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Exact simulation of Gaussian Boson Sampling in polynomial space and exponential time

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Pith's one-line read Classical algorithm samples GBS exactly in polynomial space

desk verdict First polynomial-memory GBS sampler with PNR detectors; the chain-rule construction is sound, but the 'exact' label is an overstatement once the finite photon cutoff is in play. read the letter →

arxiv 1908.08068 v3 pith:KNF4TYFE submitted 2019-08-21 quant-ph

Nicolás Quesada , Juan Miguel Arrazola This is my paper · ORCID
classification quant-ph
keywords GaussianBosonSamplingclassicalsimulationhafnianchainruleconditionalcovariancematrixthresholddetectorsnon-negativekernel
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 introduces a classical algorithm that samples exactly from the distribution produced by Gaussian Boson Sampling (GBS), in which a multi-mode Gaussian state is measured with photon-number detectors. The method applies the chain rule of probability, sampling each mode's photon number conditional on the previously sampled numbers, and it computes each conditional probability from the reduced covariance matrix of the modes already processed. For $m$ modes and $N$ detected photons the algorithm runs in $O(m^2)$ space and $O(mN^3 2^N)$ time, so for a fixed number of modes it is, up to constants, as costly as computing a single GBS output probability. The same conditioning strategy covers GBS with threshold detectors, displacements, and linear combinations of Gaussian states, and it yields a polynomial-time approximate sampler when the kernel matrix is non-negative. This matters because claims that GBS is hard to simulate classically are asymptotic; the algorithm gives a concrete benchmark of what classical simulation actually costs.

What carries the argument

The load-bearing object is the reduced covariance matrix $Q^{(k)}=V^{(k)}+1/2$ and the kernel matrix $A^{(k)}=X(1-(Q^{(k)})^{-1})$ for the first $k$ modes. The chain rule of probability, $p(s_1,\dots,s_m)=\prod_{k=1}^m p(s_k|s_1,\dots,s_{k-1})$, is the engine: each conditional probability is a ratio of two marginal probabilities of the form in Eq. (8), and each marginal is a hafnian whose matrix is obtained from $A^{(k)}$ by repeating rows and columns according to the photon pattern. The hafnian, the sum over perfect matchings that counts the terms in the output probability, is the only exponential ingredient, since Gaussian conditioning keeps all reduced matrices polynomial in size. For the non-negative case, the hafnian is replaced by the Monte Carlo identity $\mathrm{Haf}(A)=\mathbb{E}[\det W]$ for a random skew-symmetric $W$, which supplies the polynomial-time approximate variant.

What would settle it

Run the algorithm on a small GBS instance, say four modes with fixed squeezing and a known interferometer, where exact output probabilities can be enumerated by brute force, and increase the maximum photon number $n_{\max}$. The central claim requires the total variation distance between the sampled and exact distributions to go to zero as $n_{\max}$ grows; if the distance saturates above zero, or if the $n_{\max}$ needed for a fixed accuracy forces a runtime larger than $O(mN^3 2^N)$ as the mode count grows, the exactness claim fails.

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Extended reading notes

Core claim

The central claim is that sampling from the GBS distribution reduces to a sequence of conditional probability computations. Writing $S^{(k)}=(s_1,\dots,s_k)$ for a partial pattern, the paper uses $p(S^{(k)})=\mathrm{Haf}(A_s^{(k)})/(\sqrt{\det Q^{(k)}}\,s_1!\cdots s_k!)$, where $Q^{(k)}=V^{(k)}+1/2$ is built from the reduced covariance matrix $V^{(k)}$ of the first $k$ modes and $A^{(k)}=X(1-(Q^{(k)})^{-1})$ is the corresponding kernel matrix; the hafnian $\mathrm{Haf}$ is the sum over perfect matchings that gives the output probability. Because Gaussian marginals are Gaussian, every reduced matrix is obtained efficiently, and the algorithm samples each $s_k$ from $p(s_k|s_1,\dots,s_{k-1})$ by computing two such marginal quantities. The paper argues this gives exact sampling with $O(m^2)$ space and $O(mN^3 2^N)$ time, where $N$ is the total number of detected photons, and extends the same chain-rule construction to threshold detectors, displaced Gaussian states, and linear combinations of Gaussian states. For non-negative kernel matrices, the hafnian is replaced by a randomized estimator of the form $\mathbb{E}[\det W]$, giving an approximate polynomial-time sampler.

