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Cited by 36Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyOrbital Angular Momentum in Optics

Quantum walks of two correlated photons in a 2D synthetic lattice

Open access
Esposito, Chiara, Barros, Mariana R., Durán Hernández, Andrés, Carvacho, Gonzalo, Di Colandrea, Francesco, Barboza, Raouf, Cardano, Filippo, Spagnolo, Nicolò +2 more·2022·npj Quantum Information
DOI 10.1038/s41534-022-00544-0
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Abstract

Quantum walks represent paradigmatic quantum evolutions, enabling powerful applications in the context of topological physics and quantum computation. They have been implemented in diverse photonic architectures, but the realization of two-particle dynamics on a multidimensional lattice has hitherto been limited to continuous-time evolutions. To fully exploit the computational capabilities of quantum interference it is crucial to develop platforms handling multiple photons that propagate across multidimensional lattices. Here, we report a discrete-time quantum walk of two correlated photons in a two-dimensional lattice, synthetically engineered by manipulating a set of optical modes carrying quantized amounts of transverse momentum. Mode-couplings are introduced via the polarization-controlled diffractive action of thin geometric-phase optical elements. The entire platform is compact, efficient, scalable, and represents a versatile tool to simulate quantum evolutions on complex lattices. We expect that it will have a strong impact on diverse fields such as quantum state engineering, topological quantum photonics, and Boson Sampling.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper reports a discrete-time quantum walk of two correlated photons in a two-dimensional synthetic lattice encoded in transverse optical momentum, implemented with g-plates and tunable waveplates. It demonstrates single- and two-photon walks up to three steps with high similarity to theory and nonclassical correlations.

Key findings

  • A balanced 2D discrete-time quantum walk with one-step operator \(U = T_y(\pi)C(\pi/4)T_x(\pi)C(\pi/4)\) was implemented with coherent light, one-photon, and two-photon inputs, spanning 2, 6, and 12 modes for the first, second, and third step, respectively.
  • For a single photon injected at (1, 0) with polarization \(D\), the third-step similarity was \(S^{(3)}_{1p}=0.9773 \pm 0.0002\); classical-light similarities were ~0.999, ~0.996, and ~0.988 for 1–3 steps.
  • For two photons injected at (-1, 0) and (1, 0) with polarizations \(A\) and \(D\), the Hong–Ou–Mandel visibility was \(v=0.95 \pm 0.02\), and the third-step similarity was \(S^{(3)}_{2p}=0.914 \pm 0.002\); two-photon similarities were 0.9756(6), 0.9743(7), and 0.914(2) for 1–3 steps.
  • A nonclassicality witness \(V(m_1,m_2)<0\) was violated by maximum values of 96, 14, and 11 standard deviations for 1, 2, and 3 quantum-walk steps, respectively.

Methods

  • Design: discrete-time quantum walk on a 2D square lattice; walker positions are optical modes \(|m,n\rangle\) carrying transverse momentum \((m\Delta k_\perp, n\Delta k_\perp)\), the coin is encoded in circular polarization, g-plates implement spin-dependent shifts \(T_x(\delta)\) and \(T_y(\delta)\), and tunable liquid-crystal waveplates implement coin rotations \(C(\omega)\). Steps were measured by switching individual g-plates between \(\delta=\pi\) and \(\delta=0\).
  • Data/sample: photon pairs were generated by parametric down-conversion in a BBO crystal pumped by a pulsed laser at \(\lambda=392.5\) nm, producing photons at \(\lambda=785\) nm; single-mode fibers, polarization controllers, and delay lines were used, with Hong–Ou–Mandel interference for temporal synchronization. Two-photon inputs were injected at (-1, 0) and (1, 0); classical laser light was also used.
  • Measurement/analysis: output modes were mapped by a lens system and a micro-lens array (100 micro-lenses, ~5 mm focal length, 250 μm separation, 15 μm waist, coupling efficiency improved from 0.1 to 0.75) into an 8×8 multimode fiber array with 250 μm pitch connected to single-photon avalanche photodiodes and coincidence electronics; a CCD camera was used for classical light. Agreement was quantified by similarity metrics, and nonclassical correlations were tested with the inequality \(V(m_1,m_2)<0\).

Limitations

  • Stated: the number of steps currently realizable is essentially limited by optical losses, mainly photon reflections at each plate; losses are ~15% and can be reduced to at least ~5% by adding anti-reflection coatings.
  • Stated: slight differences between theoretical and experimental distributions are due to experimental imperfections such as inaccuracies of g-plate tuning values and their relative horizontal alignment.
  • Evident from the text: the experimental demonstration is limited to three steps and two photons, with no larger multiphoton or higher-step instances shown; the theoretical two-photon distribution also uses the measured HOM visibility \(c_0=0.95\) to account for partial distinguishability.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1038/s41534-022-00544-0

Cites

60 of 60 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

14 of 17 cited sources checked · 7 supported · 0 flagged · accuracy not yet measured

  • Premisecitation

    The implemented split-step quantum walk is proved to realize the Grover search algorithm in the high step-number limit.

