Pith. sign in

REVIEW 4 major objections 4 minor 54 references

Coherently mitigating boson samplers with stochastic errors

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Stochastic fabrication errors in boson samplers can be averaged away: as the number of coherently averaged noisy unitaries grows, the output distribution converges to the ideal one.

desk verdict A real generalization of Arkhipov's robustness bound to vacuum-heralded networks, with an honest application to boson sampling; the convergence claim is conditional on zero-mean noise, which the paper should flag more clearly. read the letter →

arxiv 2505.00102 v1 pith:PFOH6TMM submitted 2025-04-30 quant-ph

Deepesh Singh , Ryan J. Marshman , Nathan Walk , Jens Eisert , Timothy C. Ralph , Austin P. Lund This is my paper · ORCID
classification quant-ph
keywords bosonsamplingquantumerrormitigationunitaryaveragingstochasticerrorsvacuumheraldingSchur-Weyldualitylinearopticalcomputingtracedistancebound
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

Boson samplers are vulnerable to random fabrication errors that make each implemented interferometer deviate from the intended unitary. This paper claims those stochastic deviations can be mitigated by running $N$ noisy copies of the device inside a passive linear-optical network that coherently averages their unitaries and heralds on vacuum auxiliary modes. It proves that the total-variation distance between the averaged sampler's output distribution and the ideal boson sampling distribution is bounded by a quantity that vanishes as $N\to\infty$, provided the per-beam-splitter noise is unbiased and independent. If correct, this gives a concrete error-mitigation strategy for sampling experiments, producing an error-suppressed quantum state rather than a classical estimate, at the price of an exponentially small success probability.

What carries the argument

The engine is the unitary averaging framework: $N$ copies of the noisy $m$-mode interferometer $U_j$ are interleaved with $N$-dimensional discrete Fourier transforms, encoding each input mode into $N$ modes, and after propagation the $N-1$ auxiliary modes per input are heralded onto vacuum. This passively implements the generally non-unitary averaged operator $U_{\mathrm{avg}}=(1/N)\sum_j U_j$ on the original modes with probability $p_{\mathrm{post}}$. The proof ingredient is a Schur-Weyl duality bound: the $n$-boson homomorphism $\phi$ satisfies $\|\phi(A)-\phi(B)\| \le n k^{n-1}\|A-B\|$ because the symmetric-subspace norm is controlled by the full tensor-power norm, and the total-variation distance between distributions is bounded by the norm distance between normalized output states. Theorem 1 then converts this into the trace-distance bound involving $A p_A^{-1/2n}$ and $B p_B^{-1/2n}$. The noise model enters only through $\mathbb{E}(U_i)=(1-\nu/2)^d U$, which is what makes the infinite-$N$ average coincide with the target after renormalization.

What would settle it

Fabricate many copies of the same interferometer with controlled per-beam-splitter parameter noise, characterize each noisy unitary by full process tomography, and form the empirical mean $(1/N)\sum_j U_j$; if the renormalized mean is not proportional to the target unitary as $N$ grows, or if the total-variation distance between the averaged and ideal distributions does not decrease with increasing $N$ at fixed noise variance, the central convergence claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 1: for any two invertible vacuum-heralded linear transformations $A,B\in\mathrm{GL}(m,\mathbb{C})$ with success probabilities $p_A,p_B>0$, the total-variation distance between the corresponding $n$-photon output distributions obeys $\|D_A-D_B\| \le n k^{n-1}\|A p_A^{-1/2n} - B p_B^{-1/2n}\|_{\mathrm{op}}$, where $k$ bounds the normalized operator norms. Applying this to $A=U$ and $B=U_{\mathrm{avg}}=(1/N)\sum_j U_j$, the unitary-averaging network's effective transformation, and using the noise model $\mathbb{E}(U_i)=(1-\nu/2)^d U$, the paper concludes that in the limit $N\to\infty$ the bound vanishes: the mitigated distribution converges to the ideal boson sampling distribution. For finite $N$ the theorem supplies a quantitative rate in terms of the distance between the target and the averaged unitary, scaled by the photon number and the normalization factor $k^{n-1}$. The cost is that this is a probabilistic protocol: under a uniform-depth implementation the success probability is lower-bounded by $(1-\nu/2)^{2dn}$, decaying exponentially in photon number and interferometer depth.

