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Simulating lossy and partially distinguishable quantum optical circuits: theory, algorithms and applications to experiment validation and state preparation

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that coarse-grained photon-number distributions of Gaussian states—including lossy, spectrally impure, and partially distinguishable systems—can be computed in exponential, rather than combinatorial, time via a blocked…

desk verdict A solid algorithmic advance for coarse-grained photon-number statistics of Gaussian states; the core results hold up, and only the off-diagonal density-matrix section leans on an external proof. read the letter →

arxiv 2412.17742 v3 pith:I5JVXA3L submitted 2024-12-23 quant-ph

Jacob F.F. Bulmer , Javier Martínez-Cifuentes , Bryn A. Bell , Nicolás Quesada This is my paper · ORCID
classification quant-ph
keywords Gaussianstatesphoton-numberstatisticsloopHafniancoarse-graineddistributionsbosonsamplingfinite-differencesieveheraldednon-Gaussianspectralimpurity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks how to compute photon-number statistics for realistic quantum-optical circuits, where loss, spectral impurity, and partial distinguishability turn pure states into mixed Gaussian states and detectors see only coarse-grained bins of photons. The authors prove that coarse-grained photon-number distributions, together with photon-number moments and cumulants and the density-matrix elements of heralded non-Gaussian states, can be computed in $O((N M^3 + N^2 \log N) \prod_l (b_l+1))$ time instead of the combinatorial $O(N 2^{2N} \prod_l \binom{|\Lambda_l|+b_l-1}{b_l})$ time of summing over every compatible detection pattern. The engine is a loop-Hafnian master theorem that packages the photon-number generating function into a closed exponential, followed by a finite-difference sieve that extracts individual binned probabilities. If correct, the results make exact validation of Gaussian and Fock boson samplers practical at scales that previously needed Monte Carlo estimation, and they speed up noisy state-preparation simulations such as approximate GKP-state generation under loss and spectral impurity by about three orders of magnitude. The weakest spot is a master-theorem extension needed for off-diagonal density-matrix elements, which the paper flags as previously assumed and now supported by a cited proof.

What carries the argument

The load-bearing object is the blocked loop Hafnian, a coarse-grained version of the loop Hafnian—a graph invariant that counts single-pair matchings with loops and encodes photon-number statistics of Gaussian states. The key identity is the loop-Hafnian master theorem, which sums the weighted loop Hafnians of all photon patterns into one closed rational-exponential function $q(A,\gamma,z)$. The paper then applies a finite-difference sieve: for a polynomial $f_N(A,\gamma,w)$ of degree $N$ in $L$ block variables, the repeated divided-difference operator $D_{w_j}^{(b_j)}$ returns exactly the coefficient belonging to the binned pattern $b$, so $\mathrm{lhaf}_{\Lambda}(A,\gamma,b) = \prod_{j=1}^{L} D_{w_j}^{(b_j)} f_N(A,\gamma,w)$. The function $f_N$ is assembled from power traces $\mathrm{tr}([XA]^k)$ and displacement terms $\gamma^T[XA]^{k-1}X\gamma$, giving the $O(N M^3 + N^2 \log N)$ per-evaluation cost that replaces the expensive direct sum over loop Hafnians.

What would settle it

For a small system, say a two-mode squeezed state with two internal spectral modes under loss, compute every density-matrix element of the heralded state up to $n_{\mathrm{cutoff}}=5$ using the finite-difference formula, Eq. (103), and compare with the brute-force combinatorial blocked loop Hafnian, Eq. (54); any disagreement beyond numerical tolerance would falsify the speedup and the density-matrix formulas. Alternatively, test the master theorem itself by picking a random symmetric $4\times4$ matrix $A$ and vector $\gamma$ that do not come from a Gaussian state and evaluating both sides of Eq. (24) at several points $z$; a mismatch would invalidate Section VII while leaving the Gaussian-state probability results intact.

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Extended reading notes

Core claim

The central discovery is a closed-form generating identity, the loop-Hafnian master theorem, and its use to define the blocked loop Hafnian. For a Gaussian state with adjacency matrix $A$ and loop vector $\gamma$, the theorem states that $\sum_{n} \mathrm{lhaf}(A_{n\oplus n},\gamma_{n\oplus n}) \prod_i z_i^{n_i}/n_i! = \exp(\tfrac12 \gamma^T[I-D(z)XA]^{-1}D(z)X\gamma)/\sqrt{\det[I-D(z)XA]}$. Grouping detectors into blocks $\Lambda_l$ and replacing $z_i$ by one common variable $w_j$ inside each block turns this sum into a generating function in the block variables; applying finite-difference operators $D^{(b_j)}_{w_j}$ to the function $f_N(A,\gamma,w)$ isolates the probability of the coarse-grained pattern $b$. The paper proves that the time complexity drops from the combinatorial number of compatible patterns to $O((N M^3 + N^2 \log N) \prod_j (b_j+1))$, and it uses the same machinery for internal-mode (spectrally impure) Gaussian states, for coarse-grained moments and cumulants, and for off-diagonal elements of states heralded by fine- or coarse-grained measurements.