    “Here we implement a different split-step quantum walk, that is proved to realize the Grover search algorithm in the high step-number limit 53 .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [53] Di Franco, C., Mc Gettrick, M. & Busch, T. Mimicking the probability distribution of a two-dimensional grover walk with a single-qubit coin. Phys. Rev. Lett. 106, 080502 (2011). source

      The paper’s proof shows that its alternate-direction walk reproduces the Grover walk’s spatial probability distribution for specified initial conditions. It also says the Grover walk can be used to implement two-dimensional Grover search, but it does not prove that the alternate walk realizes the search algorithm in a high-step-number limit. (LLM, conf=0.87)

      Experimental — accuracy not yet measured
  • Findingcitation

    The presence of nonclassical correlations in the output distributions is witnessed by applying the non-classicality test.

    “The presence of nonclassical correlations in the output distributions is witnessed by applying the non-classicality test of refs.48,51 , given by:”
    Quote in the paperSubstantial claim2 of 2 sources checked
    • cites [48] Peruzzo, A. et al. Quantum walks of correlated photons. Science 329, 1500–1503 (2010). source
    • cites [51] Bromberg, Y., Lahini, Y., Morandotti, R. & Silberberg, Y. Quantum and classical correlations in waveguide lattices. Phys. Rev. Lett. 102, 253904 (2009). source
    • Rests on an assumption: The two-photon state at the input of the quantum walk is separable.
  • Findingown data

    For the experimental distribution at the third step, the similarity value S(3)2p = 0.914 ± 0.002.

    “For the experimental distribution at the third step, we obtain a similarity value S ð3Þ 2p ¼ 0:914 ± 0:002”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The two-photon state at the input of the quantum walk is separable.
  • Findingreasoning

    The number of steps that can be currently realized is essentially limited by the optical losses.

    “The number of steps that can be currently realized is essentially limited by the optical losses, which are mainly due to photon reflections at each plate.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The optical losses are mainly due to photon reflections at each plate.
  • Findingown data

    The single-photon experimental distributions show high agreement with theoretical predictions, with similarity S(3)1p = 0.9773 ± 0.0002 at the last step.

    “The similarity value of the last step S ð3Þ 1p ¼ 0:9773 ± 0:0002 shows a high agreement with the expected distribution.”
    Quote in the paperNot a substantial claim
  • Classified as probably not a substantial claim. 14
  • May combine two claims. 8

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Two-dimensional quantum walks display a richer landscape of topological features when compared to the 1D case.

    “Furthermore, twodimensional (2D) QWs display a richer landscape of topological features when compared to the 1D case 11,32 .”
    Quote in the paperNot a substantial claim2 of 2 sources checked
    • cites [11] Kitagawa, T. Topological phenomena in quantum walks: elementary introduction to the physics of topological phases. Quantum Inf. Process 11, 1107–1148 (2012). source
    • cites [32] Asboth, J. K. & Edge, J. M. Edge-state-enhanced transport in a two-dimensional quantum walk. Phys. Rev. A 91, 022324 (2015). source

      The Introduction (p. 1) describes additional topology in both 1D and 2D quantum walks, and Section C (p. 7) identifies two topological invariants for a 2D walk. The paper supports the existence of varied topological features in 2D, but does not establish that 2D walks have a richer topological landscape than 1D walks. (LLM, conf=0.82)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The platform builds on a recent approach to the simulation of single-particle 2D quantum walks using coherent laser light.

    “Our platform builds on a recent approach to the simulation of single-particle 2D QWs using coherent laser light 22 .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [22] D’Errico, A. et al. Two-dimensional topological quantum walks in the momentum space of structured light. Optica 7, 108 (2020). source

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Recent works reported 2D continuous-time quantum walks of correlated photons, relying on arrays of coupled waveguides, and important results have also been achieved by means of superconductive quantum processors.

    “Recent works reported 2D continuous-time quantum walks of correlated photons, relying on arrays of coupled waveguides 21,38,52 , and important results have been also achieved by means of superconductive quantum processors 6 .”
    Quote in the paperNot a substantial claim3 of 4 sources checked
    • cites [21] Ehrhardt, M. et al. Exploring complex graphs using three-dimensional quantum walks of correlated photons. Sci. Adv. 7, eabc5266 (2021). source
    • cites [38] Jiao, Z.-Q. et al. Two-dimensional quantum walks of correlated photons. Optica 8, 1129–1135 (2021). source
    • cites [52] Qiang, X. et al. Implementing graph-theoretic quantum algorithms on a silicon photonic quantum walk processor. Sci. Adv. 7, eabb8375 (2021). source