Load-bearing premise

The entire convergence argument rests on the hardware noise being zero-mean, independent, and identically distributed across beam splitter parameters; if fabrication errors carry a systematic bias or correlations, the averaged unitary converges to the wrong interferometer and the mitigated distribution misses the ideal one.

Editorial extensions

If this is right

  • The protocol turns stochastic unitary errors into heralded loss while outputting a genuine error-suppressed quantum state, so the improved state can be reused in further computation rather than only yielding expectation values.
  • The success probability is at least $(1-\nu/2)^{2dn}$ in the uniform-depth implementation; for small per-element noise this leaves an intermediate-scale regime where the method is practical despite the exponential cost.
  • Theorem 1 extends the known robustness bound for unitary-only interferometers to vacuum-heralded networks, quantifying how close the output distributions of any two invertible heralded interferometers are.
  • By choosing the encoding and decoding matrices, the same network realizes linear combinations of unitaries with passive optics and vacuum heralding, which also covers any $m\times m$ matrix as a combination of at most four unitaries.
  • With $N$ nominally identical chips, the protocol is a fabrication-repeatability witness: identical chips leave the ancilla modes in vacuum, while any chip-to-chip discrepancy produces nonzero ancilla photon counts.

Reading between the lines

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

  • If fabrication drift is systematic rather than zero-mean, a modified protocol that first estimates the biased mean unitary, or that calibrates the noise before averaging, would be needed; the unbiased-noise premise is what makes $U_{\mathrm{avg}}$ point at $U$.
  • The $\mathrm{GL}(m,\mathbb{C})$ invertibility assumption, needed for the Schur-Weyl decomposition, is likely a technical rather than physical restriction: small generic perturbations of an invertible target remain invertible, so the bound should extend by continuity to near-non-invertible cases.
  • A finite-$N$ version of the bound could serve as a manufacturing acceptance test: compare the operator norm of $U_{\mathrm{avg}}/p_{\mathrm{post}}^{1/2n}$ to $U$ to certify chips before running a full boson sampling experiment.
  • Where particle loss rather than parameter noise dominates, combining unitary averaging with loss-tolerant encodings could yield a combined mitigation bound; the paper notes loss is easier to detect but does not prove such a combined result.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a unitary averaging protocol for mitigating stochastic unitary errors in boson sampling. The main theoretical result is Theorem 1, an upper bound on the total variation distance between the output distributions of two invertible vacuum-heralded linear optical transformations, derived using Schur-Weyl duality. The authors apply this bound to the averaged unitary U_avg = (1/N)Σ U_i and claim that in the N→∞ limit the output distribution converges to the ideal boson sampling distribution, with a success probability lower-bounded by an expression exponential in the photon number and interferometer depth. The paper also discusses applications to linear combinations of unitaries and benchmarking fabrication repeatability.

Significance. If the convergence claim can be made rigorous, the work is significant: it extends quantum error mitigation from expectation-value estimation to sampling tasks, provides a quantitative distance bound for vacuum-heralded linear optical networks, and offers connections to LCU and photonic quantum computing. The proof of Theorem 1 appears largely sound, and the numerical demonstrations support the qualitative behavior. However, the central convergence statement currently relies on treating a second-order Taylor approximation as an exact proportionality, and there are inconsistencies in the application of Theorem 1. These issues are fixable, and the paper has the potential to be a useful contribution after revision.

