Kostant relation in filtered randomized benchmarking for passive bosonic devices

David Amaro-Alcal\'a

arxiv: 2511.00842 · v2 · submitted 2025-11-02 · 🪐 quant-ph

Kostant relation in filtered randomized benchmarking for passive bosonic devices

David Amaro-Alcal\'a This is my paper

Pith reviewed 2026-05-18 01:54 UTC · model grok-4.3

classification 🪐 quant-ph

keywords randomized benchmarkingbosonic devicesfilter functionsKostant relationscharacter filtersphoton lossquantum characterizationimmanants

0 comments

The pith

Character filters from Kostant relations cut computational cost and keep variance constant in bosonic randomized benchmarking.

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

The paper introduces filter functions based on immanants and characters of the special unitary group to simplify randomized benchmarking for passive bosonic devices. These filters avoid computing Clebsch-Gordan coefficients and give simple closed-form variance expressions. The character filter in particular stays efficient and maintains constant low variance. A reader would care because the filters work on exactly the same measurement data as earlier protocols, and numerical checks show that weak coherent states plus intensity measurements still give reliable estimates even when photon loss or gain occurs.

Core claim

We introduce two filter functions for bosonic randomized benchmarking: one using immanants and the other using characters of the special unitary group. These avoid Clebsch-Gordan coefficients and produce simple variance expressions. The character filter is efficient to compute and exhibits constant low variance. The filters operate on the same data as the original proposal. Numerical evidence indicates that weak coherent states combined with intensity measurements yield estimates close to the ideal case without loss or gain.

What carries the argument

The character filter derived via the Kostant relation for the special unitary group, which isolates the relevant representation content while delivering constant low variance without extra coefficients.

If this is right

The filters require no new experimental data beyond what the original bosonic randomized benchmarking already collects.
The character filter variance stays low and independent of certain input parameters.
Weak coherent states and intensity measurements produce estimates close to the lossless case even when photon loss and gain are present.
Avoiding Clebsch-Gordan coefficients reduces the overall computational cost of the benchmarking protocol.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar character-based filters might simplify randomized benchmarking protocols for other continuous-variable or multi-mode systems.
The approach could reduce overhead in real-time device characterization on hardware platforms that already record intensity data.
Representation-theoretic identities like the Kostant relation may yield further simplifications when applied to noise characterization in quantum optics.

Load-bearing premise

The new filters can be applied directly to the same measurement data as the original bosonic randomized benchmarking proposal without introducing additional systematic errors from the filtering process itself.

What would settle it

An experiment on a passive bosonic device that applies the character filter to data with controlled photon loss and finds variance rising with mode number or deviating from the predicted constant low value would falsify the central claim.

read the original abstract

We aim to reduce the cost of the current bosonic randomized benchmarking proposal. To do this, we introduce two filter functions: one uses immanants, the other uses characters of the special unitary group. These filters avoid computing Clebsch-Gordan coefficients and yield simple variance expressions. The character filter is not only efficient to compute, but also has a constant, low variance. Our filters rely on the same data as the original proposal. We also discuss an example with photon loss and gain. Our numerical evidence shows that a scheme using weak coherent states and intensity measurements can yield estimates close to those obtained without loss or gain. Our work could support simpler platform characterization and streamline data analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Character and immanant filters cut the cost of bosonic RB by skipping Clebsch-Gordan coefficients, but the loss/gain claims need explicit checks that the post-processing stays unbiased.