      The Introduction reports quantum walks in arrays of evanescently coupled waveguides, and the References list a study of correlated photon pairs in two-dimensional waveguide arrays. But this paper’s own Results describe a reconfigurable silicon photonic circuit, not a 2D coupled-waveguide array, so it supports the background claim rather than serving as an example of that implementation. The superconducting-processor clause is attached to a different citation. (LLM, conf=0.85)

      Experimental — accuracy not yet measured
    • cites [6] Gong, M. et al. Quantum walks on a programmable two-dimensional 62-qubit superconducting processor. Science 372, 948–952 (2021). source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    By controlling the parameters, diverse single-particle quantum walks mimicking periodically driven Chern insulators have been reported.

    “By controlling these parameters, diverse single-particle QWs mimicking periodically driven Chern insulators have been reported 22 .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [22] D’Errico, A. et al. Two-dimensional topological quantum walks in the momentum space of structured light. Optica 7, 108 (2020). source

      Section B (p. 3) reports an implemented quantum-walk protocol that realizes a periodically driven Chern insulator and has different topological phases as δ changes. The Conclusion (p. 7) says parameter control allows several walk architectures, but identifies the Chern-insulator simulation as the protocol presented here; it does not report diverse Chern-insulator-mimicking walks. (LLM, conf=0.83)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    A large variety of quantum walks can be implemented with this platform by tuning the parameters δ and ω.

    “A large variety of QWs can be implemented with this platform, by tuning the parameters δ and ω, as already shown in ref. 22 .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [22] D’Errico, A. et al. Two-dimensional topological quantum walks in the momentum space of structured light. Optica 7, 108 (2020). source

      The Conclusion (p. 7) says the platform can study several quantum-walk architectures by dynamically controlling parameters including optical retardation δ. It does not identify ω as a tunable parameter or say that varying δ and ω together implements those walks. (LLM, conf=0.85)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Quantum search algorithms based on quantum walks overcome their classical counterparts solely when the spatial dimension of the lattice is equal or greater than two.

    “As an example, quantum search algorithms based on QWs overcome their classical counterparts solely when the spatial dimension of the lattice is equal or greater than two 31 .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [31] Tulsi, A. Faster quantum-walk algorithm for the two-dimensional spatial search. Phys. Rev. A 78, 012310 (2008). source

      The Introduction (p. 1) and Discussion (p. 5) show that the paper’s quantum-walk algorithm speeds up two-dimensional spatial search; the Introduction also says it applies in higher dimensions. But our copy does not establish the claim’s exclusive threshold: it does not show that quantum-walk search fails to outperform classical search in one dimension. (LLM, conf=0.84)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

    Interpretationreasoning

    The approach can represent a promising platform for the implementation of Boson Sampling and Gaussian Boson Sampling experiments in large optical lattices.

    “Furthermore, given the possibility to exploit multiphoton inputs and to control the performed transformation, this approach can also represent a promising platform for the implementation of Boson Sampling and Gaussian Boson Sampling experiments in large optical lattices 30 .”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [30] Brod, D. J. et al. Photonic implementation of boson sampling: a review. Adv. Photonics 1, 034001 (2019). source

      Section 4 (Experimental Boson Sampling; page unavailable) reports both boson sampling and Gaussian boson sampling in a chip with an array of 12 coupled waveguides. It also discusses control of optical transformations in other interferometer designs. The review does not establish that the particular approach in the citing sentence can implement both experiments in large optical lattices. (LLM, conf=0.84)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Premisecitation

    The presented platform is compact, flexible, and enables the implementation of a large variety of different topological quantum walks.

    “The presented platform is compact, flexible, and enables the implementation of a large variety of different topological quantum walks 22 .”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [22] D’Errico, A. et al. Two-dimensional topological quantum walks in the momentum space of structured light. Optica 7, 108 (2020). source

  • May combine two claims.

    Findingnone given

    The system is based on a synthetic 2D lattice made of internal modes of a single optical beam, as opposed to real-space neighboring lattice sites, and implements discrete-time evolutions that can be actively controlled and easily reconfigured.

    “We stress that our system is based on a very different approach, exploiting a synthetic 2D lattice made of internal modes of a single optical beam, as opposed to real-space neighboring lattice sites and implementing discrete-time evolutions that can be actively controlled and easily reconfigured.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingnone given

    The platform was employed to implement a balanced quantum walk U with single-and two-photon inputs, up to three time-steps.

    “We have employed our platform to implement a balanced quantum walk U with single-and two-photon inputs, up to three time-steps.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    The adoption of the micro-lens array enables an improvement in the coupling efficiency from 0.1 to 0.75.