Load-bearing premise

The computation of off-diagonal density-matrix elements of heralded non-Gaussian states assumes the loop-Hafnian master theorem holds for arbitrary symmetric matrices and vectors, not only for matrices that describe physical Gaussian states; the paper states that an earlier version treated this as an assumption and that a cited proof now supports it.

Editorial extensions

If this is right

  • Exact validation of Gaussian boson samplers becomes cheaper: the paper computes the total photon-number distribution of the 216-mode Borealis configuration in about one second, roughly fifteen times faster than phase-space sampling with $2.4\times 10^6$ samples, and with no sampling error.
  • Noisy state preparation can be simulated at new scales: the density matrix of an approximate GKP state up to Fock cutoff 26, under loss and two spectral modes, is computed about three orders of magnitude faster with the finite-difference sieve than with the combinatorial blocked loop Hafnian.
  • Coarse-grained photon-number moments and cumulants inherit the speedup, so genuine correlations of Gaussian photonic states can be extracted without expanding sums over combinations of modes.
  • Fock-state inputs through lossy linear circuits also reduce to blocked loop Hafnians, improving on permanent-based sums and enabling exact binned distributions for Fock boson sampling validation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the authors do not explore automatic differentiation in this paper, but since $f_N(A,\gamma,w)$ is built from matrix powers and traces, the finite-difference sieve should be differentiable; this would turn the exact probability formulas into gradients for inverse design of noisy photonic circuits.
  • Editorial inference: because the cost scales with the product $(b_l+1)$, the method is most favorable when coarse-graining is strong; a natural extension is to threshold detectors, where each block has $b_l \in \{0,1\}$, possibly combined with the fully distinguishable internal-mode simplifications in appendix D for further polynomial speedups.
  • Editorial inference: the same blocked-loop-Hafnian formalism could be applied to other bosonic problems with naturally coarse-grained final states, such as vibronic spectra or Franck-Condon factors, where loop Hafnians already appear and where binning over final vibrational states is common.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops classical algorithms for computing coarse-grained photon-number distributions, photon-number moments and cumulants, and heralded density-matrix elements for Gaussian states affected by loss, spectral impurity, and partial distinguishability. The central technical ingredients are a loop-Hafnian master theorem derived from the normalization of Gaussian Fock probabilities (Eq. (24)), an O(N M^3 + N^2 log N) algorithm for the auxiliary polynomial coefficient f_N (Appendix A), a finite-difference sieve for coefficient extraction (Sec. V), and a blocked loop-Hafnian formula for coarse-grained probabilities (Eqs. (58)-(59)). These tools are then applied to off-diagonal density-matrix elements of heralded non-Gaussian states, using an extension of the master theorem to arbitrary symmetric matrices and loop vectors that is cited to the external preprint [70]. The numerical section validates the methods against analytic total-photon-number distributions and phase-space Monte Carlo, including a configuration of the Borealis experiment.

Significance. The Gaussian-state results are potentially significant: they replace a combinatorial sum over detection patterns with a product of (b_l+1) evaluations of a polynomial-coefficient routine, and the paper gives a careful derivation of the coefficient algorithm together with numerical validation against analytic formulas and independent Monte Carlo methods. The strengths include the self-contained derivation of the master theorem from the normalization of Gaussian Fock probabilities, the detailed proof and complexity analysis of the f-function algorithm in Appendix A, the reproducible numerical demonstrations, and the public code release. The heralded-state and GKP/Fock-state-preparation claims, however, are conditional on the arbitrary-matrix generalization of the loop-Hafnian master theorem, which is not proved in the manuscript and is delegated to an external preprint; the state-preparation applications in Sec. VIII should be read with that caveat.