read the letter

The main takeaway is that this paper introduces two filters—one based on immanants, the other on characters of the special unitary group—to simplify randomized benchmarking for passive bosonic devices. The character filter is presented as especially useful because it is cheap to compute and delivers a constant, low variance while working on the same intensity measurement data as the original protocol. It also applies the Kostant relation to avoid Clebsch-Gordan coefficients and produce simple variance expressions. That is the concrete advance over prior bosonic RB work. The numerical example with weak coherent states under photon loss and gain is a practical touch, showing estimates that stay reasonably close to the ideal case. This matters for real devices where some loss is unavoidable. The approach keeps the experimental overhead unchanged, which is a clear plus for experimental groups. The softer part is the handling of non-unitary channels. The central claim that the filtered estimator remains unbiased and keeps constant variance rests on the filter commuting appropriately with loss and gain. The abstract gives numerical support, but without the full derivations or explicit checks against Fock truncation or POVM details, it is not yet clear whether hidden bias or variance inflation appears. If those assumptions hold, the method is solid; if not, the “close estimates” result weakens. This is aimed at theorists and experimentalists working on bosonic quantum optics and hardware characterization. A reader who already knows standard RB and wants a lighter computational path for bosonic modes will find the filters directly useful. I would send it to peer review. The contribution is narrow and targeted, the numerics are there, and referees can verify the representation-theory steps and the loss/gain preservation in one round.

Referee Report

2 major / 2 minor

Summary. The paper proposes two new filter functions—one based on immanants and one on characters of the special unitary group—derived via the Kostant relation to reduce the computational cost of bosonic randomized benchmarking for passive devices. These filters avoid explicit Clebsch-Gordan coefficients, rely on the same measurement data as prior bosonic RB protocols, and are claimed to produce simple variance expressions, with the character filter having constant low variance. The work includes a discussion and numerical example of photon loss and gain, showing that weak coherent states combined with intensity measurements can produce estimates close to the lossless/gainless case.

Significance. If the derivations and numerical results hold, the filters could meaningfully lower the barrier to practical bosonic device characterization by enabling simpler post-processing of existing intensity data while maintaining low variance even under loss/gain. The explicit use of representation-theoretic tools (immanants/characters) to bypass Clebsch-Gordan coefficients is a constructive technical contribution if the variance claims are fully substantiated.

major comments (2)

[Abstract / character filter section] Abstract and the section presenting the character filter: the central claim that this filter 'has a constant, low variance' and yields 'simple variance expressions' is load-bearing for the efficiency argument, yet the manuscript provides no explicit variance formula, derivation steps, or proof that the variance is independent of sequence length. Without these, it is impossible to confirm the claimed constancy or to check for hidden parameter dependence.
[Numerical evidence / loss-gain example] The numerical evidence paragraph on photon loss and gain: the statement that weak coherent states and intensity measurements 'can yield estimates close to those obtained without loss or gain' is presented as supporting applicability of the filters to non-unitary channels, but no quantitative error bars, bias analysis, or demonstration that the Kostant-derived post-processing preserves the expectation value under the loss/gain map are supplied. This leaves open whether the filter introduces additional systematic error when the underlying SU representation interacts with the non-unitary POVM.

minor comments (2)

Clarify the precise definition of the intensity measurement POVM and the Fock-space truncation used in the numerical example so that readers can reproduce the 'close estimates' result.
Add a short comparison table or plot contrasting the computational cost (e.g., number of Clebsch-Gordan evaluations avoided) between the new filters and the original bosonic RB protocol.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the changes we will make in revision.

read point-by-point responses

Referee: [Abstract / character filter section] Abstract and the section presenting the character filter: the central claim that this filter 'has a constant, low variance' and yields 'simple variance expressions' is load-bearing for the efficiency argument, yet the manuscript provides no explicit variance formula, derivation steps, or proof that the variance is independent of sequence length. Without these, it is impossible to confirm the claimed constancy or to check for hidden parameter dependence.

Authors: We acknowledge that the explicit variance formula, derivation steps, and demonstration of independence from sequence length were not presented with sufficient detail in the main text. The character filter variance follows from the orthogonality of irreducible characters and the Kostant relation, which yields an expression depending only on the representation dimension and the noise parameters rather than sequence length. In the revised manuscript we will add a dedicated paragraph (or short appendix) that spells out the variance calculation step by step and proves its constancy under the protocol assumptions. This revision will directly substantiate the central claim. revision: yes
Referee: [Numerical evidence / loss-gain example] The numerical evidence paragraph on photon loss and gain: the statement that weak coherent states and intensity measurements 'can yield estimates close to those obtained without loss or gain' is presented as supporting applicability of the filters to non-unitary channels, but no quantitative error bars, bias analysis, or demonstration that the Kostant-derived post-processing preserves the expectation value under the loss/gain map are supplied. This leaves open whether the filter introduces additional systematic error when the underlying SU representation interacts with the non-unitary POVM.