Load-bearing premise

The algorithm is exact only if the photon number in every mode is drawn from its full, unbounded range of possibilities; any practical maximum photon number turns the simulation into an approximation, and the stated runtime treats that maximum as a fixed constant rather than as a quantity that must grow with the number of modes.

Editorial extensions

If this is right

  • For a fixed number of modes, exact classical simulation of GBS costs the same order as computing a single output probability, up to constant factors.
  • The $O(m^2)$ space requirement removes the exponential memory bottleneck of earlier threshold-detector simulation algorithms.
  • The same chain-rule sampler handles GBS with threshold detectors, displaced Gaussian states, and linear combinations of Gaussian states, with the time cost changed only by known prefactors.
  • When the kernel matrix is non-negative, approximate sampling runs in polynomial time; the paper's benchmark reports errors up to 228% in the normalization proxy, so the polynomial speedup carries a substantial accuracy cost.

Reading between the lines

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

  • Because the only property the sampler needs is efficient marginals, the method should extend to any state family with efficiently computable reduced descriptions, not just Gaussian states; the paper notes the Gaussian property but does not pursue this general criterion.
  • The runtime is driven by the number of detected photons, so the hardest instances are high-brightness outputs; low-photon samples, even in many modes, should be classically cheap to produce.
  • The non-negative case's practicality is gated by the $1/\sqrt{M}$ Monte Carlo convergence of the hafnian estimator; a variance-reduced estimator or importance-sampling scheme would be a natural development, and the paper's own error benchmarks show why it is needed.
  • One can compute the Appendix B bound in polynomial time, so an instance-specific choice of maximum photon number could certify a target total-variation error; the tradeoff between that cutoff, the mode count, and the runtime is a direct next calculation.
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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

3 major / 3 minor

Summary. The paper presents a classical sampling algorithm for Gaussian Boson Sampling (GBS). The method samples modes sequentially using the chain rule of probability, computing each conditional probability from the reduced covariance matrix of the modes already sampled and the corresponding hafnian. The authors claim exact sampling in O(m^2) space and O(m N^3 2^N) time, where N is the number of detected photons, and argue that for fixed mode number the cost is equivalent to that of computing output probabilities up to constant factors. They extend the method to threshold detectors, displaced GBS, linear combinations of Gaussian states, and a polynomial-time approximate variant for non-negative kernel matrices. The paper also reports an implementation in The Walrus library and benchmarks it on Haar-random interferometers and Erdős–Rényi graph encodings.

Significance. If the central claims were fully established, the paper would be a useful contribution: it would show that classical simulation of GBS has space complexity only quadratic in the mode number and time complexity governed by the cost of hafnian computation, placing GBS simulation on a footing comparable to Boson Sampling. The paper also provides a publicly available implementation, benchmarks with concrete runtimes, and a clear numerical demonstration of the challenges facing the approximate non-negative-kernel variant. These strengths make the work potentially valuable to the quantum-simulation and quantum-advantage community. However, the exactness and complexity claims are not supported as stated because the implemented algorithm truncates the photon-number distribution, and the Appendix A proof underlying the polynomial-time variant contains a logical gap.