major comments (4)
  1. [Sec. IV, Eq. (40)] Equation (40) drops the vacuum factor Pr(0|A,gamma) that is present in Eqs. (30) and (33). As written, the right-hand side equals f_N(A_Y,gamma_Y), not Pr(N_Y=N,N_Z=0|A,gamma), since q_N(A,gamma,0)=1 and the coefficient f_N is not a probability. The numerical implementations in Sec. VIII presumably include the correct normalization, but the displayed formula is inconsistent with the derivation and should be corrected.
  2. [Sec. VII A, Eqs. (72)-(86) and (89)] The off-diagonal density-matrix formulas depend on the loop-Hafnian master theorem for arbitrary symmetric matrices A' and vectors gamma', as the authors explicitly acknowledge in the paragraph preceding Eq. (89). The proof of this arbitrary-matrix extension is only cited to the external preprint [70] and is not reproduced or verified in the manuscript. Since Eqs. (92), (102), (103) and the GKP/Fock simulations in Sec. VIII all rely on this theorem, the state-preparation claims are not self-contained. The authors should either include a proof or state the precise theorem from [70] and confirm that it applies to the complex symmetric matrices A' with the arbitrary fill entries constructed in Sec. VII A.
  3. [Sec. VII A, Eqs. (72)-(86)] The construction of the permutation matrix P and the extended matrices A', gamma' is only illustrated for a single 4x4 example and then asserted for the general case. Because this construction is load-bearing for every density-matrix element formula in Sec. VII, a complete constructive proof with explicit indexing for general n and m, including the odd-T padding by the block (1), is needed. The current treatment leaves too much to inspection, especially the claimed ability to fill arbitrary entries of A' while preserving the identity in Eqs. (72) and (73).
  4. [Sec. V and Sec. VI, finite-difference sieve] The finite-difference sieve evaluates f_N(A,gamma,w) at arbitrary nodes w_j = v_j + m(u_j-v_j), while the derivation of the master theorem in Sec. III is phrased for valid Gaussian states and does not explicitly supply the analytic-continuation or formal-power-series argument needed to justify those evaluations for arbitrary complex nodes. If the arbitrary-matrix theorem of [70] is invoked for this purpose, that dependence should be stated clearly in Secs. V and VI as well as in Sec. VII; otherwise a short argument showing that Eq. (24) holds as a formal power series identity in z near z=0 should be added.
minor comments (5)
  1. [Sec. VIII, Eqs. (108) and (110)] The quantity F_Fock = sqrt(<n|rho|n>) is called a fidelity, but for a pure target state |n> the standard fidelity is <n|rho|n>; the displayed quantity is the square-root fidelity. This terminology should be clarified, and the same comment applies to F_GKP.
  2. [Sec. V, Eq. (44)] The kth finite-difference operator is defined for k >= 1, but Eq. (48) and later formulas use D^{(0)}_{z_i} when some n_i = 0. The k=0 case should be defined explicitly as the identity operator (or, equivalently, as evaluation at an arbitrary point), since the displayed formula in Eq. (44) gives P(v_j) rather than the identity.
  3. [Sec. VIII, Figs. 5 and 8] The phase-space Monte Carlo runtimes are reported as single values without error bars or repeated-run statistics, despite the fact that the phase-space estimates themselves carry sampling uncertainty. Reporting the variance over runs or a statistical uncertainty on the runtime would make the comparison more informative.
  4. [References, [70]] Reference [70] is an arXiv preprint; the manuscript should give the version or date and, if available, a journal reference, since the arbitrary-matrix master theorem is a load-bearing external result.
  5. [Sec. VI C, Eq. (68)] The text says that for cumulants one uses only the j=1 term of the expansion in Eq. (38), but the notation ilde q_N is introduced without explaining that this is a different generating function, not simply a truncation of q_N. A sentence clarifying the distinction would prevent confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: derivations are self-contained and numerical targets are external.

full rationale

The paper's central loop-Hafnian master theorem (Eq. 24) is derived from the normalization condition of known Gaussian-state Fock probabilities (Eq. 17), not assumed as the target result. The coarse-grained blocked-Hafnian formulas (Eqs. 58-59) follow by applying a standard finite-difference sieve to the generating function, and the complexity reduction is a comparison against the combinatorial definition. The Sec. VII off-diagonal density-matrix formulas require extending the master theorem to arbitrary symmetric matrices A' and vectors gamma'; the paper explicitly flags this and cites an external proof by Tarasov [70], which is independent support rather than a self-citation. Total-photon-number results are validated against the independent analytic formula of Ref. [44] and against Borealis experimental data, while Fock/GKP state simulations use target states from the literature. Self-citations (e.g., Refs. [11, 52, 53]) supply standard background formulas or code-backed algorithms whose correctness is not the paper's claimed output; none of the load-bearing steps reduces by construction to an input or fitted parameter. Residual concerns about the arbitrariness of A', gamma' and analytic continuation are correctness risks, not circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim depends on the loop Hafnian master theorem, the finite-difference sieve, and the assumption that the master theorem extends to arbitrary symmetric matrices (for off-diagonal elements). The numerical examples choose physical parameters such as squeezing, spectral purity, and loss but do not fit them to target outputs. No new physical entities are introduced; the blocked loop Hafnian is a new mathematical object, not a postulated physical entity.

assumptions (3)
  • ad hoc to paper The loop Hafnian master theorem extends to arbitrary symmetric matrices A and vectors gamma, not only those corresponding to valid Gaussian states.
    This is invoked in Sec. VII.A when computing off-diagonal density matrix elements with matrices A' and vectors gamma' that need not represent Gaussian states. The authors state that this was an assumption in a previous draft and is now proven in Ref. [70].
  • domain assumption The photon-number statistics of Gaussian states are given by the loop Hafnian formula (Eq. 11).
    Taken from Refs. [24,52] as the starting point; used throughout.
  • standard math Standard linear algebra and combinatorial identities (Sylvester determinant identity, multinomial theorem, integer partition sums, Dirichlet's divisor-type estimates) are valid.
    Used in Secs. III, IV, V and Appendix A.