major comments (4)
  1. [Section III, Eqs. (3.7)-(3.9); Section IV.B, Eq. (4.12)] The convergence claim in Eq. (4.12) is not justified as written. Equation (3.7) is a second-order Taylor approximation yielding E(y) ≈ (1−ν/2)y0, and Eq. (3.9) inherits this approximation. In Section IV.B this approximate proportionality is treated as exact: Eq. (4.8) uses (1−ν/2)^{dn} and p_uni = (1−ν/2)^{2dn}, and Eq. (4.12) concludes the TVD is ≤ 0. For Gaussian noise the exact expectation is E[cos(θ+δ)] = cosθ e^{−ν/2}, with analogous expressions for sin and e^{iφ}, so the exact proportionality is E(U_i) = e^{−νd/2} U and the associated success probability is e^{−νdn}. With the stated p_uni, the normalized averaged unitary converges to (e^{−νd/2}/(1−ν/2)^d) U, leaving an O(ν²) residual, so the limit in Eq. (4.12) is not zero. Please either use exact Gaussian moments throughout (with p_uni = e^{−νdn}) or explicitly state that the convergence is up to O(ν²) and adjust Eq. (4.12) accordingly.
  2. [Section IV.B, Eq. (4.11)] The constant k in Eq. (4.11) is defined as max(∥U∥, ∥U_avg p_post^{−1/2n}∥), but Theorem 1 requires k to be the maximum of the norms of the two normalized matrices appearing in the difference, namely max(∥U∥, ∥U_avg p_uni^{−1/2n}∥) when B is normalized by p_uni. Since p_post ≥ p_uni, we have p_post^{−1/2n} ≤ p_uni^{−1/2n}, so the defined k can be smaller than the required constant and the stated bound may fail. The normalizations p_uni and p_post should be used consistently throughout the application of the theorem.
  3. [Section III, Eq. (3.4); Section VII.C] The mitigation result depends critically on the zero-mean assumption E(δθ)=E(δφ)=0 in Eq. (3.4). If fabrication noise contains a systematic component, Eq. (3.9) fails and the averaged unitary converges to a biased matrix that the scalar normalization p_uni cannot correct; the TVD in Eq. (4.12) would then tend to a positive value, not zero. Section VII.C itself lists systematic deviations and process drift as sources of fabrication variability, so the paper should explicitly state this limitation and discuss whether a calibration or bias-estimation step is needed before the protocol can be applied to real devices.
  4. [Section IV.B and Theorem 1] Theorem 1 requires A,B ∈ GL(m,C), and the paper assumes U_avg is invertible for finite N without proof. This is not automatic for an average of unitaries. If U_avg is singular, the theorem does not apply and the convergence conclusion is not established. The authors should either condition the statement on invertibility of U_avg or argue that the singular set has measure zero under the continuous Gaussian noise model, so that the assumption holds with probability one.
minor comments (4)
  1. [Section IV.B, Eq. (4.12)] Since a norm is nonnegative, the inequality ∥D_U − D_Uavg∥ ≤ 0 is equivalent to equality; the text should say '= 0' rather than '≤ 0'.
  2. [Section VII.A] The complexity expression 'O(log(n)n32n)' appears to be a typesetting error; it should presumably be O(log(n) n^3 2^n) or similar. Please check the formula.
  3. [Section V.B, Eq. (5.12)] The norm signs in Eq. (5.12) appear garbled in the manuscript; please ensure the displayed equation is typeset correctly.
  4. [Section III, Eq. (3.4)] The phrase 'Without loss of generality' before specifying Gaussian noise is misleading, because the exact form of Eq. (3.9) depends on the Gaussian moment structure; higher moments of a general zero-mean distribution would enter at O(ν²). This is related to the first major comment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: convergence follows from the stated zero-mean noise model and the self-contained Schur-Weyl bound, not from fitting or a self-citation chain.

full rationale

The paper's central convergence claim is a conditional mathematical consequence of its explicit noise model. Eq. (3.9), E(U_i) = (1 - nu/2)^d U, is derived by Taylor expansion of the i.i.d. zero-mean Gaussian perturbations in Eq. (3.4); Eq. (4.9) fixes p_uni by normalization; substituting these into the Theorem 1 bound (Eq. (4.10)) makes the right-hand side of Eq. (4.11) vanish in the N -> infinity limit, yielding Eq. (4.12). This is a derivation, not a fitted parameter renamed as a prediction: nu is an assumed hardware parameter, no data are fit, and the convergence is proven from stated assumptions. Theorem 1 itself is proved in Section V via Schur-Weyl duality and is a genuine generalization of Arkhipov's bound [35]; the proof is self-contained. The unitary averaging protocol is taken from self-cited prior work [15-18], but it appears as a building block with an explicit construction in the paper, and the new bound and boson-sampling application do not reduce to those citations. The well-known sensitivity to systematic fabrication bias is a limitation of the zero-mean noise model, not circularity, since the paper's own Sec. VII.C lists systematic deviations and process drift as sources. No circular step meeting the required evidence standard was found.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new physical entities. It relies on standard linear optics, the boson sampling formula, Schur-Weyl duality, and a specific model of zero-mean fabrication noise. The most fragile premises are the unbiased noise model and the noiseless encoding/decoding assumption.

