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arxiv: 2607.06040 · v1 · pith:FLT4I457 · submitted 2026-07-07 · quant-ph · cs.ET

Hybrid quantum floating-point method for sharp arithmetic

Gabriele Agliardi , Enrico Prati This is my paper

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classification quant-ph cs.ET
keywords quantum arithmeticfloating-pointhybrid quantum-classicalerror propagationquantum computingrandom variable encoding
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The pith

Classical offset slashes quantum arithmetic error by up to 89%

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

This paper introduces the Classically-Enriched Floating Point Variable (CEFV), a hybrid quantum-classical encoding for real-valued random variables on a quantum register. A CEFV stores a quantum register of qubit values alongside a classical register holding four parameters: an offset (a), a scaling factor (b), and two tolerance bounds (epsilon-minus, epsilon-plus). The offset shifts the representable grid away from zero, allowing a fixed qubit budget to cover a narrower, more relevant range with higher density. The scaling factor stretches or compresses that grid. The tolerances track how far the true target random variable can stray from the nearest representable value. The paper defines addition and multiplication operations on CEFVs that guarantee no overflow: the output scaling factor is automatically adjusted by a power of two to use all available output qubits while keeping the result in range. After each operation, the tolerances are propagated forward so the output CEFV carries an honest error bar. The central claim is that the classical offset, which costs no quantum gates for addition because it is summed classically, densifies the representable grid in the region where the data actually lives, and this dramatically reduces the accumulated rounding error after repeated additions compared to mono-quantum floating variables (MFV), which lack an offset.

Core claim

The paper's central object is the CEFV encoding, and its core result is that introducing a classically stored offset into a quantum floating-point representation narrows the tolerance window around arithmetic results. In the paper's benchmark, a 3-qubit register encoding a random variable taking values 6 or 7 with equal probability is summed with itself six times. Without an offset, the tolerance grows to epsilon-plus = 28; with an offset, it stays at epsilon-plus = 3, an 89% reduction. The correctness of the addition (Proposition S2.9) and multiplication (Proposition S2.17) is proven: for any inputs compatible with the input CEFVs, the output state is guaranteed to be compatible with the ar

What carries the argument

CEFV (Classically-Enriched Floating Point Variable): a quantum register of n qubits paired with classical parameters (offset a, scaling factor b, tolerances epsilon-minus and epsilon-plus). The encoding maps basis state |z> to value x = a + b*z. Addition and multiplication are implemented via semi-boolean polynomial evaluation (SBPEval), where input qubit weights are rounded to integers and the output scaling factor is auto-tuned to a power-of-two multiple of a reference value b_lead to prevent overflow and minimize rounding.

If this is right

  • Quantum algorithms requiring repeated arithmetic on real-valued distributions, such as option pricing or quantum amplitude estimation for integration, could achieve materially higher output precision on near-term hardware with limited qubit counts.
  • The principle of classically storing a range-shifting offset to densify the representable grid is not specific to floating-point arithmetic; it could be applied to any quantum encoding where the data occupies a narrow band far from zero, including fixed-point and amplitude-encoded representations.
  • The tolerance-propagation framework provides a built-in error accounting system: any quantum algorithm using CEFVs can report a guaranteed error bound on its output without needing separate error analysis, which is useful for certification of quantum computation results.

Where Pith is reading between the lines

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

  • If the offset advantage is real and general, then any quantum algorithm that chains many additions (e.g., iterative solvers, Monte Carlo path averaging) would benefit from switching to CEFV encoding, because the error compounding rate per operation would be lower. The paper demonstrates this for six additions on one distribution; the implicit claim is that the mechanism (denser grid near the data)
  • The auto-tuning of the output scaling factor to a power-of-two multiple of b_lead suggests a natural extension: a compiler-level pass that, given a sequence of CEFV operations, optimizes the choice of b_lead at each step to minimize cumulative tolerance growth across the entire circuit, not just per-operation.
  • The fact that offsets add no quantum gate overhead for addition but only modest overhead for multiplication (linear terms only, not the quadratic cross-term) implies that CEFVs are most cost-effective in addition-heavy circuits; the break-even point where multiplication overhead cancels the precision gain would depend on the ratio of additions to multiplications in a given algorithm.

Load-bearing premise

The 89% error reduction is demonstrated on a single example: a 3-qubit register encoding a variable taking values 6 or 7, summed with itself six times. The claim that this advantage generalizes to other input distributions, register sizes, and operation sequences rests on the assumption that the offset's grid-densification benefit persists broadly, which is not systematically benchmarked in the paper.