major comments (3)
  1. [II.B, Steps 1–3; Appendix B] The algorithm as described is not an exact sampler from the GBS distribution. In Step 1, and analogously in Steps 2 and 3, the algorithm computes p(s_k = 0), ..., p(s_k = nmax) and samples from the resulting truncated distribution, rather than from the full unbounded photon-number distribution. Equation (11) is exact only if every conditional distribution is sampled over its full support. Appendix B explicitly acknowledges this by bounding the total-variation distance of the truncated distribution, but a bound on the approximation error does not establish exactness. The manuscript should either provide a procedure that samples exactly from the unbounded support or revise the title, abstract, and Section II.B to describe an approximate sampler with a controlled error, and should state precisely how the remaining probability mass (the event mapped to ∅ in Appendix B) is treated in the sampling step.
  2. [II.C, Eqs. (12)–(13)] The complexity analysis treats nmax as a fixed constant and absorbs 2^{nmax} into the O-constant, but this is not legitimate for the stated claims. If the sampler is required to approximate the GBS distribution within a fixed total-variation error, Eq. (B2) forces nmax to grow with the number of modes m (and with the state parameters), so the per-mode cost contains a factor 2^{nmax} that depends on m. Consequently, the claimed O(m N^3 2^N) complexity for an exact or fixed-error sampler is not established. The paper should state the complexity as a function of nmax and the target error, or restrict the exactness/complexity claim to a regime where m is fixed and nmax is chosen accordingly, with that dependence made explicit.
  3. [Appendix A, Eqs. (A3)–(A9)] The proof that a non-negative proper covariance matrix Q leads to a non-negative kernel matrix A contains a logical gap. The identification Q^{-1} = 1 - O and the comparison with the M-matrix decomposition Q^{-1} = λ0 1 - N, setting λ0 = 1, assumes part of what is to be proved. The chain of inequalities in Eq. (A9) is in the Loewner (positive-semidefinite) order, and positive-definiteness bounds do not imply entrywise non-negativity of O. Since the polynomial-time approximate algorithm of Section II.D.4 depends on this lemma, the proof needs to be replaced by a valid argument or a precise reference establishing the claim.
minor comments (3)
  1. [III.B, Fig. 2 and text] The reported average error ε = 63% and maximum error ε = 228% in the approximate non-negative-kernel algorithm are very large; the text acknowledges this, but the figure caption and the phrase 'approximate sampling algorithm' should more prominently warn that the sampled distribution can be far from the target GBS distribution at the tested parameters.
  2. [II.C, paragraph after Eq. (13)] The statement that 'for all values of N for which hafnians can be computed in practice, it is preferable to employ the algorithm of Ref. [38]' would benefit from a quantitative comparison or a reference to the actual crossover point, since it is not obvious from the asymptotic exponents alone.
  3. [Appendix B, Eq. (B2)] The notation 'S > nmax' is introduced informally; a short formal definition (e.g., S in N^m with at least one coordinate exceeding nmax) would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

Exact-sampling derivation is independent; the nonnegative-kernel lemma in Appendix A proves its conclusion by setting N=O in the M-matrix decomposition.

  1. self definitional [Appendix A, Eqs. (A6)-(A8) and the concluding sentence]
    "Comparing Eq. (A3) and Eq. (A6), we can identify λ0 = 1 (A7) N = O. (A8) However, for this identification to be correct, it remains to show that λ0 = 1 ≥ max{eigvals(O)}. This is easily seen from the following chain of inequalities ... Therefore, we conclude that O has non-negative entries."

    The cited M-matrix theorem only asserts the existence of some λ0 and N with N≥0 and λ0≥max(eig(N)); it does not force λ0=1. Identifying λ0=1 makes N=O by direct comparison with Q^{-1}=I-O, so the desired statement 'O has non-negative entries' is exactly the non-negativity of the N that was inserted by the identification. The inequality chain that follows gives only operator bounds on I−Q^{-1}, not entrywise non-negativity, so it does not break the circularity. This lemma is load-bearing only for the polynomial-time approximate algorithm for non-negative kernel matrices in Sec. II.D.4, not for the exact sampler.

