---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
PlanS_conform: '1'
_id: '21917'
abstract:
- lang: eng
  text: A defining feature of quantum many-body systems is the exponential scaling
    of the Hilbert space with the number of degrees of freedom. This exponential complexity
    naïvely renders a complete state characterization, for instance via the complete
    set of bipartite Renyi entropies for all disjoint regions, a challenging task.
    Recently, a compact way of storing subregions' purities by encoding them as amplitudes
    of a fictitious quantum wave function, known as entanglement feature, was proposed.
    Notably, the entanglement feature can be a simple object even for highly entangled
    quantum states. However the complexity and practical usage of the entanglement
    feature for general quantum states has not been explored. In this work, we demonstrate
    that the entanglement feature can be efficiently learned using only a polynomial
    amount of samples in the number of degrees of freedom through the so-called tensor
    cross interpolation (TCI) algorithm, assuming it is expressible as a finite bond
    dimension MPS. We benchmark this learning process on Haar and random MPS states,
    confirming analytic expectations. Applying the TCI algorithm to quantum eigenstates
    of various one dimensional quantum systems, we identify cases where eigenstates
    have entanglement feature learnable with TCI. We conclude with possible applications
    of the learned entanglement feature, such as quantifying the distance between
    different entanglement patterns and finding the optimal one-dimensional ordering
    of physical indices in a given state, highlighting the potential utility of the
    proposed purity interpolation method.
acknowledgement: "We acknowledge useful discussions with Richard Küng\r\non the interpolation
  methods and error spreading, Ilia\r\nA. Luchnikov, Margarita Davydova, and, in particular,
  Hiroshi Shinaoka, Marc Ritter, Yuriel Nuñez\r\nfor useful discussions about TCI
  and the various\r\nworkarounds within the TensorCrossInterpolation.jl\r\nlibrary.
  We also acknowledge the comments of anonymous Referee B, that encouraged us to expand
  the\r\nmanuscript with discussion of additional applications\r\nof entanglement
  feature in Section 4.3. M.S. acknowledges discussions with D. V. Savostyanov at
  the 2nd\r\nInternational Quantum Tensor Networks (IQTN) plenary meeting at Flatiron
  Institute’s Center for Computational Quantum Physics (CCQ) for introduction\r\nto
  the TCI approach. D.K and M.S. acknowledge support by the European Research Council
  (ERC) under We acknowledge useful discussions with Richard Küng\r\non the interpolation
  methods and error spreading, Ilia\r\nA. Luchnikov, Margarita Davydova, and, in particular,
  Hiroshi Shinaoka, Marc Ritter, Yuriel Nuñez\r\nfor useful discussions about TCI
  and the various\r\nworkarounds within the TensorCrossInterpolation.jl\r\nlibrary.
  We also acknowledge the comments of anonymous Referee B, that encouraged us to expand
  the\r\nmanuscript with discussion of additional applications\r\nof entanglement
  feature in Section 4.3. M.S. acknowledges discussions with D. V. Savostyanov at
  the 2nd\r\nInternational Quantum Tensor Networks (IQTN) plenary meeting at Flatiron
  Institute’s Center for Computational Quantum Physics (CCQ) for introduction\r\nto
  the TCI approach. D.K and M.S. acknowledge support by the European Research Council
  (ERC) under We acknowledge useful discussions with Richard Küng\r\non the interpolation
  methods and error spreading, Ilia\r\nA. Luchnikov, Margarita Davydova, and, in particular,
  Hiroshi Shinaoka, Marc Ritter, Yuriel Nuñez\r\nfor useful discussions about TCI
  and the various\r\nworkarounds within the TensorCrossInterpolation.jl\r\nlibrary.
  We also acknowledge the comments of anonymous Referee B, that encouraged us to expand
  the\r\nmanuscript with discussion of additional applications\r\nof entanglement
  feature in Section 4.3. M.S. acknowledges discussions with D. V. Savostyanov at
  the 2nd\r\nInternational Quantum Tensor Networks (IQTN) plenary meeting at Flatiron
  Institute’s Center for Computational Quantum Physics (CCQ) for introduction\r\nto
  the TCI approach. D.K and M.S. acknowledge support by the European Research Council
  (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant
  Agreement No. 850899).\r\nR.V. acknowledges partial support from the US Department
  of Energy, Office of Science, Basic Energy\r\nSciences, under award No. DE-SC0023999,
  and the\r\nSwiss National Science Foundation (grant 10008234).\r\nThis research
  was supported in part by grant NSF\r\nPHY-2309135 to the Kavli Institute for Theoretical\r\nPhysics
  (KITP)"
article_number: '2114'
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Dmytro
  full_name: Kolisnyk, Dmytro
  id: 530a7320-5355-11ee-ae5a-82a46997aaa7
  last_name: Kolisnyk
  orcid: 0000-0002-8612-8202
- 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: Romain
  full_name: Vasseur, Romain
  last_name: Vasseur
- first_name: Maksym
  full_name: Serbyn, Maksym
  id: 47809E7E-F248-11E8-B48F-1D18A9856A87
  last_name: Serbyn
  orcid: 0000-0002-2399-5827
citation:
  ama: Kolisnyk D, Medina Ramos RA, Vasseur R, Serbyn M. Tensor cross interpolation
    of purities in quantum many-body systems. <i>Quantum</i>. 2026;10. doi:<a href="https://doi.org/10.22331/q-2026-05-22-2114">10.22331/q-2026-05-22-2114</a>
  apa: Kolisnyk, D., Medina Ramos, R. A., Vasseur, R., &#38; Serbyn, M. (2026). Tensor
    cross interpolation of purities in quantum many-body systems. <i>Quantum</i>.
    Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.
    <a href="https://doi.org/10.22331/q-2026-05-22-2114">https://doi.org/10.22331/q-2026-05-22-2114</a>
  chicago: Kolisnyk, Dmytro, Raimel A Medina Ramos, Romain Vasseur, and Maksym Serbyn.
    “Tensor Cross Interpolation of Purities in Quantum Many-Body Systems.” <i>Quantum</i>.
    Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften,
    2026. <a href="https://doi.org/10.22331/q-2026-05-22-2114">https://doi.org/10.22331/q-2026-05-22-2114</a>.
  ieee: D. Kolisnyk, R. A. Medina Ramos, R. Vasseur, and M. Serbyn, “Tensor cross
    interpolation of purities in quantum many-body systems,” <i>Quantum</i>, vol.
    10. Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften,
    2026.
  ista: Kolisnyk D, Medina Ramos RA, Vasseur R, Serbyn M. 2026. Tensor cross interpolation
    of purities in quantum many-body systems. Quantum. 10, 2114.
  mla: Kolisnyk, Dmytro, et al. “Tensor Cross Interpolation of Purities in Quantum
    Many-Body Systems.” <i>Quantum</i>, vol. 10, 2114, Verein zur Förderung des Open
    Access Publizierens in den Quantenwissenschaften, 2026, doi:<a href="https://doi.org/10.22331/q-2026-05-22-2114">10.22331/q-2026-05-22-2114</a>.
  short: D. Kolisnyk, R.A. Medina Ramos, R. Vasseur, M. Serbyn, Quantum 10 (2026).
corr_author: '1'
date_created: 2026-05-26T19:39:12Z
date_published: 2026-05-22T00:00:00Z
date_updated: 2026-06-02T09:15:13Z
day: '22'
ddc:
- '530'
department:
- _id: MaSe
- _id: GradSch
doi: 10.22331/q-2026-05-22-2114
ec_funded: 1
external_id:
  arxiv:
  - '2503.17230'
file:
- access_level: open_access
  checksum: f8ce78607ad06120cdf894dc8cef55da
  content_type: application/pdf
  creator: dernst
  date_created: 2026-06-02T09:12:11Z
  date_updated: 2026-06-02T09:12:11Z
  file_id: '21939'
  file_name: 2026_Quantum_Kolisnyk.pdf
  file_size: 3284798
  relation: main_file
  success: 1
file_date_updated: 2026-06-02T09:12:11Z
fulldoi: https://doi.org/10.22331/q-2026-05-22-2114
has_accepted_license: '1'
intvolume: '        10'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '05'
oa: 1
oa_version: Published Version
project:
- _id: 23841C26-32DE-11EA-91FC-C7463DDC885E
  call_identifier: H2020
  grant_number: '850899'
  name: 'Non-Ergodic Quantum Matter: Universality, Dynamics and Control'
publication: Quantum
publication_identifier:
  eissn:
  - 2521-327X
publication_status: published
publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
quality_controlled: '1'
status: public
title: Tensor cross interpolation of purities in quantum many-body systems
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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 10
year: '2026'
...
---
_id: '9760'
abstract:
- lang: eng
  text: "The quantum approximate optimization algorithm (QAOA) is a prospective near-term
    quantum algorithm due to its modest circuit depth and promising benchmarks. However,
    an external parameter optimization required in the QAOA could become a performance
    bottleneck. This motivates studies of the optimization landscape and search for
    heuristic ways of parameter initialization. In this work we visualize the optimization
    landscape of the QAOA applied to the MaxCut problem on random graphs, demonstrating
    that random initialization of the QAOA is prone to converging to local minima
    with suboptimal performance. We introduce the initialization of QAOA parameters
    based on the Trotterized quantum annealing (TQA) protocol, parameterized by the
    Trotter time step. We find that the TQA initialization allows to circumvent\r\nthe
    issue of false minima for a broad range of time steps, yielding the same performance
    as the best result out of an exponentially scaling number of random initializations.
