Two-Particle Bosonic-Fermionic Quantum Walk via Integrated Photonics
Open accessAbstract
Quantum walk represents one of the most promising resources for the simulation of physical quantum systems, and has also emerged as an alternative to the standard circuit model for quantum computing. Here we investigate how the particle statistics, either bosonic or fermionic, influences a two-particle discrete quantum walk. Such an experiment has been realized by exploiting polarization entanglement to simulate the bunching-antibunching feature of noninteracting bosons and fermions. To this scope a novel three-dimensional geometry for the waveguide circuit is introduced, which allows accurate polarization independent behavior, maintaining remarkable control on both phase and balancement.
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Written from the paper’s full text.
TL;DR
The paper experimentally implements a four-step discrete quantum walk in an integrated femtosecond-laser-written waveguide circuit and uses polarization-entangled photon pairs to simulate bosonic, fermionic, and anyonic two-particle statistics. Measured output distributions agree with theory with similarities above 0.97.
Key findings
- The device is polarization insensitive: single-photon distributions for horizontal, vertical, diagonal, and antidiagonal inputs match theory, with mean similarity \(S_{1ph}=0.992\pm0.002\); supplementary table values range from \(0.983\pm0.002\) to \(0.997\pm0.002\).
- Two-photon entangled states reproduce bosonic and fermionic walks: triplet \(|\Psi^+\rangle\) (\(\phi=0\)) gives \(S_{bos}=0.982\pm0.002\), and singlet \(|\Psi^-\rangle\) (\(\phi=\pi\)) gives \(S_{fer}=0.973\pm0.002\).
- Anyonic states \(|\Psi_\phi\rangle\) with \(\phi=\pi/4,\pi/2,3\pi/4\) show mixed bunching/antibunching, with similarities \(S^{\pi/4}_{any}=0.987\pm0.002\), \(S^{\pi/2}_{any}=0.988\pm0.001\), and \(S^{3\pi/4}_{any}=0.980\pm0.002\).
- The fermionic walk-position distribution has nonzero diagonal elements even though the probability of two fermions emerging from the same output spatial mode vanishes, due to the mapping between output ports and lattice positions.
Methods
- A four-step discrete quantum walk was implemented as a network of directional couplers in borosilicate glass using femtosecond laser writing; a three-dimensional tilted geometry (waveguides at \(62^\circ\), \(11\,\mu\text{m}\) spacing, interaction length \(2.1\,\text{mm}\)) gave polarization-insensitive balanced splitters, with a \(32\,\text{mm}\)-long chip.
- Polarization-entangled photon pairs at \(\lambda=806\,\text{nm}\) were generated by type-II non-collinear spontaneous parametric down conversion in a \(1.5\,\text{mm}\) BBO crystal pumped by a CW diode laser at \(P=50\,\text{mW}\); waveplates and a liquid-crystal phase shifter prepared Bell states, and interference filters set \(\Delta\lambda=6\,\text{nm}\).
- Output photons were collected by multimode fibers and detected via single-photon counting modules for single counts and two-photon coincidences; measured distributions were compared with theory using a similarity \(S\), a generalized fidelity, while changing the Bell-state phase \(\phi\) selected bosonic, fermionic, or anyonic behavior.
Limitations
- The paper does not state a dedicated limitations section. Evident from the text: the demonstration is limited to a four-step discrete quantum walk and a small integrated network, with no demonstration of scaling to many steps.
- Evident from the text: bosonic and fermionic statistics are simulated via polarization entanglement of photons rather than actual non-interacting bosonic or fermionic particles, and anyonic behavior is measured only for selected phases \(\phi=\pi/4,\pi/2,3\pi/4\).
- Stated in the paper: residual birefringence and slightly different guided modes for the two polarizations cause residual polarization dependence in directional couplers; the work mitigates this with the tilted three-dimensional geometry but does not claim perfect elimination.
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Claims the paper makes
4 of 8 cited sources checked · 0 supported · 0 flagged · accuracy not yet measured
- Findingcitation§ Results
By considering a generic phase φ different from 0 and π, it is possible to simulate the behaviour of the quantum walk of two anyons.
“Furthermore, by considering a generic phase φ (dif- ferent from 0 and π), it is possible to simulate the be- haviour of the quantum walk of two anyons, particles with a non semi-integer spin that represent a generaliza- tion of fermions and bosons [30].”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites [30] source
The Introduction (p. 1) describes a continuous-time quantum walk of a particle on a one-dimensional lattice, and the Discussion (p. 7) addresses disorder and decoherence. Searches for anyons and a generic phase φ yielded no passage about simulating a two-anyon walk, so our copy does not verify the attributed claim. (LLM, conf=0.74)
Experimental — accuracy not yet measured
- Findingcitation§ Introduction
By changing the symmetry of the entangled state, one can simulate the quantum walk of two particles with bosonic or fermionic statistics (integer or semi-integer spin).