What would settle it

If one constructs a repeated-addition scenario where the target values are uniformly or near-uniformly distributed across a wide range centered at zero (so that the offset provides no grid-densification benefit), the CEFV tolerance should be no better than the MFV tolerance, and the 89% reduction claim would not hold.

Figures

Figures reproduced from arXiv: 2607.06040 by Enrico Prati, Gabriele Agliardi.

Figure 1
Figure 1. Figure 1: The top row contains a visual representation of a CEFV. In the rows below, some examples of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of the choice of blead on errors, for the sum of two CEFVs of size n, in an output register of same size n = 8. In the top plot, b1 is kept equal to 1, b2 varies from 10−3 to 104 , and blead in [1, 2]. The color intensity represents the relative error, namely ϵ ++ϵ − b12n+b22n . In the bottom plot, a focus on three specific choices of blead, namely b1 (i.e., 1), b2, and the optimal bopt. multiplicat… view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative distribution function for repeated applications of the sum. Left plot: the initial [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

There are several possible ways to encode random variables in a quantum state. The basis encoding of bit strings has paramount importance because it allows to load the values of a random variable through the superposition of corresponding basis states, and to then exploit quantum parallelism in processing algorithms. The basis encoding offers a natural way to represent an unsigned integer random variable, and extends to signed integers, as well as to fixed-point and floating-point variables. Each quantum representation of fractional numbers, however, involves a trade-off between accuracy and depth of manipulation circuits. Here, an efficient hybrid quantum-classical representation of quantum floating points is introduced. It combines a quantum register containing the values, with a classical register storing global information about the variable, namely the range and approximation tolerances. The sum and product operations are defined, in such a way as to ensure they are performed without overflow. By taking advantage of the stored classical information, the precision degradation that occurs due to rounding after repeated data manipulations, can be significantly reduced compared to known strategies. Ad hoc examples show up to around $90\%$ reduction in approximation, compared to previous techniques, after repeated additions. The method finds application in many algorithms of practical relevance and constitutes a significant advance in the design of arithmetic circuits with low depth and high accuracy.

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.

Referee Report

1 major / 7 minor

Summary. This manuscript introduces Classically-Enriched Floating Point Variables (CEFVs), a hybrid quantum-classical encoding for floating-point numbers on quantum registers. The CEFV augments a quantum register storing integer values with classical side information: an offset $a$, a scaling factor $b$, and upper/lower tolerance parameters $ε^±$. The authors define addition (Algorithm S1) and multiplication (Algorithm S2) operations on CEFVs, prove that both operations are overflow-free and correctly propagate tolerances (Propositions S2.9 and S2.17), and benchmark the approach against mono-quantum floating variables (MFV) from Ref. [17]. The headline empirical result is an 89% reduction in approximation error after six repeated additions on a 3-qubit register, attributed to the classical offset enabling a denser representable grid in the relevant range.

Significance. The CEFV representation is a natural and well-motivated extension of mono-quantum coding (Ref. [17]). The correctness proofs for addition and multiplication (Props. S2.9, S2.17) are the core technical contribution and appear sound: the tolerance propagation is derived from the encoding structure, not fitted to data, and the no-overflow guarantee is properly established via the weight-bounding condition in Algorithm S1 (Step 5). The observation that classical offsets and scaling factors can be adjusted without quantum circuit manipulation — enabling zero-overhead ClassicalAdd and ClassicalProd operations — is a genuine practical advantage. The framework is falsifiable: the tolerance bounds are constructive and can be checked against any concrete input. However, the empirical claim of ~90% error reduction rests on a single engineered example, which limits the demonstrated significance.

