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REVIEW 3 major objections 5 minor 29 references

Autocallable Options Pricing with Integration-Based Exponential Amplitude Loading

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that a refined integration-based exponential amplitude loading reduces the T-depth of the payoff-loading module in an autocallable pricing circuit from about 2.1e3 to about 40, a roughly 50x reduction, for a 20-step…

desk verdict A genuinely new partial-exponential amplitude-loading trick and transparent T-depth accounting, but the correctness of the pricing mapping is unverified and the 50x claim lacks a QSP baseline. read the letter →

arxiv 2507.19039 v1 pith:76BDANID submitted 2025-07-25 quant-ph cs.ETq-fin.PR

Francesca Cibrario , Ron Cohen , Emanuele Dri , Christian Mattia , Or Samimi Golan , Tamuz Danzig , Giacomo Ranieri , Hanan Rosemarin
show 3 more authors
Davide Corbelletto Amir Naveh Bartolomeo Montrucchio
This is my paper · ORCID
classification quant-phcs.ETq-fin.PR MSC 81P6891G20
keywords autocallableoptionsquantumderivativepricingintegration-basedexponentialamplitudeloadingT-depthreductioniterativeestimationpath-dependentderivativesencodingfinance
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

This paper claims to remove the main bottleneck in quantum pricing of autocallable options by improving how the option's nontrivial put payoff is loaded into a quantum amplitude. The central quantitative claim is that the improved 'partial exponential integration' amplitude loading cuts the T-depth of the payoff module from about 2.1e3 to about 40 for a representative instance (T=20 timesteps, d=3 assets, total estimation error 2e-3), a roughly 50x reduction, while the other modules stay essentially unchanged. If correct, this matters because T-depth is a dominant fault-tolerant cost and payoff encoding was previously the part that resisted easy optimization; the result shifts the practical bottleneck to Gaussian state preparation and arithmetic. The paper supports the claim with a full circuit construction, a complexity decomposition, and simulations on up to 33 qubits whose results converge to the classically estimated payoff as precision grows.

What carries the argument

The central object is the partial exponential state preparation followed by a comparator-as-integrator. The register r is prepared in the state sum_{r=x0}^{x1} $\sqrt$($e^{{a r}}$/Z)|r> using Ry rotations with angles theta_i = 2 arctan($e^{{a 2^i/2}}$); when x1 - x0 is a power of two this is done on the least significant bits plus an in-place addition, and otherwise exact amplitude amplification with a Grover comparator oracle is used. The comparator then produces the integration amplitude of Eq. (15). The key mechanism is that restricting the exponential to the payoff interval keeps the normalization denominator $e^{{a(x1+1)}}$ - $e^{{a x0}}$ comparable to the numerator, avoiding the huge ~ $e^{{a 2^n}}$ normalization of full-domain loading. Because the exponential state preparation can run in parallel with the Gaussian and arithmetic blocks, the serial T-depth of the module is governed almost entirely by the controlled integration comparator.

What would settle it

Derive the full amplitude of the circuit as a function of the final log-return l_T: prepare the partial exponential state on [x0, x1], apply the comparator, then the indicator Ry rotation, and simplify the resulting amplitude of the target qubit. Compare this symbolically with the intended rescaled payoff $R^{{-1}}$[V($e^{{l_T}}$ - r_{Tmin} - K)] over the interval l_T in [ln b, ln K]; any disagreement means the circuit estimates the wrong expectation regardless of T-depth. A circuit-level check would synthesize the payoff module for one fixed l_T, run it with many shots on a simulator, and test whether the measured |1> amplitude matches the classically computed rescaled payoff.