“By changing the symmetry of entanglement we can simulate the quan- tum dynamics of the walks of two particles with bosonic or fermionic statistic.”
- Also stated in: § Introduction · plain
“By changing the symmetry of the entangled state we can simulate the quantum walk of two particles with integer or semi- integer spin [29].”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [29] source
The discussion of the quantum walk on Gₙ (pp. 1–2) describes a particle moving on a graph, while the discussion of symmetry (p. 2) concerns imperfections in the graph. Searches for entanglement symmetry and two-particle bosonic or fermionic walks found no passage addressing the attributed claim. (LLM, conf=0.76)
Experimental — accuracy not yet measured
- Also stated in: § Introduction · plain
- Premisecitation§ Results
The expected probability to measure two fermions over the same output spatial mode is vanishing.
“In fact, the expected probability to measure two fermions over the same output spatial mode is vanishing [29].”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [29] source
The discussion on pp. 1–3 concerns the probability of finding a single particle at a graph node and how localization affects its propagation. The passage on p. 4 mentions small numbers of excitations but does not address two fermions occupying the same output spatial mode. Searches for that phrase and for fermions and the Pauli exclusion principle found no supporting passage. (LLM, conf=0.76)
Experimental — accuracy not yet measured
- Findingcitation§ Supplementary Information
Birefringence of these waveguides is on the order of B = 7×10−5.
“Birefringence of these waveguides is on the order of B = 7× 10−5, as characterized in [1].”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites [1] source
The cited paper discusses theoretical quantum-walk wires and scattering widgets (III. UNIVERSAL GATE SET, p. 3; IV. MOMENTUM FILTERING, p. 5). Searches for waveguide birefringence, polarization characterization, and B = 7×10−5 found no passage reporting that measurement. (LLM, conf=0.76)
Experimental — accuracy not yet measured
- Findingreasoning§ Results
Some diagonal elements of the fermionic two-particle walk are nonzero both in the theoretical and experimental distribution.
“Note that some of the diagonal ele- ments of the fermionic two-particle walk are nonzero both in the theoretical and experimental distribution.”
Quote in the paperNot a substantial claim
- Findingown data§ Results
There exists an angle for which the ratio between the two coupling coefficients is one, i.e. the coupler becomes polarization insensitive.
“In particular there exists an angle for which the ratio between the two coefficients is one, i.e. the coupler becomes polarization insensitive.”
Quote in the paperNot a substantial claim - Findingown data§ Results
The particle symmetry affects the quantum walks.
“In Fig. 4 we observe how the particle symmetry affects the quantum walks.”
Quote in the paperSubstantial claim - Findingnone given§ Abstract
The experiment was realized by exploiting polarization entanglement to simulate the bunching-antibunching feature of non-interacting bosons and fermions.
“Such experiment has been realized by exploiting polarization entanglement to simulate the bunching-antibunching feature of non interacting bosons and fermions.”
Quote in the paperSubstantial claim - Findingown data§ Results
The mean similarity between experimental single-photon distributions and expected distributions over tested input polarization states is S1ph = 0.992 ± 0.002.
“The mean value over the tested input polarization states is S1ph = 0.992± 0.002.”
Quote in the paperSubstantial claim - Findingown data§ Results
The ratio between the measured coupling coefficients for the two polarizations (CV and CH) depends on the angle θ between the waveguides.
“It can be observed that the ratio between the measured coupling coefficients for the two polarizations (CV and CH) depends on θ, as shown in Fig. 2b.”
Quote in the paperNot a substantial claim - Findingown data§ Results
The experimental data for anyonic quantum walks with φ = π/4, π/2, 3π/4 agree with theoretical distributions with similarities Sany = 0.987 ± 0.002, 0.988 ± 0.001, and 0.980 ± 0.002 respectively.
“S π/4 any = 0.987 ± 0.002, S π/2 any = 0.988± 0.001 and S 3π/4 any = 0.980± 0.002 for the anyonic quantum walks with φ = π 4 , π 2 , 3 4 π, respectively.”
Quote in the paperSubstantial claim - Findingown data§ Results
The experimental data for bosonic and fermionic quantum walks agree with theoretical distributions with similarities Sbos = 0.982 ± 0.002 and Sfer = 0.973 ± 0.002.