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Cite this review

Pith. "Pith review of Simulating lossy and partially distinguishable quantum optical circuits: theory, algorithms and applications to experiment validation and state preparation." pith.science (2026). https://pith.science/paper/I5JVXA3L

@misc{pith2026241217742,
  author       = {Pith},
  title        = {Pith review of: Simulating lossy and partially distinguishable quantum optical circuits: theory, algorithms and applications to experiment validation and state preparation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5JVXA3L}},
  note         = {Machine review of arXiv:2412.17742}
}
read the original abstract

To understand quantum optics experiments, we must perform calculations that consider the principal sources of noise, such as losses, spectral impurity and partial distinguishability. In both discrete and continuous variable systems, these can be modeled as mixed Gaussian states over multiple modes. The modes are not all resolved by photon-number measurements and so require calculations on coarse-grained photon-number distribution. Existing methods can lead to a combinatorial explosion in the time complexity, making this task unfeasible for even moderate sized experiments of interest. In this work, we prove that the computation of this type of distributions can be done in exponential time, providing a combinatorial speedup. We develop numerical techniques that allow us to determine coarse-grained photon number distributions of Gaussian states, as well as density matrix elements of heralded non-Gaussian states prepared in the presence of spectral impurity and partial distinguishability. These results offer significant speed-up and accuracy improvements to validation tests of both Fock and Gaussian boson samplers that rely on binned probability distributions. Moreover, they pave the way to a more efficient simulation of realistic photonic circuits, unlocking the ability to perform exact calculations at scales which were previously out of reach. In addition to this, our results, including loop Hafnian master theorems, may be of interest to the fields of combinatorics and graph theory.

Figures

Figures reproduced from arXiv: 2412.17742 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Illustration of the problem of computing general coarse-grained probabilities in the context of Gaussian Boson [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Setup for the generation of an arbitrary Fock state, [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Fidelity between the noisy (including photon loss and spectral impurity) state, ˆϱ [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Setup for the generation of the approximate GKP [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Time of computation of ˆϱ [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Fidelity between the noisy (including photon loss and [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Wigner function of the noisy (including photon loss and spectral impurity) state ˆϱ [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Total photon number distribution Pr( [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 8
Figure 8. Figure 8: shows the comparison of the time it takes to compute Pr(N|A) using both phase space methods and Eq. (40) (we show the results of ten runs of this equa￾tion), for ξ = 0.89 and ηeff = 0.36. We used the values M ∈ {16, 32, 64, 128, 256}, which correspond to [PITH_FULL_IM…
Figure 10
Figure 10. Figure 10: FIG. 10. Total photon-number distribution for one of the [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Illustration of an interferometric setup where the input light has a different state in the temporal degree of freedom [PITH_FULL_IMAGE:figures/full_fig_p032_11.png]

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Derivation of the Loop Hafnian Generating Function for Arbitrary Symmetric Matrices via Gaussian Integration

    quant-ph 2025-07 accept novelty 6.0 of 10

    The loop hafnian generating function is proven valid for all symmetric complex matrices via Gaussian integration, removing a quantum-optics restriction.

Reference graph

Works this paper leans on

88 extracted references · 65 canonical work pages · cited by 1 Pith paper

  1. [70]

    K. K. Sabapathy, H. Qi, J. Izaac, and C. Weedbrook, Pro- duction of photonic universal quantum gates enhanced by machine learning, Physical Review A100, 012326 (2019)

    work page 2019

  2. [1]

    Serafini, Quantum continuous variables: a primer of theoretical methods (CRC press, 2017)

    A. Serafini, Quantum continuous variables: a primer of theoretical methods (CRC press, 2017)

    2017

  3. [2]

    Ferraro, S

    A. Ferraro, S. Olivares, and M. G. Paris, Gaussian states in continuous variable quantum information, arXiv preprint quant-ph/0503237 (2005). 23

    arXiv 2005

  4. [3]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garc ´ ıa-Patr´ on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012)

    2012

  5. [4]

    Asavanant and A

    W. Asavanant and A. Furusawa, Optical Quantum Com- puters (AIP Publishing LLC, 2022)

    2022

  6. [5]

    Barnett and P

    S. Barnett and P. M. Radmore, Methods in theoretical quantum optics, Vol. 15 (Oxford University Press, 2002)

    2002

  7. [6]

    Gerry and P

    C. Gerry and P. L. Knight, Introductory quantum optics (Cambridge university press, 2005)

    2005

  8. [7]

    Dodonov, O

    V. Dodonov, O. Man’ko, and V. Man’ko, Multidimen- sional hermite polynomials and photon distribution for polymode mixed light, Phys. Rev. A 50, 813 (1994)

    1994

Show all 88 references
  1. [8]

    Krenn, X

    M. Krenn, X. Gu, and A. Zeilinger, Quantum exper- iments and graphs: Multiparty states as coherent su- perpositions of perfect matchings, Phys. Rev. Lett. 119, 240403 (2017)

    2017

  2. [9]

    X. Gu, M. Erhard, A. Zeilinger, and M. Krenn, Quan- tum experiments and graphs ii: Quantum interference, computation, and state generation, Proceedings of the National Academy of Sciences 116, 4147 (2019)

    2019

  3. [10]

    Quesada, Franck-condon factors by counting perfect matchings of graphs with loops, The Journal of chemical physics 150, 164113 (2019)