Authors: The referee is correct that the current numerical example is qualitative and lacks quantitative error bars or an explicit bias analysis. We will revise the relevant paragraph to report standard deviations from repeated Monte Carlo runs and to include a brief argument that, for weak coherent states and intensity measurements, the linear character filter preserves the expectation value to first order in the loss/gain strength because the POVM acts diagonally in the photon-number basis. A complete bias analysis for arbitrary non-unitary maps lies beyond the scope of the present work; we will therefore add an explicit statement of this limitation while retaining the illustrative example. revision: partial

Circularity Check

0 steps flagged

No significant circularity; filters derived from independent representation-theoretic constructions

full rationale

The paper introduces immanant and character filters for bosonic randomized benchmarking via the Kostant relation, yielding simple variance expressions and constant low variance for the character filter. These are new mathematical post-processing steps applied to the same intensity measurement data as the original protocol. Numerical evidence for weak coherent states under loss/gain is presented as separate verification. No quoted step reduces a claimed prediction or variance result to a fitted parameter from the same data, nor does any load-bearing claim rest on a self-citation chain that is itself unverified; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard group theory constructs and the assumption that the original benchmarking data remains valid under filtering; no new physical entities or heavily fitted parameters are introduced in the abstract.

axioms (1)

domain assumption Filters can be applied to the same data as the original bosonic randomized benchmarking proposal.
Explicitly stated in the abstract as a key feature of the approach.

pith-pipeline@v0.9.0 · 5635 in / 1154 out tokens · 34855 ms · 2026-05-18T01:54:44.018398+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We propose an improved protocol that maintains the simplicity of the original method. Our method eliminates the need for photon-number-resolving or homodyne detection and removes the dependence on permanents and Clebsch-Gordan coefficients in the data analysis.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

Masada, K

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, Continuous-variable entan- glement on a chip, Nat. Photonics9, 316–319 (2015)

work page 2015
[2]

Takeda and A

S. Takeda and A. Furusawa, Universal quantum comput- ing with measurement-induced continuous-variable gate sequence in a loop-based architecture, Phys. Rev. Lett. 119, 120504 (2017)

work page 2017
[3]

Yonezu, Y

K. Yonezu, Y. Enomoto, T. Yoshida, and S. Takeda, Time-domain universal linear-optical operations for uni- versal quantum information processing, Phys. Rev. Lett. 131, 040601 (2023)

work page 2023
[4]

Fukui and S

K. Fukui and S. Takeda, Building a large-scale quantum computer with continuous-variable optical technologies, J. Phys. B: At. Mol. Opt. Phys.55, 012001 (2022)

work page 2022
[5]

Emerson, R

J. Emerson, R. Alicki, and K. ˙Zyczkowski, Scalable noise estimation with random unitary operators, J. Opt. B: Quantum Semiclassical Opt.7, S347–S352 (2005)

work page 2005
[6]

Muller, S

E. Magesan, J. M. Gambetta, and J. Emerson, Scal- able and robust randomized benchmarking of quan- tum processes, Phys. Rev. Lett.106, 10.1103/phys- revlett.106.180504 (2011)

work page doi:10.1103/phys- 2011
[7]

Randomized benchmarking of quantum gates,

E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quan- tum gates, Phys. Rev. A77, 10.1103/physreva.77.012307 (2008)

work page doi:10.1103/physreva.77.012307 2008
[8]

Amaro-Alcal´ a, B

D. Amaro-Alcal´ a, B. C. Sanders, and H. de Guise, Ran- domised benchmarking for universal qudit gates, New J. Phys.26, 073052 (2024)

work page 2024
[9]