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Extended reading notes

Core claim

On its own terms, the paper establishes that the integration-based exponential amplitude loading introduced for rainbow options can be repaired so that it does not degrade the normalization factor. Instead of preparing an exponential state over the full 2^n grid and paying for unused normalization, the algorithm loads the exponential only over the interval [x0, x1] that corresponds to the put's nonzero payoff region. The amplitude after the comparator is Eq. (15): 0 below x0, $\sqrt$(($e^{{a(x+1)}}$ - $e^{{a x0}}$)/($e^{{a(x1+1)}}$ - $e^{{a x0}}$)) on the interval, and 1 above x1. With the appropriate additional Ry remapping, this amplitude is rescaled into the discounted put payoff, and the full autocallable pricing circuit is assembled from Gaussian log-return loading, in-place addition, barrier and binary comparators in log-return space, the partial exponential state preparation running in parallel, and the integration comparator serially. Complexity analysis for the representative setting gives a payoff amplitude-loading T-depth of about 40, against about 2.1e3 for the Quantum Signal Processing baseline.

Load-bearing premise

The load-bearing premise is an unstated algebraic identity: the amplitude that the partial exponential integration produces, after the extra Ry rotation that remaps it, has to equal the rescaled put payoff at every point in the nonzero-payoff range, and the paper never writes this equation out.

Editorial extensions

If this is right

  • With the payoff amplitude-loading T-depth down to about 40 from about 2.1e3, that module ceases to be the dominant T-depth contributor, so further end-to-end improvements must come from Gaussian state preparation and quantum arithmetic.
  • The same restricted-interval exponential loading can repair the normalization degradation in the earlier rainbow-option integration method, improving the accuracy of that algorithm as well.
  • The full circuit synthesis and IQAE experiments provide a validation template: compare the quantum result against a classical benchmark sharing the same Gaussian discretization and fixed-point precision, then verify that the quantum estimate falls inside the IQAE confidence interval.
  • For multi-asset best-of or worst-of autocallables, the per-asset log-return and exponential preparation blocks can run in parallel, so the roughly 50x reduction in the payoff module should persist while arithmetic and comparator depth scale with the number of assets and time steps.
  • The complexity decomposition gives explicit formulas linking each error source (truncation, discretization, approximation, arithmetic, amplitude loading) to the required T-depth, making the trade-offs reproducible for other error budgets and instance sizes.

Reading between the lines

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

  • The unstated algebraic identity behind the Ry remapping, once written out, likely generalizes beyond the put payoff: any monotone payoff that can be rescaled to sit inside an exponential partial integral could be loaded this way, covering barrier and digital components in other structured products.
  • If the T-depth reduction survives synthesis at scale, the practical resource barrier for quantum advantage in autocallable pricing shifts from payoff encoding to Gaussian preparation and arithmetic, so those modules are the natural next optimization targets.
  • A direct empirical check of the paper's two claims would be to scale the same simulation comparison to the T=20, d=3 instance used for the complexity numbers, rather than the small single-asset, three-timestep instance used in the experiments.
  • The alternative mentioned in the paper, loading the exponential over the entire payoff range to avoid the extra indicator Ry, could be benchmarked against the remapped version on the same instance to see which approach wins on total T-depth once normalization and circuit width are both counted.
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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 / 5 minor

Summary. The paper proposes a quantum algorithm for pricing autocallable options based on an improved 'integration-based exponential amplitude loading' method. The key idea is to load a partial exponential state over a restricted interval [x0,x1] instead of the full register, which is claimed to avoid the normalization-factor degradation of the earlier method in [7]. The authors provide a T-depth complexity analysis for a setting with T=20 timesteps, d=3 assets, and total error epsilon=2e-3, claiming a ~50x reduction in the payoff-component T-depth (from ~2.1e3 to ~40) compared with the earlier integration method. They also present simulation results on up to 33 qubits using HPC simulators to validate the full algorithm against classical benchmarks.