    N. Quesada, Franck-condon factors by counting perfect matchings of graphs with loops, The Journal of chemical physics 150, 164113 (2019)

    2019

  4. [11]

    J. F. Bulmer, B. A. Bell, R. S. Chadwick, A. E. Jones, D. Moise, A. Rigazzi, J. Thorbecke, U.-U. Haus, T. Van Vaerenbergh, R. B. Patel,et al., The boundary for quantum advantage in gaussian boson sampling, Science advances 8, eabl9236 (2022)

    2022

  5. [12]

    Cardin and N

    Y. Cardin and N. Quesada, Photon-number moments and cumulants of Gaussian states, Quantum 8, 1521 (2024)

    2024

  6. [13]

    Roeland, S

    G. Roeland, S. Kaali, V. R. Rodriguez, N. Treps, and V. Parigi, Mode-selective single-photon addi- tion to a multimode quantum field, arXiv preprint arXiv:2110.14488 (2021)

    2021 arXiv

  7. [14]

    O. F. Thomas, W. McCutcheon, and D. P. McCutcheon, A general framework for multimode gaussian quantum optics and photo-detection: Application to hong–ou– mandel interference with filtered heralded single photon sources, APL Photonics 6, 040801 (2021)

    2021

  8. [15]

    Quesada, Very Nonlinear Quantum Optics, Ph.D

    N. Quesada, Very Nonlinear Quantum Optics, Ph.D. the- sis, University of Toronto (2015)

    2015

  9. [16]

    Tiedau, T

    J. Tiedau, T. J. Bartley, G. Harder, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, Scalability of para- metric down-conversion for generating higher-order fock states, Phys. Rev. A 100, 041802 (2019)

    2019

  10. [17]

    Sonoyama, K

    T. Sonoyama, K. Takahashi, T. Sano, T. Suzuki, T. No- mura, M. Yabuno, S. Miki, H. Terai, K. Takse, W. Asa- vanant, et al. , Generation of multi-photon fock states at telecommunication wavelength using picosecond pulsed light, arXiv preprint arXiv:2405.06567 (2024)

    2024

  11. [18]

    Konno, W

    S. Konno, W. Asavanant, F. Hanamura, H. Nagayoshi, K. Fukui, A. Sakaguchi, R. Ide, F. China, M. Yabuno, S. Miki, et al. , Logical states for fault-tolerant quan- tum computation with propagating light, Science 383, 289 (2024)

    2024

  12. [19]

    M. Endo, K. Takahashi, T. Nomura, T. Sonoyama, M. Yabuno, S. Miki, H. Terai, T. Kashiwazaki, A. Inoue, T. Umeki, et al. , Optically-sampled superconducting- nanostrip photon-number resolving detector for non- classical quantum state generation, arXiv preprint arXiv:2405.06901 (2024)

    2024 arXiv

  13. [20]

    Asavanant and A

    W. Asavanant and A. Furusawa, Multipartite continuous-variable optical quantum entanglement: Generation and application, Physical Review A 109, 040101 (2024)

    2024

  14. [21]

    Kawasaki, R

    A. Kawasaki, R. Ide, H. Brunel, T. Suzuki, R. Nehra, K. Nakashima, T. Kashiwazaki, A. Inoue, T. Umeki, F. China, et al. , High-rate generation and state to- mography of non-gaussian quantum states for ultra- fast clock frequency quantum processors, arXiv preprint arXiv:2402.17...

    2024 arXiv

  15. [22]

    Wiesner, H

    F. Wiesner, H. M. Chrzanowski, G. Pieplow, T. Schr¨ oder, A. Pappa, and J. Wolters, The influence of experimen- tal imperfections on photonic ghz state generation, arXiv preprint arXiv:2406.18257 (2024)

    2024 arXiv

  16. [23]

    Y.-S. Ra, A. Dufour, M. Walschaers, C. Jacquard, T. Michel, C. Fabre, and N. Treps, Non-gaussian quan- tum states of a multimode light field, Nature Physics 16, 144 (2020)

    2020

  17. [24]

    Kruse, C

    R. Kruse, C. S. Hamilton, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Detailed study of gaussian bo- son sampling, Physical Review A 100, 032326 (2019)

    2019

  18. [25]

    C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Gaussian boson sampling, Physical review letters 119, 170501 (2017)

    2017

  19. [26]

    imperfect multimode sources, losses, manufacturing imperfections in passive interfer- ometers and so on

    We considered 50 samples for each of the parameters p, ηeff. All the density matrices involved in the calcula- tion of the fidelity were obtained using Eq. (103). The maximum value of the fidelity is FGKP = 0 .96 for p = 1 .0, ηeff = 1 .0. For decreasing transmission, or incre...