    Moreover, we demonstrate that the optimal value of the time step coincides with
    the point of proliferation of Trotter errors in quantum annealing. Our results
    suggest practical ways of initializing QAOA protocols on near-term quantum devices
    and reveal new connections between QAOA and quantum annealing."
acknowledgement: We would like to thank D. Abanin and R. Medina for fruitful discussions
  and A. Smith and I. Kim for valuable feedback on the manuscript. We acknowledge
  support by the European Research Council (ERC) under the European Union’s Horizon
  2020 research and innovation program (Grant Agreement No. 850899).
article_number: '491'
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Stefan
  full_name: Sack, Stefan
  id: dd622248-f6e0-11ea-865d-ce382a1c81a5
  last_name: Sack
  orcid: 0000-0001-5400-8508
- first_name: Maksym
  full_name: Serbyn, Maksym
  id: 47809E7E-F248-11E8-B48F-1D18A9856A87
  last_name: Serbyn
  orcid: 0000-0002-2399-5827
citation:
  ama: Sack S, Serbyn M. Quantum annealing initialization of the quantum approximate
    optimization algorithm. <i>Quantum</i>. 2021;5. doi:<a href="https://doi.org/10.22331/Q-2021-07-01-491">10.22331/Q-2021-07-01-491</a>
  apa: Sack, S., &#38; Serbyn, M. (2021). Quantum annealing initialization of the
    quantum approximate optimization algorithm. <i>Quantum</i>. Verein zur Förderung
    des Open Access Publizierens in den Quantenwissenschaften. <a href="https://doi.org/10.22331/Q-2021-07-01-491">https://doi.org/10.22331/Q-2021-07-01-491</a>
  chicago: Sack, Stefan, and Maksym Serbyn. “Quantum Annealing Initialization of the
    Quantum Approximate Optimization Algorithm.” <i>Quantum</i>. Verein zur Förderung
    des Open Access Publizierens in den Quantenwissenschaften, 2021. <a href="https://doi.org/10.22331/Q-2021-07-01-491">https://doi.org/10.22331/Q-2021-07-01-491</a>.
  ieee: S. Sack and M. Serbyn, “Quantum annealing initialization of the quantum approximate
    optimization algorithm,” <i>Quantum</i>, vol. 5. Verein zur Förderung des Open
    Access Publizierens in den Quantenwissenschaften, 2021.
  ista: Sack S, Serbyn M. 2021. Quantum annealing initialization of the quantum approximate
    optimization algorithm. Quantum. 5, 491.
  mla: Sack, Stefan, and Maksym Serbyn. “Quantum Annealing Initialization of the Quantum
    Approximate Optimization Algorithm.” <i>Quantum</i>, vol. 5, 491, Verein zur Förderung
    des Open Access Publizierens in den Quantenwissenschaften, 2021, doi:<a href="https://doi.org/10.22331/Q-2021-07-01-491">10.22331/Q-2021-07-01-491</a>.
  short: S. Sack, M. Serbyn, Quantum 5 (2021).
corr_author: '1'
date_created: 2021-08-01T22:01:21Z
date_published: 2021-07-01T00:00:00Z
date_updated: 2026-10-19T22:31:11Z
day: '01'
ddc:
- '530'
department:
- _id: GradSch
- _id: MaSe
doi: 10.22331/Q-2021-07-01-491
ec_funded: 1
external_id:
  arxiv:
  - '2101.05742'
  isi:
  - '000669830600001'
file:
- access_level: open_access
  checksum: 9706c2bb8e748e9b5b138381995a7f6f
  content_type: application/pdf
  creator: cchlebak
  date_created: 2021-08-06T06:44:31Z
  date_updated: 2021-08-06T06:44:31Z
  file_id: '9774'
  file_name: 2021_Quantum_Sack.pdf
  file_size: 2312482
  relation: main_file
file_date_updated: 2021-08-06T06:44:31Z
fulldoi: https://doi.org/10.22331/Q-2021-07-01-491
has_accepted_license: '1'
intvolume: '         5'
isi: 1
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
project:
- _id: 23841C26-32DE-11EA-91FC-C7463DDC885E
  call_identifier: H2020
  grant_number: '850899'
  name: 'Non-Ergodic Quantum Matter: Universality, Dynamics and Control'
publication: Quantum
publication_identifier:
  eissn:
  - 2521-327X
publication_status: published
publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
quality_controlled: '1'
related_material:
  record:
  - id: '14622'
    relation: dissertation_contains
    status: public
scopus_import: '1'
status: public
title: Quantum annealing initialization 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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 5
year: '2021'
...