Significance. If the central identity connecting the integration amplitude to the rescaled payoff holds, the proposed partial-exponential loading is a plausible and transparent improvement over the authors' earlier integration method, with a clean error budget and explicit T-depth formulas. The paper ships a Qmod-based implementation and reports HPC simulations, and the convergence study comparing the quantum circuit against a classically discretized closed form is good practice. However, the quantitative headline claim is conditional on an unstated algebraic identity, is not validated by the experiments (which test the baseline III-A method), and is compared against the authors' prior method rather than against the QSP approach named as state-of-the-art.

major comments (3)
  1. [Section III-C / Algorithm 1, step 11 / Eq. (15)] The paper never writes the explicit equation connecting the integration amplitude of Eq. (15) to the rescaled put payoff. Specifically, it does not define the parameter a, the integer values x0 and x1 in terms of the fixed-point representation of l_T, or the Ry rotation angle that maps the integration amplitude to the amplitude-mapped payoff. Because the amplitude in Eq. (15) is a function of e^{a x}, it can equal the payoff amplitude only if a equals the fixed-point scale and x0, x1 correspond to the minimum log-return and ln K; otherwise, no single Ry remapping can correct the mismatch, and the circuit would estimate a different expected value. Please provide the complete algebraic derivation and specify all parameters.
  2. [Section V, first paragraph] The experiments validate only the baseline method III-A ('The full algorithm ... is based on the method described in III-A'), not the improved partial-exponential method III-B that underlies the ~50x T-depth reduction. Consequently, the central quantitative claim has no numerical or simulated support. Please either validate III-B (at least for a case where [x0,x1] has power-of-two size) or explicitly qualify the 50x claim as an unvalidated theoretical estimate and provide a separate small-scale test of the partial exponential state preparation and the integration comparator.
  3. [Section IV-E and abstract/conclusion] The stated ~50x reduction in amplitude-loading T-depth is computed against the authors' earlier integration method [7] (from ~2.1e3 to ~40), but the abstract and conclusion claim improvement over 'state-of-the-art' and the QSP method [6]. No explicit QSP amplitude-loading T-depth baseline is given, so the comparison with [6] is not demonstrated. Please provide the QSP baseline T-depth under the same error budget and problem parameters, or rephrase the claims to refer specifically to [7].
minor comments (5)
  1. [Section II-A, payoff formula] The displayed payoff condition for the short knock-in put is written as 'rt < b, forall t in [0,T]', but the surrounding text and Algorithm 1 use 'barrier crossed at least once' (exists t). The displayed formula should be corrected to match the intended definition.
  2. [Introduction] The name 'Stamatopoulus' is a typo for 'Stamatopoulos'.
  3. [Section III-B] The phrase 'assuming it is continuous' is vague; please specify that the relevant range is a contiguous interval of the integer register.
  4. [Section IV-E] The sentence 'from ~2.1e3 to ~40 T-depth' appears without a derivation of the numerical values; please state the value of m and the error allocation used to obtain these numbers so the reader can reproduce the comparison.
  5. [Table II] The 'Notional value (V) 18$' entry uses an unusual unit presentation; please clarify the currency and formatting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the T-depth and amplitude-loading derivations are self-contained algebraic results, and the self-citations to [7] are not load-bearing.

full rationale

Walking the derivation chain: the integration amplitude loading formula (Eq. 13) follows directly from the exponential state preparation and comparator construction; the improved partial-exponential version (Eqs. 14-15) is an explicit algebraic generalization with no fitted parameters; the T-depth analysis (Dtot, DAL, Dexp) is assembled from tabled building-block depths and standard error bounds; and the comparison with [6] changes only the amplitude-loading module while keeping the other modules fixed. None of these steps is equivalent to its input by construction. The paper's benchmark against its own prior work [7] is a normal methodological inheritance: [7] is cited as the source of the exponential loading technique, and the formulas are restated in the paper rather than imported as an unverified uniqueness theorem. The experimental section's statement that the full algorithm is based on Section III-A rather than III-B is a validation gap, not circularity. The verbal description in Section III-C of how the exponential reference maps to the rescaled put payoff is under-specified and could be a correctness risk, but the paper contains no equation showing that the payoff equals the loading amplitude by construction, so this is not a circular step either. Overall, the central T-depth reduction claim is derived from explicit circuit decompositions and independent complexity formulas; no prediction reduces to a fit or to a self-citation chain.