Jafarzadeh, Y.-D

M. Jafarzadeh, Y.-D. Wu, Y. R. Sanders, and B. C. Sanders, Randomized benchmarking for qudit Clifford gates, New J. Phys.22, 063014 (2020)

work page 2020
[10]

Helsen, I

J. Helsen, I. Roth, E. Onorati, A. Werner, and J. Eisert, General framework for randomized benchmarking, PRX Quantum3, 020357 (2022)

work page 2022
[11]

Arienzo, D

M. Arienzo, D. Grinko, M. Kliesch, and M. Heinrich, Bosonic randomized benchmarking with passive trans- formations, PRX Quantum6, 020305 (2025)

work page 2025
[12]

Wilkens, M

J. Wilkens, M. Ioannou, E. Derbyshire, J. Eis- ert, D. Hangleiter, I. Roth, and J. Haferkamp, Benchmarking bosonic and fermionic dynamics (2024), arXiv:2408.11105 [quant-ph]

work page arXiv 2024
[13]

Valiant, The complexity of computing the permanent, Theor

L. Valiant, The complexity of computing the permanent, Theor. Comput. Sci.8, 189–201 (1979)

work page 1979
[14]

B¨ urgisser, The computational complexity of im- manants, SIAM J

P. B¨ urgisser, The computational complexity of im- manants, SIAM J. Comput.30, 1023–1040 (2000)

work page 2000
[15]

Kostant, Immanant inequalities and 0-weight spaces, J

B. Kostant, Immanant inequalities and 0-weight spaces, J. Am. Math. Soc.8, 181 (1995)

work page 1995
[16]

J. Lin, B. Buonacorsi, R. Laflamme, and J. J. Wallman, On the freedom in representing quantum operations, New J. Phys.21, 023006 (2019)

work page 2019
[17]

Fulton,Young Tableaux: With Applications to Repre- sentation Theory and Geometry(Cambridge University Press, 1996)

W. Fulton,Young Tableaux: With Applications to Repre- sentation Theory and Geometry(Cambridge University Press, 1996)

work page 1996
[18]

Amaro-Alcal´ a, D

D. Amaro-Alcal´ a, D. Spivak, and H. de Guise, Sum rules in multiphoton coincidence rates, Phys. Lett. A384, 126459 (2020)

work page 2020
[19]

Proctor, K

T. Proctor, K. Rudinger, K. Young,et al., What random- ized benchmarking actually measures, Phys. Rev. Lett. 119, 130502 (2017)

work page 2017
[20]

J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A94, 052325 (2016)

work page 2016
[21]

Hashim, R

A. Hashim, R. K. Naik, A. Morvan, J.-L. Ville, B. Mitchell, J. M. Kreikebaum, M. Davis, E. Smith, C. Iancu, K. P. O’Brien, I. Hincks, J. J. Wallman, J. Emerson, and I. Siddiqi, Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor, Phys. Rev. X11, 041039 (2021)

work page 2021
[22]

Raczka and A

R. Raczka and A. O. Barut,Theory of group representa- tions and applications(World Scientific Publishing Com- pany, 1986)

work page 1986
[23]

D. E. Littlewood,The theory of group characters and ma- trix representations of groups(American Mathematical, 1977)

work page 1977
[24]

wolframcloud.com/FunctionRepository/resources/ Immanant/, accessed: 2025-10-07

Wolfram, Immanant,https://resources. wolframcloud.com/FunctionRepository/resources/ Immanant/, accessed: 2025-10-07

work page 2025
[25]

de Guise, D

H. de Guise, D. Spivak, J. Kulp, and I. Dhand, D- functions and immanants of unitary matrices and sub- matrices, J. Phys. A: Math. Theor.49, 09LT01 (2016)

work page 2016
[26]

de Guise, O

H. de Guise, O. Di Matteo, and L. L. S´ anchez-Soto, Sim- ple factorization of unitary transformations, Phys. Rev. A97, 022328 (2018)

work page 2018
[27]

(A⊙B) i,j :=A i,jBi,j

work page
[28]