full rationale

The central exact GBS simulation algorithm is not circular. It samples modes sequentially via the genuine chain rule p(S)=∏p(s_k|s_<k), evaluating each conditional from the reduced covariance matrix with the known hafnian output-probability formula; no parameter is fitted to the target distribution and the complexity bound follows from the published hafnian runtime. The displaced and threshold extensions cite self-authored formulas (Refs. [26,37,39,40]), but those are parameter-free published identities used as evidence, not conclusions imported to make the present result true. The main caveats are not circularity: the algorithm as written truncates each mode at nmax, so 'exactness' holds only in the nmax→∞ limit, and the stated O(mN^3 2^N) hides nmax in the constant, a correctness/complexity gap acknowledged by Appendix B. One genuine circular step appears in Appendix A, where the nonnegative-kernel lemma is concluded by identifying λ0=1 and N=O in the M-matrix decomposition; this is a self-definitional local step. Since that lemma supports only the secondary polynomial-time approximate variant and not the central exact sampler, the overall circularity is moderate.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central algorithm uses standard hafnian probability formulas and Gaussian marginal facts; no parameter is fitted to the target distribution. The approximate algorithm introduces user-chosen parameters nmax and M. The paper also postulates, via a defective proof, the lemma that nonnegative proper Q implies nonnegative kernels, which is load-bearing for polynomial runtime.

free parameters (2)
  • nmax = not specified in the main text; chosen to satisfy the Appendix B total-variation bound
    Truncation cutoff for the per-mode photon-number distribution. The exactness of the algorithm depends on taking it to infinity; in benchmarks it is finite.
  • M = 1000 in Section III.B
    Number of determinant samples used to estimate each hafnian in the approximate nonnegative-kernel algorithm. The error scales as 1/sqrt(M).
assumptions (5)
  • domain assumption GBS output probabilities are given by p(S)=Haf(A_S)/(sqrt(det Q) prod s_i!) for zero-mean Gaussian states.
    Taken from Ref. [10]; the sampling algorithm computes conditional probabilities using this formula on reduced covariance matrices.
  • standard math The marginal of a Gaussian state on a subset of modes is Gaussian with covariance matrix given by the corresponding principal submatrix.
    Used in Section II.B to justify replacing Q by Q(k) in Eq. (8).
  • domain assumption Hafnians of 2N by 2N matrices can be evaluated in O(N^3 2^N) time (Ref. [37]), and loop hafnians and torontonians have equivalent complexity.
    Used in Section II.C to derive runtime T(N,m)=O(m N^3 2^N).
  • domain assumption For nonnegative matrices A, Haf(A)=E[det(W)] with W_ij=G_ij sqrt(a_ij) (Barvinok estimator), enabling polynomial-time estimation.
    Used in Section II.D.4 and Appendix A for the approximate nonnegative-kernel algorithm.
  • ad hoc to paper If Q is a nonnegative proper quantum covariance matrix, then the reduced kernel matrices A^{(k)}_s are nonnegative (Appendix A).
    This is the lemma whose proof identifies N=O in the M-matrix decomposition, a step that appears circular; it is load-bearing for the polynomial-time claim.

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Pith. "Pith review of Exact simulation of Gaussian Boson Sampling in polynomial space and exponential time." pith.science (2026). https://pith.science/paper/KNF4TYFE

@misc{pith2026190808068,
  author       = {Pith},
  title        = {Pith review of: Exact simulation of Gaussian Boson Sampling in polynomial space and exponential time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KNF4TYFE}},
  note         = {Machine review of arXiv:1908.08068}
}
read the original abstract

We introduce an exact classical algorithm for simulating Gaussian Boson Sampling (GBS). The complexity of the algorithm is exponential in the number of photons detected, which is itself a random variable. For a fixed number of modes, the complexity is in fact equivalent to that of calculating output probabilities, up to constant prefactors. The simulation algorithm can be extended to other models such as GBS with threshold detectors, GBS with displacements, and sampling linear combinations of Gaussian states. In the specific case of encoding non-negative matrices into a GBS device, our method leads to an approximate sampling algorithm with polynomial runtime. We implement the algorithm, making the code publicly available as part of Xanadu's The Walrus library, and benchmark its performance on GBS with random Haar interferometers and with encoded Erd\H{o}s-Renyi graphs.

Figures

Figures reproduced from arXiv: 1908.08068 by the authors.