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

No free parameters are fitted to data; model parameters are given inputs. Five assumptions are needed: the Black-Scholes model, the Gaussian loader, exact amplitude amplification, the T-depth tables, and the unproven payoff-mapping identity. No new physical or computational entities are introduced.

assumptions (5)
  • domain assumption Black-Scholes dynamics with constant drift and volatility (Eqs 1-2) are the correct return model
    The pricing algorithm is built on this model; the paper does not justify it.
  • domain assumption The variational Gaussian loader from [5] approximates the standard normal within the stated error budget when pre-trained as described
    Section II-B adopts the re-parameterization of [5] without independent verification.
  • domain assumption Exact amplitude amplification (Long [17], McArdle et al. [18]) prepares the partial exponential state with a comparator oracle and frees the auxiliary qubit
    Section III-B, Figure 2b uses this algorithm as a black box.
  • domain assumption The T-depth formulas in Table I (sourced from [20]-[24]) are correct for the respective gates and comparators
    The entire complexity comparison rests on these values.
  • ad hoc to paper The integration amplitude ratio in Eq (15) coincides with the rescaled payoff after the described Ry remapping
    Section III-C asserts the mapping without a derivation; this is load-bearing for pricing correctness.

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Cite this review

Pith. "Pith review of Autocallable Options Pricing with Integration-Based Exponential Amplitude Loading." pith.science (2026). https://pith.science/paper/76BDANID

@misc{pith2026250719039,
  author       = {Pith},
  title        = {Pith review of: Autocallable Options Pricing with Integration-Based Exponential Amplitude Loading},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/76BDANID}},
  note         = {Machine review of arXiv:2507.19039}
}
read the original abstract

We present a comprehensive quantum algorithm tailored for pricing autocallable options, offering a full implementation and experimental validation. Our experiments include simulations conducted on high-performance computing (HPC) hardware, along with an empirical analysis of convergence to the classically estimated value. Our key innovation is an improved integration-based exponential amplitude loading technique that reduces circuit depth compared to state-of-the-art approaches. A detailed complexity analysis in a relevant setting shows an approximately 50x reduction in T-depth for the payoff component relative to previous methods. These contributions represent a step toward more efficient quantum approaches to pricing complex financial derivatives.

Figures

Figures reproduced from arXiv: 2507.19039 by the authors.

Figure 1
Figure 1. Integration amplitude loading circuit for a generic [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. State preparation of exponential on a specific interval: [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Expected payoff for the following classical models: [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

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Works this paper leans on

29 extracted references · 12 canonical work pages

  1. [7]

    Quantum amplitude loading for rainbow options pricing,

    F. Cibrario, O. S. Golan, G. Ranieri, E. Dri, M. Ippoliti, R. Cohen, C. Mattia, B. Montrucchio, A. Naveh, and D. Corbelletto, “Quantum amplitude loading for rainbow options pricing,” in 2024 IEEE International Conference on Quantum Computing and Engineering (QCE). IEEE, Sep. 2024, p. 211–220. [Online]. Available: http://dx.doi.org/10.1109/QCE60285.2024.00034

    arXiv 2024

  2. [6]

    Derivative Pricing using Quantum Signal Processing,

    N. Stamatopoulos and W. J. Zeng, “Derivative Pricing using Quantum Signal Processing,” Quantum, vol. 8, p. 1322, Apr. 2024. [Online]. Available: https://doi.org/10.22331/q-2024-04-30-1322

    doi:10.22331/q-2024-04-30-1322 2024

  3. [1]

    Quantum computing for finance: Overview and prospects,

    R. Or ´us, S. Mugel, and E. Lizaso, “Quantum computing for finance: Overview and prospects,” Reviews in Physics , vol. 4, p. 100028, Nov

    work page

  4. [2]

    Quantum computing for finance: State-of-the-art and future prospects,

    D. J. Egger, C. Gambella, J. Marecek, S. McFaddin, M. Mevissen, R. Raymond, A. Simonetto, S. Woerner, and E. Yndurain, “Quantum computing for finance: State-of-the-art and future prospects,” IEEE Transactions on Quantum Engineering , vol. 1, p. 1–24, 2020. [Online]. Available: http://dx.doi.org/10.1109/TQE.2020.3030314

    arXiv 2020

  5. [3]