“The experimental data can be compared with the the- oretical distributions by the similarity obtaining Sbos = 0.982 ± 0.002 and Sfer = 0.973 ± 0.002 for the bosonic and fermionic quantum walk”
Quote in the paperSubstantial claim - Findingown data§ Results
The obtained results are in good agreement with the expected behaviours.
“The obtained results are in good agreement with the ex- pected behaviours.”
Quote in the paperSubstantial claim - Findingreasoning§ Discussion
Exploiting the different statistics of singlet and triplet entangled states, the scheme allowed simulation of how symmetric and antisymmetric particles travel through the quantum walk.
“Exploiting the different statistics of singlet and triplet en- tangled states, such scheme allowed us to simulate how symmetric and antisymmetric particles travels through the quantum walk.”
Quote in the paperSubstantial claim - Premisecitation§ Introduction
Femtosecond laser technology can produce high-quality waveguides able to support polarization-entangled photon states.
“In a previous work we have demonstrated that fem- tosecond laser technology can produce high-quality waveguides able to support polarization entangled pho- ton states [25, 26].”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning§ Introduction
When several directional couplers are cascaded in a DC array for a quantum walk, small differences in splitting ratios accumulate and affect the indistinguishability of the two polarizations.
“When several devices are cascaded, as in the case of a DC array for implement- ing a quantum walk, small differences in the splitting ratios would accumulate and in the end affect the indis- tinguishability of the two polarizations.”
Quote in the paperSubstantial claim - Premiseown data§ Results
Anyonic systems exhibit both bunching and anti-bunching behaviours (diagonal and off-diagonal elements in the final distribution).
“These systems exhibit both bunching and anti-bunching be- haviours (i.e. diagonal and off-diagonal elements in the final distribution).”
Quote in the paperSubstantial claim - Premisereasoning§ Supplementary Information
In an array with an even (odd) number of steps, the output ports J are grouped into the even (odd) final positions j of the walker.
“Then it is clear that in an array with an even (odd) num- ber of steps the output ports J are grouped into the even (odd) final positions j of the walker.”
Quote in the paperSubstantial claim - Interpretationreasoning§ Supplementary Information
The antisymmetry for fermions in the same lattice site k is provided by the different internal coin state, with one particle in |U〉 and the other in |D〉, giving a total antisymmetric wavefunction.
“the antisymmetry for fermions in the same lattice site k is provided by the different internal coin state, one particle is in the |U〉 state and the other one in the |D〉 state with a total wavefunction of the form:”
Quote in the paperSubstantial claim - Interpretationreasoning§ Supplementary Information
The diagonal elements of the two-particle fermionic distribution are non-vanishing because of the relation between the output ports of the BS array and the positions of the walker.
“These relations between the output probabilities of the BS array and the sites of the QW cause the diagonal elements of the two-particle fermionic distribution to be non-vanishing, as clearly shown in Fig. 6(b).”
Quote in the paperSubstantial claim - Interpretationnone given§ Supplementary Information
The dynamics of a two-particle quantum walk cannot be reconstructed by multiplying the output probabilities of two single-particle walks.
“Note that the dynam- ics of two-particle quantum walk cannot be reconstructed by multiplying the output probabilities (|UJK |2) of two single-particle walks.”
Quote in the paperSubstantial claim
- May combine two claims. 8
- Classified as probably not a substantial claim. 7
May combine two claims.
Findingown data§ IntroductionThe guided modes for the two polarizations are slightly different, resulting in residual polarization dependence in the properties of the fabricated directional couplers.
“Anyway the guided modes for the two polarizations are still slightly different (as shown in Fig. 2a) and this results in a resid- ual polarization dependence in the properties of the fab- ricated DCs.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Findingnone given§ IntroductionThe fermionic and bosonic behaviours drastically influence the dynamics of their quantum walk on the line.
“The fermionic and bosonic behaviors drastically influence the dynamics of their quantum walk on the line.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Findingown data§ ResultsThe measured bosonic and fermionic two-particle distributions compared with expected ones are reported in Fig. 4a-b for |Φ+〉 and |Ψ−〉.
“The measured bosonic and fermionic distributions compared with expected ones are reported in Fig. 4a-b for |Φ+〉 and |Ψ−〉.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown data§ Supplementary InformationThe single-photon distributions for horizontal, vertical, diagonal and antidiagonal polarized light injected in modes kA and kB are polarization independent, as confirmed by similarities.