H. Wang, J. Qin, X. Ding, M.-C. Chen, S. Chen, X. You, Y.-M. He, X. Jiang, L. You, Z. Wang, C. Schneider, J. J. Renema, S. H¨ ofling, C.-Y. Lu, and J.-W. Pan, Boson sampling with 20 input photons and a 60-mode interfer- ometer in a 10 14-dimensional hilbert space, Phys. Rev. Lett.123, 250503 (2019)

work page 2019
[29]

A. Alex, M. Kalus, A. Huckleberry, and J. von Delft, A numerical algorithm for the explicit calculation of su(n) and SL(n,C)sl(n, c) clebsch–gordan coefficients, J. Phys. A: Math. Theor.52, 10.1063/1.3521562 (2011). 9 Appendix A: Example Kostant’s relation for SU(3) In this appendix, we illustrate several cases of the gen- eral result, labeled Kostant’s r...

work page doi:10.1063/1.3521562 2011

[1] [1]

Masada, K

G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, Continuous-variable entan- glement on a chip, Nat. Photonics9, 316–319 (2015)

work page 2015

[2] [2]

Takeda and A

S. Takeda and A. Furusawa, Universal quantum comput- ing with measurement-induced continuous-variable gate sequence in a loop-based architecture, Phys. Rev. Lett. 119, 120504 (2017)

work page 2017

[3] [3]

Yonezu, Y

K. Yonezu, Y. Enomoto, T. Yoshida, and S. Takeda, Time-domain universal linear-optical operations for uni- versal quantum information processing, Phys. Rev. Lett. 131, 040601 (2023)

work page 2023

[4] [4]

Fukui and S

K. Fukui and S. Takeda, Building a large-scale quantum computer with continuous-variable optical technologies, J. Phys. B: At. Mol. Opt. Phys.55, 012001 (2022)

work page 2022

[5] [5]

Emerson, R

J. Emerson, R. Alicki, and K. ˙Zyczkowski, Scalable noise estimation with random unitary operators, J. Opt. B: Quantum Semiclassical Opt.7, S347–S352 (2005)

work page 2005

[6] [6]

Muller, S

E. Magesan, J. M. Gambetta, and J. Emerson, Scal- able and robust randomized benchmarking of quan- tum processes, Phys. Rev. Lett.106, 10.1103/phys- revlett.106.180504 (2011)

work page doi:10.1103/phys- 2011

[7] [7]

Randomized benchmarking of quantum gates,

E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Randomized benchmarking of quan- tum gates, Phys. Rev. A77, 10.1103/physreva.77.012307 (2008)

work page doi:10.1103/physreva.77.012307 2008

[8] [8]

Amaro-Alcal´ a, B

D. Amaro-Alcal´ a, B. C. Sanders, and H. de Guise, Ran- domised benchmarking for universal qudit gates, New J. Phys.26, 073052 (2024)

work page 2024

[9] [9]

Jafarzadeh, Y.-D

M. Jafarzadeh, Y.-D. Wu, Y. R. Sanders, and B. C. Sanders, Randomized benchmarking for qudit Clifford gates, New J. Phys.22, 063014 (2020)

work page 2020

[10] [10]

Helsen, I

J. Helsen, I. Roth, E. Onorati, A. Werner, and J. Eisert, General framework for randomized benchmarking, PRX Quantum3, 020357 (2022)

work page 2022

[11] [11]

Arienzo, D

M. Arienzo, D. Grinko, M. Kliesch, and M. Heinrich, Bosonic randomized benchmarking with passive trans- formations, PRX Quantum6, 020305 (2025)

work page 2025

[12] [12]

Wilkens, M

J. Wilkens, M. Ioannou, E. Derbyshire, J. Eis- ert, D. Hangleiter, I. Roth, and J. Haferkamp, Benchmarking bosonic and fermionic dynamics (2024), arXiv:2408.11105 [quant-ph]

work page arXiv 2024

[13] [13]