free parameters (1)
  • noise variance ν = 0.01 in numerics
    Variance of the zero-mean Gaussian (or symmetric) noise on each beam-splitter parameter (Eq. (3.4)). It is an input to the error model, not fitted to data; the convergence claim holds for any small ν, but the success probability and the bound depend on it.
assumptions (7)
  • standard math Schur-Weyl duality gives the decomposition of V^⊗n into irreps of GL(m,C) and S_n, with the n-photon space as the symmetric subspace.
    Invoked in Sec. V A, Eq. (5.2)-(5.3), to bound ‖φ(A)-φ(B)‖ ≤ ‖A^⊗n - B^⊗n‖.
  • standard math The boson sampling probability for a linear transformation M is P_M(x) = |Perm(M_x)|^2 / (∏ x_i!).
    Used throughout, Eq. (2.1)-(2.2), following Aaronson-Arkhipov [23].
  • standard math Any m-mode unitary can be decomposed into beam splitters and phase shifters via Reck or Clements decompositions.
    Underlies the error model in Sec. III, Eq. (3.1)-(3.5), with the uniform-depth assumption.
  • domain assumption The noise on each beam-splitter parameter is i.i.d., zero-mean, and small, with E(δ^2)=ν and E(δ^3)=0.
    Eq. (3.4). This gives E(U_i) = (1-ν/2)^d U, the linchpin of the convergence claim. A biased or correlated noise process would break the argument.
  • domain assumption The encoding and decoding DFT networks in the UA protocol are noiseless, or their errors are suppressed to first order.
    Stated in Sec. VI, cited to the self-authored Ref. [17]; not derived in this paper. The convergence result assumes ideal encoding/decoding.
  • ad hoc to paper The average matrix U_avg = (1/N)∑ U_i is invertible and belongs to GL(m,C).
    Assumed in Sec. IV B ('The requirement of invertibility...') to apply Schur-Weyl; the average of N unitaries can be singular for finite N, and the paper does not condition on the invertible event.
  • standard math The sample mean of i.i.d. bounded random matrices converges almost surely to the expectation.
    Implicit in Sec. IV B where U_avg → E(U_i) as N→∞, used to conclude convergence of the bound to zero.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Coherently mitigating boson samplers with stochastic errors." pith.science (2026). https://pith.science/paper/PFOH6TMM

@misc{pith2026250500102,
  author       = {Pith},
  title        = {Pith review of: Coherently mitigating boson samplers with stochastic errors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFOH6TMM}},
  note         = {Machine review of arXiv:2505.00102}
}
read the original abstract

Sampling experiments provide a viable route to show quantum advantages of quantum devices over classical computers in well-defined computational tasks. However, quantum devices such as boson samplers are susceptible to various errors, including stochastic errors due to fabrication imperfections. These cause the implemented unitary operations to deviate randomly from their intended targets, following distributions with finite variance. Whilst full-scale quantum error correction remains challenging in the near term, quantum error mitigation schemes have been devised to estimate expectation values, but it is unclear how these schemes would work for sampling experiments. In this work, we demonstrate that, given access to multiple stochastic unitaries, it is possible to mitigate the effect of these errors in sampling experiments. We adopt the unitary averaging protocol which employs multiple stochastic boson samplers to generate a distribution that approximates the ideal boson sampler distribution as the number of samplers increases. We derive a rigorous upper bound on the trace distance between the output probability distributions induced by invertible vacuum-heralded networks based on the Schur-Weyl duality. This result can be seen concretely as an error mitigation scheme in sampling experiments against stochastic errors. On a broader level, it suggests a path towards understanding error mitigation for sampling experiments and developing analysis tools for photonic circuits incorporating measurements and feed-forward. We further provide other applications of unitary averaging, including its use in implementing the linear combination of unitaries and benchmarking fabrication repeatability in linear optics.

Figures

Figures reproduced from arXiv: 2505.00102 by the authors.