Figure 1
Figure 1. FIG. 1. Running times of the exact simulation algorithm [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Running times of the approximate simulation al [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Bound on the total variation distance for a GBS [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

54 extracted references · 46 canonical work pages · cited by 2 Pith papers

  1. [1]

    This is done by computing each probability p(s1 = 0),p (s1 = 1),...,p (s1 = nmax) up to a maximum photon numbernmax and sampling from the resulting distribution

    Compute A(1) and sample the number of photons s1 in the first mode from the distribution p(s1) = 1√ det(Q(1)) Haf(A(1) s1 ) s1! . This is done by computing each probability p(s1 = 0),p (s1 = 1),...,p (s1 = nmax) up to a maximum photon numbernmax and sampling from the resulting distribution. The pa- rameter nmax must be chosen to ensure that the resulting d...

    work page

  2. [2]

    Compute A(2) and, as before, sample the number of photons s2 in the second mode from the condi- tional distribution p(s2|s∗

    work page

  3. [3]

    , (9) where s∗ 1 is fixed from the result of the previous step and p(s∗ 1,s 2) is defined as in Eq. (8)

    work page

  4. [4]

    The condi- tional distribution of the photon number sk for the k-th mode is p(sk|s∗ 1,...,s ∗ k−1) = p(s∗ 1,s∗ 2,...,s ∗ k−1,sk) p(s∗ 1,...,s ∗ k−1)

    Repeat this procedure for all modes. The condi- tional distribution of the photon number sk for the k-th mode is p(sk|s∗ 1,...,s ∗ k−1) = p(s∗ 1,s∗ 2,...,s ∗ k−1,sk) p(s∗ 1,...,s ∗ k−1) . (10) The outputs (s∗ 1,...,s ∗ k−1) are fixed from the re- sults of the previous steps, p(s∗ 1,s∗ 2,...,s ∗ k−1,sk) is given by Eq. (8), and the probability p(s∗ 1,...,s ...

    work page

  5. [5]

    The best known algorithms for computing torontonians have the same complexity as those for calculating haf- nians [26]

    Threshold detectors The first case we consider is GBS with threshold de- tectors, where output probabilities are given by p ( S(k) ) = Tor ( O(k) s ) √ det ( Q(k)), (14) where as before O(k) = 1 − ( Q(k))−1 , Tor(·) is the torontonian [26], and the entries of S(k) take the val- ues si = 0 for no detection and si = 1 for detection. The best known algorithms...

    work page

  6. [6]

    Displacements The algorithm can also be applied to GBS with dis- placements, i.e., when the Gaussian state has non-zero mean ¯α⁄= 0. We first define the following useful quan- tities: ⃗ α(k) = ( ¯α(k), [ ¯α(k) ]∗) , (15) ⃗ γ(k) = [ Q(k) ]−1[ ⃗ α(k) ]† , (16) N (k) = exp ( − 1 2⃗ α(k)[ Q(k)]−1[ ⃗ α(k)]†) √ det(Q(k))s1!··· sk! . (17) As shown in Refs. [37, 39...

    work page

  7. [7]

    The coefficientsqi can form a probability distribution, in which case ϱ is a mixture of Gaussian states, or they can be (possibly negative) real numbers

    Linear combinations of Gaussian states The algorithm can also be applied to sampling using threshold detectors for states that can be written as linear combinations of Gaussian states: ϱ = 𝓁∑ i=1 qi ϱi( ¯αi, Σi), (19) where ϱi( ¯αi, Σi) is a Gaussian state with vector of means ¯αi and covariance matrix Σi. The coefficientsqi can form a probability distribut...

    work page

  8. [8]

    An important example is when the covariance matrixQ, or equivalently Σ, is non-negative, i.e., whenQij≥ 0 for alli,j

    Non-negative kernel matrices Besides being generalizable to different states and measurements, the algorithm can have polynomial time complexity whenever output probabilities can be esti- mated efficiently. An important example is when the covariance matrixQ, or equivalently Σ, is non-negative, i.e., whenQij≥ 0 for alli,j . Indeed, as we show in Ap- pendix A...