Valiant, The complexity of computing the permanent, Theor

L. Valiant, The complexity of computing the permanent, Theor. Comput. Sci.8, 189–201 (1979)

work page 1979

[14] [14]

B¨ urgisser, The computational complexity of im- manants, SIAM J

P. B¨ urgisser, The computational complexity of im- manants, SIAM J. Comput.30, 1023–1040 (2000)

work page 2000

[15] [15]

Kostant, Immanant inequalities and 0-weight spaces, J

B. Kostant, Immanant inequalities and 0-weight spaces, J. Am. Math. Soc.8, 181 (1995)

work page 1995

[16] [16]

J. Lin, B. Buonacorsi, R. Laflamme, and J. J. Wallman, On the freedom in representing quantum operations, New J. Phys.21, 023006 (2019)

work page 2019

[17] [17]

Fulton,Young Tableaux: With Applications to Repre- sentation Theory and Geometry(Cambridge University Press, 1996)

W. Fulton,Young Tableaux: With Applications to Repre- sentation Theory and Geometry(Cambridge University Press, 1996)

work page 1996

[18] [18]

Amaro-Alcal´ a, D

D. Amaro-Alcal´ a, D. Spivak, and H. de Guise, Sum rules in multiphoton coincidence rates, Phys. Lett. A384, 126459 (2020)

work page 2020

[19] [19]

Proctor, K

T. Proctor, K. Rudinger, K. Young,et al., What random- ized benchmarking actually measures, Phys. Rev. Lett. 119, 130502 (2017)

work page 2017

[20] [20]

J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A94, 052325 (2016)

work page 2016

[21] [21]

Hashim, R

A. Hashim, R. K. Naik, A. Morvan, J.-L. Ville, B. Mitchell, J. M. Kreikebaum, M. Davis, E. Smith, C. Iancu, K. P. O’Brien, I. Hincks, J. J. Wallman, J. Emerson, and I. Siddiqi, Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor, Phys. Rev. X11, 041039 (2021)

work page 2021

[22] [22]

Raczka and A

R. Raczka and A. O. Barut,Theory of group representa- tions and applications(World Scientific Publishing Com- pany, 1986)

work page 1986

[23] [23]

D. E. Littlewood,The theory of group characters and ma- trix representations of groups(American Mathematical, 1977)

work page 1977

[24] [24]

wolframcloud.com/FunctionRepository/resources/ Immanant/, accessed: 2025-10-07

Wolfram, Immanant,https://resources. wolframcloud.com/FunctionRepository/resources/ Immanant/, accessed: 2025-10-07

work page 2025

[25] [25]

de Guise, D

H. de Guise, D. Spivak, J. Kulp, and I. Dhand, D- functions and immanants of unitary matrices and sub- matrices, J. Phys. A: Math. Theor.49, 09LT01 (2016)

work page 2016

[26] [26]

de Guise, O

H. de Guise, O. Di Matteo, and L. L. S´ anchez-Soto, Sim- ple factorization of unitary transformations, Phys. Rev. A97, 022328 (2018)

work page 2018

[27] [27]

(A⊙B) i,j :=A i,jBi,j

work page

[28] [28]

H. Wang, J. Qin, X. Ding, M.-C. Chen, S. Chen, X. You, Y.-M. He, X. Jiang, L. You, Z. Wang, C. Schneider, J. J. Renema, S. H¨ ofling, C.-Y. Lu, and J.-W. Pan, Boson sampling with 20 input photons and a 60-mode interfer- ometer in a 10 14-dimensional hilbert space, Phys. Rev. Lett.123, 250503 (2019)

work page 2019

[29] [29]

A. Alex, M. Kalus, A. Huckleberry, and J. von Delft, A numerical algorithm for the explicit calculation of su(n) and SL(n,C)sl(n, c) clebsch–gordan coefficients, J. Phys. A: Math. Theor.52, 10.1063/1.3521562 (2011). 9 Appendix A: Example Kostant’s relation for SU(3) In this appendix, we illustrate several cases of the gen- eral result, labeled Kostant’s r...

work page doi:10.1063/1.3521562 2011