Figure 1
Figure 1. Implementation of an averaged unitary action on 4 input modes. Each original input mode is encoded with N − 1 vacuum modes using N-dimensional DFT operators. The setup uses N redun￾dant copies of the m-mode unitary Ui, where 1 ≤ i ≤ N, followed by decoding with DFTs and vacuum heralding on the auxiliary modes. The modes are colour-coded to indicate which unitary they belong to, as described in Ref. [15]. A. Distribu… view at source ↗
Figure 2
Figure 2. The success probability of the unitary averaging protocol in the limit N → ∞ under the uniform depth assumption (puni) is shown as a function of the depth (d) and the number of photons (n) in a boson sampler. The variance of each tunable element of the interferometer implementing the boson sampler is fixed at ν = 0.01. Notably, puni serves as a lower bound for the true success probability, ppost, of the unitary aver… view at source ↗
Figure 3
Figure 3. Schematic representation of the quantum circuits con￾sidered in Theorem 1. (a) Circuit implementing linear transforma￾tion A ∈ GL(m, C) with unitary U and s vacuum auxiliary modes such that ∥A∥≤ 1. (b) Circuit implementing linear transformation B ∈ GL(m, C) with unitary V and t vacuum auxiliary modes such that ∥B∥≤ 1. |ψin⟩ represents the fixed input state vector in both cir￾cuits. The vacuum auxiliary modes (|0⟩ ⊗s… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparison of distribution averaging and unitary averaging protocols for noisy linear networks, averaged over 300 Monte Carlo runs. The setup consists of two single-photon inputs to a 2-mode unitary. (a) Average total variation distance (TVD) vs. the number of redundan…
Figure 5
Figure 5. Figure 5: This figure illustrates the hierarchy of quantum state prepa￾ration capabilities in linear optics. At its core, basic linear optics allows for SU(m) transformations in an m-mode interferometer. The next level, achieved by incorporating vacuum heralding on ancillary mod…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 34 canonical work pages

  1. [17]

    R. J. Marshman, D. Singh, T. C. Ralph, and A. P. Lund, Uni- tary averaging with fault and loss tolerance, Phys. Rev. A109, 062436 (2024)

    2024

  2. [1]

    Flamini, N

    F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: a review, Rep. Prog. Phys. 82, 016001 (2018)

    2018

  3. [2]

    P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear optical quantum computing with photonic qubits, Rev. Mod. Phys. 79, 135 (2007)

    2007

  4. [3]

    J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Integrated photonic quantum technologies, Nature Phot. 14, 273 (2020)

    work page 2020

  5. [4]

    Bartolucci, P

    S. Bartolucci, P. Birchall, H. Bombín, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow, Fusion- based quantum computation, Nature Comm. 14, 912 (2023)

    work page 2023

  6. [5]

    L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vin- cent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V . D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie, Quantum computational advantage with a programmable pho- tonic processor, Nature 606, 75–81 (2022)

    work page 2022

  7. [6]

    F. H. B. Somhorst, R. van der Meer, M. C. Anguita, R. Schadow, H. J. Snijders, M. de Goede, B. Kassenberg, P. Venderbosch, C. Taballione, J. P. Epping, H. H. van den Vlekkert, J. F. F. Bulmer, J. Lugani, I. A. Walmsley, P. W. H. Pinkse, J. Eisert, N. Walk, and J. J. Renema, Quantum photo-thermodynamics on a programmable photonic quantum processor, Nature ...

    work page 2023

  8. [7]

    Pirandola, J

    S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, Advances in quantum teleportation, Nature Phot. 9, 641 (2015)

    work page 2015

Show all 54 references
  1. [8]

    Burgwal, W

    R. Burgwal, W. R. Clements, D. H. Smith, J. C. Gates, W. S. Kolthammer, J. J. Renema, and I. A. Walmsley, Using an imper- fect photonic network to implement random unitaries, Opt. Expr. 25, 28236 (2017)

    2017

  2. [9]

    N. J. Russell, L. Chakhmakhchyan, J. L. O’Brien, and A. Laing, Direct dialling of haar random unitary matrices, New J. Phys. 19, 033007 (2017)

    2017

  3. [10]

    Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y . Li, J. R. McClean, and T. E. O’Brien, Quantum error mitiga- tion, Rev. Mod. Phys. 95, 045005 (2023)

    2023

  4. [11]

    Li and S

    Y . Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X 7, 021050 (2017), 1611.09301

    2017 arXiv

  5. [12]

    Temme, S

    K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Phys. Rev. Lett. 119, 180509 (2017), 1612.02058

    2017 arXiv

  6. [13]

    S. Endo, S. C. Benjamin, and Y . Li, Practical quantum error mitigation for near-future applications, Phys. Rev. X 8, 031027 (2018)

    2018

  7. [14]

    Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y . Li, J. R. McClean, and T. E. O’Brien, Quantum error mitiga- tion, Rev. Mod. Phys. 95, 045005 (2023), 2210.00921

    2023 arXiv

  8. [15]