  20. [27]

    Grier, D

    D. Grier, D. J. Brod, J. M. Arrazola, M. B. de An- drade Alonso, and N. Quesada, The complexity of bipar- tite gaussian boson sampling, Quantum 6, 863 (2022)

    2022

  21. [28]

    Deshpande, A

    A. Deshpande, A. Mehta, T. Vincent, N. Quesada, M. Hinsche, M. Ioannou, L. Madsen, J. Lavoie, H. Qi, J. Eisert, et al. , Quantum computational advantage via high-dimensional gaussian boson sampling, Science ad- vances 8, eabi7894 (2022)

    2022

  22. [29]

    B. Go, C. Oh, L. Jiang, and H. Jeong, Exploring shallow- depth boson sampling: Toward a scalable quantum ad- vantage, Physical Review A 109, 052613 (2024)

    2024

  23. [30]

    S. L. Braunstein, Squeezing as an irreducible resource, Physical Review A—Atomic, Molecular, and Optical Physics 71, 055801 (2005)

    2005

  24. [31]

    Thekkadath, M

    G. Thekkadath, M. Houde, D. England, P. Bustard, F. Bouchard, N. Quesada, and B. Sussman, Gain-induced group delay in spontaneous parametric down-conversion, Physical Review Letters 133, 203601 (2024)

    2024

  25. [32]

    Houde and N

    M. Houde and N. Quesada, Waveguided sources of con- sistent, single-temporal-mode squeezed light: The good, the bad, and the ugly, A VS Quantum Science 5 (2023)

    2023

  26. [33]

    de Guise, S.-H

    H. de Guise, S.-H. Tan, I. P. Poulin, and B. C. Sanders, Coincidence landscapes for three-channel linear optical networks, Physical Review A 89, 063819 (2014)

    2014

  27. [34]

    M. E. Villalobos, A. Valencia, and A. Botero, The quan- tum beam splitter with many partially indistinguishable photons: multiphotonic interference and asymptotic clas- sical correspondence, arXiv preprint arXiv:2312.16774 (2023)

    2023 arXiv

  28. [35]

    Nawrath, F

    C. Nawrath, F. Olbrich, M. Paul, S. Portalupi, M. Jet- ter, and P. Michler, Coherence and indistinguishability of highly pure single photons from non-resonantly and res- onantly excited telecom c-band quantum dots, Applied Physics Letters 115 (2019)

    2019

  29. [36]

    Nawrath, R

    C. Nawrath, R. Joos, S. Kolatschek, S. Bauer, P. Pruy, F. Hornung, J. Fischer, J. Huang, P. Vijayan, R. Sittig, et al. , Bright source of purcell-enhanced, triggered, sin- gle photons in the telecom c-band, Advanced Quantum Technologies 6, 2300111 (2023)

    2023

  30. [37]

    Y. Yu, S. Liu, C.-M. Lee, P. Michler, S. Reitzenstein, K. Srinivasan, E. Waks, and J. Liu, Telecom-band quan- tum dot technologies for long-distance quantum net- works, Nature Nanotechnology 18, 1389 (2023). 24

    2023

  31. [38]

    C. L. Phillips, A. J. Brash, M. Godsland, N. J. Martin, A. Foster, A. Tomlinson, R. Dost, N. Babazadeh, E. M. Sala, L. Wilson, et al. , Purcell-enhanced single photons at telecom wavelengths from a quantum dot in a photonic crystal cavity, Scientific Reports 14, 4450 (2024)

    2024

  32. [39]

    Dusanowski, C

    L. Dusanowski, C. Nawrath, S. L. Portalupi, M. Jetter, T. Huber, S. Klembt, P. Michler, and S. H¨ ofling, Optical charge injection and coherent control of a quantum-dot spin-qubit emitting at telecom wavelengths, Nature com- munications 13, 748 (2022)

    2022

  33. [40]

    Aaronson and A

    S. Aaronson and A. Arkhipov, The computational com- plexity of linear optics, in Proceedings of the forty-third annual ACM symposium on Theory of computing (2011) pp. 333–342

    2011

  34. [41]

    A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. Ralph, Boson sampling from a gaussian state, Physical review letters 113, 100502 (2014)

    2014

  35. [42]

    Seron, L

    B. Seron, L. Novo, A. Arkhipov, and N. J. Cerf, Efficient validation of boson sampling from binned photon-number distributions, arXiv preprint arXiv:2212.09643 (2022)

    2022 arXiv

  36. [43]

    Bressanini, B

    G. Bressanini, B. Seron, L. Novo, N. J. Cerf, and M. Kim, Gaussian boson sampling validation via detector binning, arXiv preprint arXiv:2310.18113 (2023)

    2023 arXiv

  37. [44]

    L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Ror- tais, T. Vincent, J. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, et al., Quantum computational advantage with a programmable photonic processor, Na- ture 606, 75 (2022)

    2022

  38. [45]

    Avagyan, E

    A. Avagyan, E. Knill, and S. Glancy, Multi-mode gaus- sian state analysis with total-photon counting, Journal of Physics B: Atomic, Molecular and Optical Physics 56, 145501 (2023)

    2023

  39. [46]

    Avagyan, E

    A. Avagyan, E. Knill, and S. Glancy, Multi-mode gaus- sian state analysis with one bounded photon counter, arXiv preprint arXiv:2404.08957 (2024)

    2024

  40. [47]