    Quantum computing for finance,

    D. Herman, C. Googin, X. Liu, Y . Sun, A. Galda, I. Safro, M. Pistoia, and Y . Alexeev, “Quantum computing for finance,” Nature Reviews Physics, vol. 5, no. 8, p. 450–465, Jul. 2023. [Online]. Available: http://dx.doi.org/10.1038/s42254-023-00603-1

    doi:10.1038/s42254-023-00603-1 2023

  6. [4]

    Quantum computing in finance: The intesa sanpaolo experience,

    R. Sotelo, D. Corbelletto, E. Dri, E. Giusto, and B. Montrucchio, “Quantum computing in finance: The intesa sanpaolo experience,” IEEE Engineering Management Review , vol. 52, no. 2, p. 9–15, Apr. 2024. [Online]. Available: http://dx.doi.org/10.1109/EMR.2024.3373796

    arXiv 2024

  7. [5]

    A threshold for quantum advantage in derivative pricing,

    S. Chakrabarti, R. Krishnakumar, G. Mazzola, N. Stamatopoulos, S. Woerner, and W. J. Zeng, “A threshold for quantum advantage in derivative pricing,” Quantum, vol. 5, p. 463, Jun. 2021. [Online]. Available: http://dx.doi.org/10.22331/q-2021-06-01-463

    doi:10.22331/q-2021-06-01-463 2021

  8. [8]

    Qmod - the quantum modeling language,

    “Qmod - the quantum modeling language,” https://docs.classiq.io/latest/ qmod-reference/, 2024

    work page 2024

Show all 29 references
  1. [9]

    Leonardo: A pan-european pre-exascale supercomputer for hpc and ai applications,

    M. Turisini, M. Cestari, and G. Amati, “Leonardo: A pan-european pre-exascale supercomputer for hpc and ai applications,” Journal of large-scale research facilities JLSRF , vol. 9, no. 1, Jan. 2024. [Online]. Available: http://dx.doi.org/10.17815/jlsrf-8-186

    2024 doi

  2. [10]

    Hull, Options, futures, and other derivatives , 7th ed., ser

    J. Hull, Options, futures, and other derivatives , 7th ed., ser. The Prentice Hall series in finance. Upper Saddle River, NJ: Pearson Prentice Hall, 2009

    2009

  3. [11]

    Quantum speedup of monte carlo methods,

    A. Montanaro, “Quantum speedup of monte carlo methods,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 471, no. 2181, p. 20150301, Sep. 2015. [Online]. Available: http://dx.doi.org/10.1098/rspa.2015.0301

    2015

  4. [12]

    Quantum computational finance: Monte carlo pricing of financial derivatives,

    P. Rebentrost, B. Gupt, and T. R. Bromley, “Quantum computational finance: Monte carlo pricing of financial derivatives,” Physical Review A , vol. 98, no. 2, Aug. 2018. [Online]. Available: http: //dx.doi.org/10.1103/PhysRevA.98.022321

    2018 doi

  5. [13]

    Option pricing using quantum computers,

    N. Stamatopoulos, D. J. Egger, Y . Sun, C. Zoufal, R. Iten, N. Shen, and S. Woerner, “Option pricing using quantum computers,” Quantum, vol. 4, p. 291, Jul. 2020. [Online]. Available: http: //dx.doi.org/10.22331/q-2020-07-06-291

    2020 doi

  6. [14]

    Quantum pricing with a smile: implementation of local volatility model on quantum computer,

    K. Kaneko, K. Miyamoto, N. Takeda, and K. Yoshino, “Quantum pricing with a smile: implementation of local volatility model on quantum computer,” EPJ Quantum Technology, vol. 9, no. 1, p. 7, Dec

  7. [15]

    Option pricing under stochastic volatility on a quantum computer,

    G. Wang and A. Kan, “Option pricing under stochastic volatility on a quantum computer,” Quantum, vol. 8, p. 1504, Oct. 2024. [Online]. Available: https://doi.org/10.22331/q-2024-10-23-1504

    2024 doi

  8. [16]

