the QAOA cost function along the\r\nnegative direction to the
    quartic order gives a lower bound of the QAOA cost\r\nfunction improvement. We
    provide physical intuition behind the analytic\r\nexpressions for the local curvature
    and quartic expansion coefficient. Our\r\nnumerical study confirms the accuracy
    of our approximations and reveals that\r\nthe obtained bound and the true value
    of the QAOA cost function gain have a\r\ncharacteristic exponential decrease with
    the number of layers $p$, with the\r\nbound decreasing more rapidly. Our study
    establishes an analytical method for\r\nrecursively studying the QAOA that is
    applicable in the regime of high circuit\r\ndepth."
article_number: '2405.10125'
article_processing_charge: No
arxiv: 1
author:
- first_name: Raimel A
  full_name: Medina Ramos, Raimel A
  id: CE680B90-D85A-11E9-B684-C920E6697425
  last_name: Medina Ramos
  orcid: 0000-0002-5383-2869
- first_name: Maksym
  full_name: Serbyn, Maksym
  id: 47809E7E-F248-11E8-B48F-1D18A9856A87
  last_name: Serbyn
  orcid: 0000-0002-2399-5827
citation:
  ama: Medina Ramos RA, Serbyn M. A recursive lower bound on the energy improvement
    of the quantum approximate optimization algorithm. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2405.10125">10.48550/arXiv.2405.10125</a>
  apa: Medina Ramos, R. A., &#38; Serbyn, M. (n.d.). A recursive lower bound on the
    energy improvement of the quantum approximate optimization algorithm. <i>arXiv</i>.
    <a href="https://doi.org/10.48550/arXiv.2405.10125">https://doi.org/10.48550/arXiv.2405.10125</a>
  chicago: Medina Ramos, Raimel A, and Maksym Serbyn. “A Recursive Lower Bound on
    the Energy Improvement of the Quantum Approximate Optimization Algorithm.” <i>ArXiv</i>,
    n.d. <a href="https://doi.org/10.48550/arXiv.2405.10125">https://doi.org/10.48550/arXiv.2405.10125</a>.
  ieee: R. A. Medina Ramos and M. Serbyn, “A recursive lower bound on the energy improvement
    of the quantum approximate optimization algorithm,” <i>arXiv</i>. .
  ista: Medina Ramos RA, Serbyn M. A recursive lower bound on the energy improvement
    of the quantum approximate optimization algorithm. arXiv, 2405.10125.
  mla: Medina Ramos, Raimel A., and Maksym Serbyn. “A Recursive Lower Bound on the
    Energy Improvement of the Quantum Approximate Optimization Algorithm.” <i>ArXiv</i>,
    2405.10125, doi:<a href="https://doi.org/10.48550/arXiv.2405.10125">10.48550/arXiv.2405.10125</a>.
  short: R.A. Medina Ramos, M. Serbyn, ArXiv (n.d.).
corr_author: '1'
date_created: 2024-07-10T13:12:09Z
date_published: 2024-05-16T00:00:00Z
date_updated: 2026-04-07T12:43:22Z
day: '16'
department:
- _id: MaSe
doi: 10.48550/arXiv.2405.10125
external_id:
  arxiv:
  - '2405.10125'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2405.10125
month: '05'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '17208'
    relation: dissertation_contains
    status: public
status: public
title: A recursive lower bound on the energy improvement of the quantum approximate
  optimization algorithm
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2024'
...