    work page

Show all 54 references
  1. [9]

    Aaronson and A

    S. Aaronson and A. Arkhipov, in Proceedings of the forty-third annual ACM symposium on Theory of com- puting (ACM, 2011) pp. 333–342

    2011

  2. [10]

    Tillmann, B

    M. Tillmann, B. Daki´ c, R. Heilmann, S. Nolte, A. Sza- meit, and P. Walther, Nature Photonics 7, 540 (2013)

    2013

  3. [11]

    J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, et al. , Science 339, 798 (2013)

    2013

  4. [12]

    M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. Aaronson, T. C. Ralph, and A. G. White, Science 339, 794 (2013)

    2013

  5. [13]

    Bentivegna, N

    M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Viggianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galv˜ ao, A. Crespi, et al. , Science Advances 1, e1400255 (2015)

    2015

  6. [14]

    Barkhofen, T

    S. Barkhofen, T. J. Bartley, L. Sansoni, R. Kruse, C. S. Hamilton, I. Jex, and C. Silberhorn, Physical Review Letters 118, 020502 (2017)

    2017

  7. [15]

    A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. Ralph, Physical Review Letters 113, 100502 (2014)

    2014

  8. [16]

    A. P. Lund, S. Rahimi-Keshari, and T. C. Ralph, Phys- ical Review A 96, 022301 (2017)

    2017

  9. [17]

    Chakhmakhchyan and N

    L. Chakhmakhchyan and N. J. Cerf, Physical Review A 96, 032326 (2017)

    2017

  10. [18]

    C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Physical Review Letters 119, 170501 (2017)

    2017

  11. [19]

    Kruse, C

    R. Kruse, C. S. Hamilton, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, arXiv:1801.07488 (2018)

    2018 arXiv

  12. [20]

    Rosenberg, A

    D. Rosenberg, A. E. Lita, A. J. Miller, and S. W. Nam, Physical Review A 71, 061803 (2005)

    2005

  13. [21]

    J. Huh, G. G. Guerreschi, B. Peropadre, J. R. McClean, and A. Aspuru-Guzik, Nature Photonics 9, 615 (2015)

    2015

  14. [22]

    J. M. Arrazola and T. R. Bromley, Physical Review Let- ters 121, 030503 (2018)

    2018

  15. [23]

    J. M. Arrazola, T. R. Bromley, and P. Rebentrost, Physical Review A 98, 012322 (2018)

    2018

  16. [24]

    Banchi, M

    L. Banchi, M. Fingerhuth, T. Babej, and J. M. Arra- zola, arXiv:1902.00462 (2019)

    2019 arXiv

  17. [25]

    Schuld, K

    M. Schuld, K. Br´ adler, R. Israel, D. Su, and B. Gupt, arXiv:1905.12646 (2019)

    2019 arXiv

  18. [26]

    Jahangiri, J

    S. Jahangiri, J. M. Arrazola, N. Quesada, and N. Kil- loran, arXiv:1906.11972 (2019)

    2019 arXiv

  19. [27]

    W. R. Clements, J. J. Renema, A. Eckstein, A. A. Valido, A. Lita, T. Gerrits, S. W. Nam, W. S. Koltham- mer, J. Huh, and I. A. Walmsley, Journal of Physics B: Atomic, Molecular and Optical Physics 51, 245503 (2018)

    2018

  20. [28]

    Paesani, Y

    S. Paesani, Y. Ding, R. Santagati, L. Chakhmakhchyan, C. Vigliar, K. Rottwitt, L. K. Oxenløwe, J. Wang, M. G. Thompson, and A. Laing, Nature Physics , 1 (2019)

    2019

  21. [29]

    Zhong, L.-C

    H.-S. Zhong, L.-C. Peng, Y. Li, Y. Hu, W. Li, J. Qin, D. Wu, W. Zhang, H. Li, L. Zhang, et al. , Science Bul- letin 64, 511 (2019). 9

    2019

  22. [30]

    Neville, C

    A. Neville, C. Sparrow, R. Clifford, E. Johnston, P. M. Birchall, A. Montanaro, and A. Laing, Nature Physics 13, 1153 (2017)