    Iterative quantum amplitude estimation,

    D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, “Iterative quantum amplitude estimation,” npj Quantum Information , vol. 7, no. 1, Mar. 2021. [Online]. Available: http://dx.doi.org/10.1038/ s41534-021-00379-1

    2021

  9. [17]

    Grover algorithm with zero theoretical failure rate,

    G. L. Long, “Grover algorithm with zero theoretical failure rate,” Physical Review A , vol. 64, no. 2, Jul. 2001. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.64.022307

    2001 doi

  10. [18]

    Quantum state preparation without coherent arithmetic,

    S. McArdle, A. Gily ´en, and M. Berta, “Quantum state preparation without coherent arithmetic,” 2022. [Online]. Available: https://arxiv. org/abs/2210.14892

    2022 arXiv

  11. [19]

    T-depth optimization for fault- tolerant quantum circuits,

    P. Niemann, A. Gupta, and R. Drechsler, “T-depth optimization for fault- tolerant quantum circuits,” in 2019 IEEE 49th International Symposium on Multiple-V alued Logic (ISMVL) . IEEE, 2019, pp. 108–113

    2019

  12. [20]

    Optimal ancilla-free clifford+ t approxima- tion of z-rotations,

    N. J. Ross and P. Selinger, “Optimal ancilla-free clifford+ t approxima- tion of z-rotations,” arXiv preprint arXiv:1403.2975 , 2014

    2014 arXiv

  13. [21]

    A logarithmic-depth quantum carry-lookahead adder,

    T. G. Draper, S. A. Kutin, E. M. Rains, and K. M. Svore, “A logarithmic-depth quantum carry-lookahead adder,” arXiv preprint quant-ph/0406142, 2004

    2004 arXiv

  14. [22]

    Advantages of using relative-phase toffoli gates with an application to multiple control toffoli optimization,

    D. Maslov, “Advantages of using relative-phase toffoli gates with an application to multiple control toffoli optimization,” Physical Review A , vol. 93, no. 2, p. 022311, 2016

    2016

  15. [23]

    Classiq: High-level quantum modeling language,

    “Classiq: High-level quantum modeling language,” https://github.com/ Classiq/classiq-library, 2024

    2024

  16. [24]

    A comprehensive review of quantum circuit optimization: Current trends and future directions,

    K. Karuppasamy, V . Puram, S. Johnson, and J. P. Thomas, “A comprehensive review of quantum circuit optimization: Current trends and future directions,” Quantum Reports , vol. 7, no. 1, 2025. [Online]. Available: https://www.mdpi.com/2624-960X/7/1/2

    2025

  17. [25]

    Quantum autocallable implementation,

    “Quantum autocallable implementation,” http://short.classiq.io/ quantum autocallable

  18. [26]

    Qmod: Expressive high-level quantum modeling,

    M. Vax, P. Emanuel, E. Cornfeld, I. Reichental, O. Opher, O. Roth, T. Michaeli, L. Preminger, L. Gazit, A. Naveh et al., “Qmod: Expressive high-level quantum modeling,” arXiv preprint arXiv:2502.19368 , 2025

    2025 arXiv

  19. [27]

    Design and synthesis of scalable quantum programs,

    T. Goldfriend, I. Reichental, A. Naveh, L. Gazit, N. Yoran, R. Alon, S. Ur, S. Lahav, E. Cornfeld, A. Elazari, P. Emanuel, D. Harpaz, T. Michaeli, N. Erez, L. Preminger, R. Shapira, E. M. Garcell, O. Samimi, S. Kisch, G. Hallel, G. Kishony, V . v. Wingerden, N. A. Rosenbloom, ...

    2024

  20. [2019]

    Available: http://dx.doi.org/10.1016/j.revip.2019.100028 10

    [Online]. Available: http://dx.doi.org/10.1016/j.revip.2019.100028 10

    2019

  21. [2022]

    Available: https://epjquantumtechnology.springeropen

    [Online]. Available: https://epjquantumtechnology.springeropen. com/articles/10.1140/epjqt/s40507-022-00125-2

    doi

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