    Dellios, P

    A. Dellios, P. D. Drummond, B. Opanchuk, R. Y. Teh, and M. D. Reid, Simulating macroscopic quantum corre- lations in linear networks, Physics Letters A 429, 127911 (2022)

    2022

  41. [48]

    Dellios, B

    A. Dellios, B. Opanchuk, M. Reid, and P. Drum- mond, Validation tests for gbs quantum computers us- ing grouped count probabilities (2023), arXiv preprint arXiv:2211.03480 (2022)

    2023 arXiv

  42. [49]

    P. D. Drummond, B. Opanchuk, A. Dellios, and M. D. Reid, Simulating complex networks in phase space: Gaussian boson sampling, Physical Review A 105, 012427 (2022)

    2022

  43. [50]

    Mart ´ ınez-Cifuentes, K

    J. Mart ´ ınez-Cifuentes, K. Fonseca-Romero, and N. Que- sada, Classical models may be a better explanation of the jiuzhang 1.0 gaussian boson sampler than its targeted squeezed light model, Quantum 7, 1076 (2023)

    2023

  44. [51]

    A. S. Dellios, M. D. Reid, and P. D. Drum- mond, Validation tests of gaussian boson samplers with photon-number resolving detectors, arXiv preprint arXiv:2411.11228 (2024)

    2024 arXiv

  45. [52]

    Mart ´ ınez-Cifuentes, H

    J. Mart ´ ınez-Cifuentes, H. de Guise, and N. Quesada, Lin- ear cross-entropy certification of quantum computational advantage in gaussian boson sampling, PRX Quantum 5, 040312 (2024)

    2024

  46. [53]

    Quesada, L

    N. Quesada, L. Helt, J. Izaac, J. Arrazola, R. Shahrokhshahi, C. Myers, and K. Sabapathy, Simulating realistic non-gaussian state preparation, Physical Review A 100, 022341 (2019)

    2019

  47. [54]

    Bj¨ orklund, B

    A. Bj¨ orklund, B. Gupt, and N. Quesada, A faster hafnian formula for complex matrices and its benchmarking on a supercomputer, Journal of Experimental Algorithmics (JEA) 24, 1 (2019)

    2019

  48. [55]

    J. J. Sylvester, Xxxix. on the equation to the secular inequalities in the planetary theory, The London, Edin- burgh, and Dublin Philosophical Magazine and Journal of Science 16, 267 (1883)

  49. [56]

    Pozrikidis, An introduction to grids, graphs, and net- works (Oxford University Press, USA, 2014)

    C. Pozrikidis, An introduction to grids, graphs, and net- works (Oxford University Press, USA, 2014)

    2014

  50. [57]

    V. V. Kocharovsky, V. V. Kocharovsky, and S. V. Tarasov, The hafnian master theorem, Linear Algebra and its Applications (2022)

    2022

  51. [58]

    com/XanaduAI/thewalrus/blob/master/octave/ loophafnian.m

    The Walrus, loophafnian (2024), https://github. com/XanaduAI/thewalrus/blob/master/octave/ loophafnian.m

    2024

  52. [59]

    Cygan and M

    M. Cygan and M. Pilipczuk, Faster exponential-time al- gorithms in graphs of bounded average degree, Informa- tion and Computation 243, 75 (2015)

    2015

  53. [60]

    J. F. Bulmer, S. Paesani, R. S. Chadwick, and N. Que- sada, Threshold detection statistics of bosonic states, Physical Review A 106, 043712 (2022)

    2022

  54. [61]

    E. T. Bax, Finite-difference algorithms for counting prob- lems, Ph.D. thesis, California Institute of Technology (1998)

    1998

  55. [62]

    H. J. Ryser, Combinatorial mathematics, Vol. 14 (Amer- ican Mathematical Soc., 1963)

    1963

  56. [63]

    D. G. Glynn, The permanent of a square matrix, Euro- pean Journal of Combinatorics 31, 1887 (2010)

    2010

  57. [64]

    Shchesnovich, On the classical complexity of sam- pling from quantum interference of indistinguishable bosons, International Journal of Quantum Information 18, 2050044 (2020)

    V. Shchesnovich, On the classical complexity of sam- pling from quantum interference of indistinguishable bosons, International Journal of Quantum Information 18, 2050044 (2020)

    2020

  58. [65]

    Kan, From moments of sum to moments of product, Journal of Multivariate Analysis 99, 542 (2008)

    R. Kan, From moments of sum to moments of product, Journal of Multivariate Analysis 99, 542 (2008)

    2008

  59. [66]

    Shi and T

    J. Shi and T. Byrnes, Effect of partial distinguishability on quantum supremacy in gaussian boson sampling, npj Quantum Information 8, 54 (2022)

    2022

  60. [67]

    H. D. Ursell, The evaluation of Gibbs’ phase-integral for imperfect gases, Math. Proc. Cambridge Philos. Soc. 23, 685–697 (1927)

    1927

  61. [68]

    R. A. Fisher and J. Wishart, The Derivation of the Pattern Formulae of Two-Way Partitions from those of Simpler Patterns, Proc. London Math. Soc. s2-33, 195 (1932)