    Singh, A

    D. Singh, A. P. Lund, and P. P. Rohde, Optical cluster-state generation with unitary averaging, Phys. Rev. A 110, 012457 (2024)

    2024

  9. [16]

    R. J. Marshman, A. P. Lund, P. P. Rohde, and T. C. Ralph, Passive quantum error correction of linear optics networks through error averaging, Phys. Rev. A 97, 022324 (2018)

    2018

  10. [18]

    S. N. Swain, R. J. Marshman, P. P. Rohde, A. P. Lund, A. S. Solntsev, and T. C. Ralph, Improving continuous-variable quan- tum channels with unitary averaging, Phys. Rev. A 110, 032622 (2024)

    2024

  11. [19]

    This approach leverages multiple stochastic unitaries to probabilistically generate improved quantum states without correcting for all error syndromes

    and exponential error suppression [20], quantum error mit- igation with classical shadows [21], and to some extent, with mitigated readout schemes [22], as they all involve the coherent manipulation of quantum states comprising multiple copies of a quantum circuit. This approa...

    2025 arXiv

  12. [20]

    W. J. Huggins, S. McArdle, T. E. O’Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Vir- tual distillation for quantum error mitigation, Phys. Rev. X11, 041036 (2021)

    2021

  13. [21]

    Koczor, Exponential error suppression for near-term quantum devices, Phys

    B. Koczor, Exponential error suppression for near-term quantum devices, Phys. Rev. X 11, 031057 (2021)

    2021

  14. [22]

    Seif, Z.-P

    A. Seif, Z.-P. Cian, S. Zhou, S. Chen, and L. Jiang, Shadow distillation: Quantum error mitigation with classical shadows for near-term quantum processors, PRX Quantum 4, 010303 (2023)

    2023

  15. [23]

    Onorati, J

    E. Onorati, J. Kitzinger, J. Helsen, M. Ioannou, A. H. Werner, I. Roth, and J. Eisert, Noise-mitigated randomized measurements and self-calibrating shadow estimation (2024), arXiv:2403.04751

    2024 arXiv

  16. [24]

    Aaronson and A

    S. Aaronson and A. Arkhipov, The computational complexity of linear optics, Th. Comp. 9, 143 (2013)

    2013

  17. [25]

    Hangleiter and J

    D. Hangleiter and J. Eisert, Computational advantage of quantum random sampling, Rev. Mod. Phys. 95, 035001 (2023)

    2023

  18. [26]

    Takagi, S

    R. Takagi, S. Endo, S. Minagawa, and M. Gu, Fundamental limits of quantum error mitigation, npj Quant. Inf. 8, 114 (2022), arXiv:2210.11505

    2022 arXiv

  19. [27]

    Y . Quek, D. S. França, S. Khatri, J. J. Meyer, and J. Eisert, Exponentially tighter bounds on limitations of quantum error mitigation, Nature Phys. 20, 1 (2024)

    2024

  20. [28]

    Zimborás, B

    Z. Zimborás, B. Koczor, Z. Holmes, E.-M. Borrelli, A. Gilyén, H.-Y . Huang, Z. Cai, A. Acín, L. Aolita, L. Banchi, F. G. S. L. Brandão, D. Cavalcanti, T. Cubitt, S. N. Filippov, G. García- Pérez, J. Goold, O. Kálmán, E. Kyoseva, M. A. C. Rossi, B. Sokolov, I. Tavernelli, and S...

    2025 arXiv

  21. [29]

    Knill, R

    E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001)

    2001

  22. [30]

    D. E. Browne and T. Rudolph, Resource-efficient linear optical quantum computation, Phys. Rev. Lett. 95, 010501 (2005)

    2005

  23. [31]

    C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, Noise thresholds for optical quantum computers, Phys. Rev. Lett. 96, 020501 (2006)

    2006

  24. [32]

    Gross, K

    D. Gross, K. Kieling, and J. Eisert, Potential and limits to cluster- state quantum computing using probabilistic gates, Phys. Rev. A 74, 042343 (2006)

    2006

  25. [33]

    A. M. Childs and N. Wiebe, Hamiltonian simulation using lin- ear combinations of unitary operations, Quant. Inf. Comp. 12, 901–924 (2012)

    2012

  26. [34]

    Y . Cai, Y . Tong, and J. Preskill, Stochastic error cancellation in analog quantum simulation, in 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024), Leibniz International Proceedings in Informatics (LIPIcs), V ol. 310, edited by F...