    “The adoption of the micro-lens array enables an improvement in the coupling efficiency from 0.1 to 0.75.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The platform realizes a three-step quantum walk dynamics with coherent light, one-photon, and two-photon inputs, spanning 2, 6, and 12 modes for the first, second, and third step, respectively.

    “Specifically, we realize a three-step quantum walk dynamics with coherent light, one-photon, and two-photon inputs, spanning 2, 6, and 12 modes for the first, second, and third step, respectively.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The paper has presented and realized a platform for the implementation of two-dimensional, multiphoton discrete-time quantum walks, demonstrating experimentally single-and two-photon operations on a 2D squared lattice.

    “We have presented and realized a platform for the implementation of two-dimensional, multiphoton discrete-time quantum walks, demonstrating experimentally single-and two-photon operations on a 2D squared lattice.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The maximum violation of the non-classicality inequality is 96, 14, and 11 standard deviations for 1, 2, and 3 quantum walk steps, respectively.

    “The obtained maximum violation of the inequality are of 96, 14, and 11 standard deviations, respectively for 1, 2, and 3 QW steps”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    Similar results were observed when injecting classical light to the quantum walk platform.

    “Similar results were observed when injecting classical light to the quantum walk platform (see Table 1).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    Optical losses can be significantly reduced from ~15% to at least ~5% by adding a standard anti-reflection coating on the plate outer surfaces.

    “However, these can be significantly reduced (from ~15% to at least ~5%) by adding a standard anti-reflection coating on the plate outer surfaces.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The optical losses are mainly due to photon reflections at each plate.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    Multiparticle regimes have been implemented in continuous-time quantum walks, even in 2D lattices.

    “Multiparticle regimes have been already implemented in continuous-time QWs [47][48][49] , even in 2D lattices 38 .”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [47][48][49]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [38] Jiao, Z.-Q. et al. Two-dimensional quantum walks of correlated photons. Optica 8, 1129–1135 (2021). source

  • May combine two claims.

    Interpretationown data

    The indistinguishability between injected photons gives rise to quantum interferences, yielding probability distributions that cannot be reproduced by classical states of light.

    “The indistinguishability between injected photons gives rise to quantum interferences, yielding in turn probability distributions that cannot be reproduced by classical states of light.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The two-photon state at the input of the quantum walk is separable.

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing). May combine two claims.

    Interpretationown data

    The proposed platform has the potential to be employed for significantly larger instances, with a high degree of control on the implemented protocol.

    “These results highlight that the proposed platform has the potential to be employed for significantly larger instances, with a high degree of control on the implemented protocol.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationown data

    The slight differences between theoretical and experimental output distributions are due to experimental imperfections such as inaccuracies of g-plates tuning values and of their relative horizontal alignment.

    “The slight differences between theoretical and experimental output distributions are due to experimental imperfections such as inaccuracies of g-plates tuning values and of their relative horizontal alignment.”
    Quote in the paperNot a substantial claim
20 of 20

  • The paper relies on this without support.

    The two-photon state at the input of the quantum walk is separable.

    “It is worth noticing that at the input of the quantum walk the two-photon state is separable.”

  • The paper relies on this without support.

    The optical losses are mainly due to photon reflections at each plate.

    “The number of steps that can be currently realized is essentially limited by the optical losses, which are mainly due to photon reflections at each plate.”
2 of 2

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Related

Authors
Esposito, C., Barros, M. R., Durán Hernández, A., Carvacho, G., Di Colandrea, F., Barboza, R., Cardano, F., Spagnolo, N., Marrucci, L., & Sciarrino, F.
Affiliations
  • Sapienza University of Rome (IT)
  • University of Naples Federico II (IT)
  • Institute for Complex Systems (IT)
Corresponding author
  • Cardano, Filippo
  • Sciarrino, Fabio
ORCID
Venue
npj Quantum Information
Publisher
Springer Science and Business Media LLC
Published
2022-03-24
Volume
8
Issue
1
ISSN
2056-6387 (ISSN-L)
DOI
10.1038/s41534-022-00544-0
OpenAlex
W4220861426
Cited by
36
References
62
Citations by year
2026: 7 · 2025: 7 · 2024: 13 · 2023: 7 · 2022: 2
FWCI
2.37
Type
journal article
Language
en
Open access
Licence
Locations
  • npj Quantum Information (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • Università Politecnica delle Marche (Università Politecnica delle Marche) (submittedVersion, cc-by)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
  • IRIS Research product catalog (Sapienza University of Rome) (submittedVersion, cc-by)
  • arXiv (Cornell University)
Funders
European Commission (PRIN 2017, HORIZON2020, 2017SRNBRK)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Orbital Angular Momentum in Optics
Keywords
discrete-time quantum walks, two-particle quantum walk, correlated photons, two-dimensional photonic lattices, synthetic lattices, Berry phase, mode coupling, quantum state engineering, topological photonics, boson sampling, photonic quantum simulation

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