“It is clear from the plot that the behaviours are polarization independent, as confirmed by the simi- larities associated to each distribution, whose values are reported in table I, both for photons injected in mode kA and kB .”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
Findingown data§ Supplementary InformationThe measured propagation losses are 0.5 dB/cm and coupling losses to single-mode fiber arrays at the input facet are about 1 dB.
“Measured propagation losses are 0.5dB/cm and coupling losses to single mode fiber arrays at the input facet are about 1dB.”
Quote in the paperNot a substantial claimMay combine two claims.
Study design and conductreasoning§ IntroductionThe integrated waveguide architecture allows concentrating a large number of optical elements on a small chip and achieves intrinsic phase stability due to the monolithic structure.
“Our approach exploits an integrated waveguide architec- ture, which allows to concentrate a large number of opti- cal elements on a small chip and to achieve intrinsic phase stability due to the monolithic structure.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Study design and conductnone given§ MethodsThe interaction region length L = 2.1 mm was chosen to obtain a balanced splitting ratio.
“The length of the interaction region is chosen as L = 2.1mm in order to obtain a balanced splitting ratio.”
Quote in the paperSubstantial claimMay combine two claims.
Study design and conductnone given§ AbstractA novel three-dimensional geometry for the waveguide circuit allows accurate polarization-independent behaviour while maintaining control on both phase and balancement.
“To this scope a novel three-dimensional geometry for the waveguide circuit is introduced, which allows accurate polarization independent behaviour, maintaining a remarkable control on both phase and balancement.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ IntroductionIn femtosecond laser written waveguides, birefringence is low and does not significantly affect the coherence of the photons.
“In these femtosecond laser written waveguides, birefringence is low and does not affect sig- nificantly the coherence of the photons.”
Quote in the paperNot a substantial claim0 of 2 sources checkedClassified as probably not a substantial claim.
Study design and conductnone given§ MethodsThe basic cell of the network acts as a Mach-Zehnder interferometer.
“The basic cell of the network, depicted in Fig. 2d, acts as a Mach-Zehnder interferom- eter.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Interpretationown data§ DiscussionThe insensitivity to photon polarization, high-accuracy phase control and intrinsic scalability of the integrated multi-DC network pave the way to further advanced investigations on complexity physics phenomena.
“The insensitivity to photon polarization, high-accuracy in the phase control and intrinsic scalability of the in- tegrated multi-DC network presented in this work, pave the way to further advanced investigations on complexity physics phenomena.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ Supplementary InformationThe probability of two photons emerging from the same output of the BS array is zero (PBSJJ = 0), yet a nonzero probability pwalkjj for the walk distribution can be observed due to the relation between outputs and walker positions.
“the symmetrization postulate of quantum mechanics makes the probability of the photons emerging from the same output of the BS array to be zero -i.e. PBSJJ = 0-, all the same, according to the relations (3) generalized to the case of two-particle walk, a probability pwalkjj 6= 0 for the walk distribution can be observed.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Premisereasoning§ Supplementary InformationThe p+ and p− probabilities respectively correspond to the probabilities of detecting two bosons or two fermions injected into modes I and J of the quantum walk.
“It is easy to show that p+(I, J ;K,L) and p−(I, J ;K,L) respectively correspond to the probabilities of detecting at positionsK and L two bosons or fermions injected into the modes I ad J of the quantum walk.”
Quote in the paperSubstantial claim
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Integrated optical circuits permitting nearest-neighbor coupling between a few tens of modes have been the key technology used in the nascent field of multiphoton quantum walks.
“Integrated optical circuits permitting nearest-neighbor coupling between a few tens of modes have been the key technology used in the nascent field of multiphoton quantum walks (20–22).”
From Two-photon quantum walk in a multimode fiber (2016) · cites this paper as [20]
The paper describes an integrated waveguide circuit with directional couplers for a two-photon quantum walk (pp. 2–3), supporting the circuit’s role as an example of the technology used. Its Fig. 2 caption specifies a four-step walk, but our copy does not establish coupling between a few tens of modes or the broader claim that such circuits were the field’s key technology. (LLM, conf=0.82)
Experimental — accuracy not yet measuredThe additional dimension allowed by the direct laser write technique has been employed to compensate for polarization dependent coupling and to alter the boundary conditions of 1D quantum walks.
“The additional dimension allowed by the direct laser write technique has also been employed to compensate for polarization dependent coupling [30] and altering the boundary conditions of 1D QWs [31].”