    2024

  27. [35]

    M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, 10th ed. (Cambridge University Press, Cambridge, 2010)

    2010

  28. [36]

    Arkhipov, BosonSampling is robust against small errors in the network matrix, Phys

    A. Arkhipov, BosonSampling is robust against small errors in the network matrix, Phys. Rev. A 92, 062326 (2015)

    2015

  29. [37]

    M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experi- mental realization of any discrete unitary operator, Phys. Rev. Lett. 73, 58 (1994). 12

    1994

  30. [38]

    W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Koltham- mer, and I. A. Walmsley, Optimal design for universal multiport interferometers, Optica 3, 1460 (2016)

    2016

  31. [39]

    Wassner, T

    C. Wassner, T. Guaita, J. Eisert, and J. Carrasco, Holonomic quantum computation: A scalable adiabatic architecture (2025), arXiv:2502.17188

    2025 arXiv

  32. [40]

    A. P. Lund, Estimating Fock-state linear optics evolution using coherent states, A VS Quant. Sc.5, 011405 (2023)

    2023

  33. [41]

    Alexander, A

    K. Alexander, A. Benyamini, D. Black, D. Bonneau, S. Bur- gos, B. Burridge, H. Cable, G. Campbell, G. Catalano, A. Ce- ballos, C.-M. Chang, S. S. Choudhury, C. Chung, F. Danesh, T. Dauer, M. Davis, E. Dudley, P. Er-Xuan, J. Fargas, A. Farsi, C. Fenrich, J. Frazer, M. Fukami, Y...

    2025

  34. [42]

    Somhorst, B

    F. Somhorst, B. Sauër, S. van den Hoven, and J. Renema, Photon- distillation schemes with reduced resource costs based on multi- photon fourier interference, Phys. Rev. Appl. 23, 044003 (2025)

    2025

  35. [43]

    T. C. Ralph, A. J. F. Hayes, and A. Gilchrist, Loss-tolerant optical qubits, Phys. Rev. Lett. 95, 100501 (2005)

    2005

  36. [44]

    A. J. F. Hayes, A. Gilchrist, and T. C. Ralph, Loss-tolerant operations in parity-code linear optics quantum computing, Phys. Rev. A 77, 012310 (2008)

    2008

  37. [45]

    H. J. Ryser, Combinatorial Mathematics, 1st ed., V ol. 14 (Math- ematical Association of America, 1963)

    1963

  38. [46]

    Seron, L

    B. Seron, L. Novo, A. Arkhipov, and N. J. Cerf, Efficient valida- tion of boson sampling from binned photon-number distributions, Quantum 8, 1479 (2024)

    2024

  39. [47]

    A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, and D. A. Mazziotti, Quantum simulation of open quantum systems using a unitary decomposition of operators, Phys. Rev. Lett.127, 270503 (2021)

    2021

  40. [48]

    Gilyén, Y

    A. Gilyén, Y . Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 (Association for Computing Machinery,...

    2019

  41. [49]

    Saghaei, P

    H. Saghaei, P. Elyasi, and R. Karimzadeh, Design, fabrication, and characterization of Mach–Zehnder interferometers, Phot. Nanostruct. Fund. Appl. 37, 100733 (2019)

    2019

  42. [50]

    Rahimi-Keshari, M

    S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, T. C. Ralph, and A. G. White, Direct characterization of linear-optical networks, Opt. Express 21, 13450 (2013)

    2013

  43. [51]

    Laing and J

    A. Laing and J. L. O’Brien, Super-stable tomography of any linear optical device (2012), arXiv:1208.2868

    2012 arXiv

  44. [52]

    F. Hoch, T. Giordani, N. Spagnolo, A. Crespi, R. Osellame, and F. Sciarrino, Characterization of multimode linear optical networks, Adv. Phot. Nex. 2, 016007 (2023), 2304.06486

    2023 arXiv

  45. [53]

    Fyrillas, O

    A. Fyrillas, O. Faure, N. Maring, J. Senellart, and N. Belabas, Scalable machine learning-assisted clear-box characterization for optimally controlled photonic circuits, Optica 11, 427 (2024)

    2024

  46. [54]

    G. Park, I. Matsumoto, T. Kiyohara, H. F. Hofmann, R. Okamoto, and S. Takeuchi, Realization of photon correlations beyond the linear optics limit, Science Adv. 9, eadj8146 (2023)

    2023

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.