    2017

  23. [31]

    Clifford and R

    P. Clifford and R. Clifford, inProceedings of the Twenty- Ninth Annual ACM-SIAM Symposium on Discrete Al- gorithms (Society for Industrial and Applied Mathemat- ics, 2018) pp. 146–155

    2018

  24. [32]

    J. J. Renema, arXiv:1911.10112 (2019)

    2019 arXiv

  25. [33]

    A. E. Moylett, R. Garc´ ıa-Patr´ on, J. J. Renema, and P. Turner, Quantum Science and Technology (2019)

    2019

  26. [34]

    Quesada, J

    N. Quesada, J. M. Arrazola, and N. Killoran, Physical Review A 98, 062322 (2018)

    2018

  27. [35]

    B. Gupt, J. M. Arrazola, N. Quesada, and T. R. Brom- ley, arXiv:1810.00900 (2018)

    2018 arXiv

  28. [36]

    B. Gupt, J. Izaac, and N. Quesada, J. Open Source Softw. 4, 1705 (2019)

    2019

  29. [37]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garc´ ıa-Patr´ on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Reviews of Modern Physics 84, 621 (2012)

    2012

  30. [38]

    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

  31. [39]

    Picinbono, IEEE Trans

    B. Picinbono, IEEE Trans. Signal Process. 44, 2637 (1996)

    1996

  32. [40]

    W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optica 3, 1460 (2016)

    2016

  33. [41]

    de Guise, O

    H. de Guise, O. Di Matteo, and L. L. S´ anchez-Soto, Physical Review A 97, 022328 (2018)

    2018

  34. [42]

    M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Physical Review Letters 73, 58 (1994)

    1994

  35. [43]

    E. R. Caianiello, Il Nuovo Cimento (1943-1954)10, 1634 (1953)

    1953

  36. [44]

    Barvinok, Combinatorics and complexity of partition functions, Vol

    A. Barvinok, Combinatorics and complexity of partition functions, Vol. 276 (Springer, 2016)

    2016

  37. [45]

    Bj¨ orklund, B

    A. Bj¨ orklund, B. Gupt, and N. Quesada, Journal of Experimental Algorithmics (JEA) 24, 11 (2019)

    2019

  38. [46]

    Bj¨ orklund, inProceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms (Soci- ety for Industrial and Applied Mathematics, 2012) pp

    A. Bj¨ orklund, inProceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms (Soci- ety for Industrial and Applied Mathematics, 2012) pp. 914–921

    2012

  39. [47]

    Quesada, The Journal of Chemical Physics 150, 164113 (2019)

    N. Quesada, The Journal of Chemical Physics 150, 164113 (2019)

    2019

  40. [48]

    Quesada, L

    N. Quesada, L. G. Helt, J. Izaac, J. M. Arrazola, R. Shahrokhshahi, C. R. Myers, and K. K. Sabapathy, Physical Review A 100, 022341 (2019)

    2019

  41. [49]

    Barvinok, Random Structures & Algorithms 14, 29 (1999)

    A. Barvinok, Random Structures & Algorithms 14, 29 (1999)

    1999

  42. [50]

    Rudelson, A

    M. Rudelson, A. Samorodnitsky, O. Zeitouni, et al., The Annals of Probability 44, 2858 (2016)

    2016

  43. [51]

    T. R. Bromley, J. M. Arrazola, S. Jahangiri, J. Izaac, N. Quesada, A. D. Gran, M. Schuld, J. Swinarton, Z. Zabaneh, and N. Killoran, arXiv:1912.07634 (2019)

    2019 arXiv

  44. [52]

    R. H. Hadfield, Nature photonics 3, 696 (2009)

    2009

  45. [53]

    Simon, N

    R. Simon, N. Mukunda, and B. Dutta, Physical Review A 49, 1567 (1994)

    1994

  46. [54]

    Poole and T

    G. Poole and T. Boullion, SIAM review 16, 419 (1974)

    1974

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.