From Quantum Walks of Correlated Photon Pairs in Two-Dimensional Waveguide Arrays (2014) · cites this paper as [31]
Femtosecond laser waveguide writing is capable of producing high-quality waveguides with low birefringence, which are suited for the propagation of polarization-encoded qubits.
“This technique is capable of producing high-quality waveguides with low birefringence, which are indeed suited for the propagation of polarization-encoded qubits, as we recently reported in refs 22,23, where a polarization-insensitive directional coupler has been demonstrated.”
From Integrated photonic quantum gates for polarization qubits (2011) · cites this paper as [23]
The polarization sensitivity/insensitivity of the directional coupler can be selected by acting on geometrical parameters and not on physical properties of the waveguides.
“It is important to note that the polarization sensitivity/insensitivity of the directional coupler can be selected by acting on geometrical parameters and not on physical properties of the waveguides22,23.”
From Integrated photonic quantum gates for polarization qubits (2011) · cites this paper as [23]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Bosonic distribution is assigned two different Bell states
MinorThe text says the measured triplet and singlet distributions are for |Ψ+〉 and |Ψ−〉, but then says Fig. 4a-b reports them for |Φ+〉 and |Ψ−〉. The bosonic state is identified differently in the two statements.
“The distribution of the triplet and singlet states |Ψ+〉 (φ = 0) and |Ψ−〉 (φ = π) emerging from the quantum walk was reconstructed”
“The measured bosonic and fermionic distributions compared with expected ones are reported in Fig. 4a-b for |Φ+〉 and |Ψ−〉.”
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“…Feynman ’s original proposal 1 , quantum simulators of physical systems are being successfully constructed with a variety of quantum architectures, such as atoms 2-5 trapped ions 6-11 , NMR 12,13 , superconducting circuits 14 as well as single photons 15-22 , which are the focus of this review.”
InfluentialQuantum walks: a comprehensive review
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“the choice of the direction the particle will move in, and the step operator, which shifts the walker in the direction fixed by the quantum coin state (the time evolution being simulated stroboscopically) [18].”
BackgroundMethodInfluentialSelf-citationUniversal Computation by Multiparticle Quantum Walk
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Self-citationFemtosecond laser micromachining for integrated quantum photonics
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From quantum optics to quantum technologies
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Universally optimal noisy quantum walks on complex networks
Caruso, F. (2014) · New Journal of Physics · doi:10.1088/1367-2630/16/5/055015 · Cited by 54 · see citing works
Integrated photonics in quantum technologies
Giordani, T., Hoch, F., Carvacho, G., Spagnolo, N., & Sciarrino, F. (2023) · Rivista Del Nuovo Cimento · doi:10.1007/s40766-023-00040-x · Cited by 54 · see citing works
Self-citationEfficient quantum simulation of fermionic and bosonic models in trapped ions
Lamata, L., Mezzacapo, A., Casanova, J., & Solano, E. (2014) · EPJ Quantum Technology · doi:10.1140/epjqt9 · Cited by 54 · see citing works
- Authors
- Sansoni, L., Sciarrino, F., Vallone, G., Mataloni, P., Crespi, A., Ramponi, R., & Osellame, R.
- Affiliations
- Sapienza University of Rome (IT)
- National Institute of Optics (IT)
- University of Padua (IT)
- Istituto di Fotonica e Nanotecnologie (IT)
- ORCID
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2012-01-05
- Volume
- 108
- Issue
- 1
- Pages
- 010502-010502
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W1990940849
- Semantic Scholar
- eb1d90c629963289aca81d1514051460022aa368
- Cited by
- 578
- References
- 32
- Citations by year
- 2026: 25 · 2025: 16 · 2024: 32 · 2023: 16 · 2022: 36 · 2021: 37 · 2020: 36 · 2019: 58 · 2018: 43 · 2017: 35 · 2016: 52 · 2015: 51 · 2014: 69 · 2013: 43 · 2012: 28
- FWCI
- 55.97
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- INO Open Portal (submittedVersion)
- IRIS Research product catalog (Sapienza University of Rome) (submittedVersion)
- Virtual Community of Pathological Anatomy (University of Castilla La Mancha) (submittedVersion)
- Padua Research Archive (University of Padova) (submittedVersion, other-oa)
- arXiv (Cornell University)
- 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 › Quantum Mechanics and Applications
- Keywords
- quantum walks, two-particle quantum walk, fermionic statistics, polarization entanglement, integrated photonics, waveguide circuits, polarization insensitivity, discrete-time quantum walks
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-07.
Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-07.
Cited sources are checked by gpt-6-sol.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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