    1932

  62. [69]

    D. Su, C. R. Myers, and K. K. Sabapathy, Conver- sion of gaussian states to non-gaussian states using photon-number-resolving detectors, Physical Review A 100, 052301 (2019)

    2019

  63. [71]

    S. V. Tarasov, Derivation of the loop hafnian generating function for arbitrary symmetric matrices via gaussian integration, arXiv preprint arXiv:2507.16100 (2025)

    2025 arXiv

  64. [72]

    event-ready

    Q. Zhang, X.-H. Bao, C.-Y. Lu, X.-Q. Zhou, T. Yang, T. Rudolph, and J.-W. Pan, Demonstration of a scheme for the generation of “event-ready” entangled photon pairs from a single-photon source, Physical Review A—Atomic, Molecular, and Optical Physics 77, 062316 (2008)

    2008

  65. [73]

    Tzitrin and J

    I. Tzitrin and J. E. Bourassa, GKP states, https://the-walrus.readthedocs.io/en/latest/ gallery/gkp.html (2024), accessed: 2024-12-14. 25

    2024

  66. [74]

    Tzitrin, J

    I. Tzitrin, J. E. Bourassa, N. C. Menicucci, and K. K. Sabapathy, Progress towards practical qubit computa- tion using approximate gottesman-kitaev-preskill codes, Physical Review A 101, 032315 (2020)

    2020

  67. [75]

    S. K. Lam, A. Pitrou, and S. Seibert, Numba: A llvm- based python jit compiler, in Proceedings of the Second Workshop on the LL VM Compiler Infrastructure in HPC (2015) pp. 1–6

    2015

  68. [76]

    Ruiz-Gonzalez, S

    C. Ruiz-Gonzalez, S. Arlt, J. Petermann, S. Sayyad, T. Jaouni, E. Karimi, N. Tischler, X. Gu, and M. Krenn, Digital discovery of 100 diverse quantum experiments with pytheus, Quantum 7, 1204 (2023)

    2023

  69. [77]

    Rudolph, Terry vs an ai, round 1: Heralding single- rail (approximate?) 4-ghz state from squeezed sources, arXiv preprint arXiv:2303.05514 (2023)

    T. Rudolph, Terry vs an ai, round 1: Heralding single- rail (approximate?) 4-ghz state from squeezed sources, arXiv preprint arXiv:2303.05514 (2023)

    2023 arXiv

  70. [78]

    Y. Yao, F. Miatto, and N. Quesada, Riemannian opti- mization of photonic quantum circuits in phase and fock space, SciPost Physics 17, 082 (2024)

    2024

  71. [79]

    Kleinpaß, T

    P. Kleinpaß, T. Walther, and E. Fitzke, Frequency- resolved simulations of highly entangled biphoton states: Beyond the single-pair approximation. ii. application to entanglement-based quantum key distribution, arXiv preprint arXiv:2412.00958 (2024)

    2024 arXiv

  72. [80]

    Steinmetz, M

    J. Steinmetz, M. Ostmann, A. Neville, B. Pankovich, and A. Sohbi, Simulating imperfect quantum opti- cal circuits using unsymmetrized bases, arXiv preprint arXiv:2412.13330 (2024)

    2024 arXiv

  73. [81]

    T. M. Apostol, Introduction to analytic number theory (Springer Science & Business Media, 2013)

    2013

  74. [82]

    OEIS Foundation Inc., The On-Line Encyclopedia of In- teger Sequences (2024), published electronically at http: //oeis.org

    2024

  75. [83]

    Piela, Ideas of quantum chemistry (Elsevier, 2006)

    L. Piela, Ideas of quantum chemistry (Elsevier, 2006)

    2006

  76. [84]

    Quesada, L

    N. Quesada, L. Helt, M. Menotti, M. Liscidini, and J. Sipe, Beyond photon pairs—nonlinear quantum pho- tonics in the high-gain regime: a tutorial, Advances in Optics and Photonics 14, 291 (2022)

    2022

  77. [85]

    Houde, W

    M. Houde, W. McCutcheon, and N. Quesada, Matrix de- compositions in quantum optics: Takagi/autonne, bloch– messiah/euler, iwasawa, and williamson, Canadian Jour- nal of Physics 102, 497 (2024)

    2024

  78. [86]

    Nijenhuis and H

    A. Nijenhuis and H. S. Wilf, Combinatorial algorithms: for computers and calculators (Elsevier, 2014)

    2014

  79. [87]

    Hage, Enumerating submultisets of multisets, Infor- mation processing letters 85, 221 (2003)

    J. Hage, Enumerating submultisets of multisets, Infor- mation processing letters 85, 221 (2003)

    2003

  80. [88]

    initial” and “final

    D.-J. Guan, Generalized gray codes with applications, in Proc. Natl. Sci. Counc. Repub. China Part A Phys. Sci. Eng., Vol. 22 (Citeseer, 1998) pp. 841–848. Appendix A: A proof of the algorithm for computing the f -function From Eq. (38) let us define gk = tr [XA]k 2k + γT[XA]k...

    1998

Pith tools

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