major comments (1)
  1. §2.5, Fig. 3: The headline 89% error reduction is demonstrated on a single ad hoc example — a 3-qubit register encoding values {6,7} summed with itself six times. This example is specifically constructed to maximize the offset's benefit: without offset, the 3-qubit register represents {0,...,7}, and after six additions the sum ranges over [36,42], forcing $b_{out}$ to be large and making the grid extremely sparse ($ε^+=28$); with offset $a=36$, the same 3 qubits densely cover the relevant range ($ε^+=3$). The paper provides no systematic evidence that this advantage persists outside this engineered regime. For instance, when the input distribution already spans the register range (e.g., uniform on {0,...,7}), or when the register is large enough that grid sparsity is not painful, the offset provides little benefit. The heat maps in Fig. 2 study $b_{lead}$ selection but do not address the
minor comments (7)
  1. §2.1: 'fomr' should be 'from' in the definition of quantum variable.
  2. §2.6: 'Converesely' should be 'Conversely'.
  3. Fig. S10 caption: panel (b) lists 'j = 3' twice instead of 'j = 3, j = 4' or similar sequential labeling.
  4. Formula box S2: The notation 'max v' for the maximum of a five-entry vector is introduced in a footnote-like remark marked (⋆), but is used in the formula above its definition. Reordering would improve readability.
  5. §S2.2, Remark S2.10: The claim that Step 3 initialization guarantees the lower bound on weight sums and that Step 5 requires few iterations is stated with 'The proof of these facts is left to the reader.' For a journal publication, these should be proven or explicitly marked as straightforward consequences with a brief argument.
  6. Fig. 2 and Fig. S9: The relationship between the main-text Fig. 2 and the extended Fig. S9 could be stated more explicitly (e.g., which panels of S9 correspond to Fig. 2).
  7. References: The Qrisp package [22] is mentioned as now including SBPEval; a brief note on whether the authors' implementation differs would help reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive assessment. The referee correctly identifies the core technical contributions (correctness proofs for addition and multiplication, the no-overflow guarantee, the zero-overhead ClassicalAdd/ClassicalProd operations) and acknowledges the framework as a natural, well-motivated, and falsifiable extension of mono-quantum coding. The sole major concern is that the headline 89% error-reduction claim rests on a single engineered example, with no systematic evidence that the advantage persists outside that regime. We agree that the current manuscript over-claims the breadth of the empirical demonstration, and we will revise accordingly: we will add systematic benchmarks across multiple input distributions and register sizes, reframe the 89% figure as an upper bound achieved in a specific regime rather than a general result, and explicitly characterize the regimes where the offset provides minimal benefit. The theoretical contributions (Propositions S2.9 and S2.17, the tolerance propagation formulas, the no-overflow condition) are independent of this empirical point and are unaffected.

read point-by-point responses

  1. Referee: The headline 89% error reduction is demonstrated on a single ad hoc example (3-qubit register, values {6,7} summed six times), specifically constructed to maximize the offset's benefit. No systematic evidence that the advantage persists outside this engineered regime. When the input distribution already spans the register range, or when the register is large enough that grid sparsity is not painful, the offset provides little benefit. The heat maps in Fig. 2 study b_lead selection but do not address the advantage of offsets across input distributions.

    Authors: The referee is correct that the example in Figure 3 is engineered to showcase the offset's benefit in a regime where it is maximally effective, and that the manuscript does not currently provide systematic evidence across input distributions and register sizes. We accept this criticism and will revise the manuscript in three concrete ways. First, we will add a systematic numerical study varying (i) the input distribution (including the uniform-on-{0,...,7} case the referee specifically suggests, as well as distributions concentrated on subranges of varying widths), (ii) the register size n (from 3 to 8 qubits), and (iii) the number of repeated additions. This will show explicitly how the offset's benefit scales with these parameters. Second, we will reframe the 89% figure throughout the manuscript (abstract, Section 2.5, Discussion) as an upper bound on the achievable error reduction in the regime where the target distribution is concentrated in a narrow range far from zero — precisely the regime where mono-quantum coding (MFV) is forced to allocate a large exponent, making the grid sparse. We will state plainly that when the input distribution already spans the full register range, the offset provides little to no benefit, as the referee anticipates. Third, we will add a discussion characterizing the regimes where CEFV's advantage over MFV is expected to be significant (narrow-range distributions, repeated operations causing range drift) versus negligible (wide distributions, large registers). We note that the theoretical contributions — the correctness proofs (Props. S2.9, S2.17), the tolerance propagation formulas, the no-overflow guarantee, and the zero-overhead ClassicalAdd/ClassicalProd operations — are independent of the empirical breadth concern and remain valid revision: yes

Circularity Check

0 steps flagged

No significant circularity found; derivation is self-contained with external building blocks.

full rationale

The paper's central derivation chain is not circular. The CEFV definition (Def S1.6) introduces the encoding x = a + bz with tolerances ε± as a straightforward affine construction. The addition algorithm (Algorithm S1) and multiplication algorithm (Algorithm S2) produce output tolerances via Formula Boxes S1 and S2, which are derived from the rounding of integer weights—a direct consequence of the algorithm's construction, not fitted parameters. The correctness proofs (Props S2.9, S2.17) derive tolerance bounds algebraically from the encoding structure and the no-overflow constraint, with no step that reduces to its own inputs. The key external building block, SBPEval, is cited from Ref [17] (Seidel et al., different authors), providing independent support. The 89% error reduction figure (§2.5) is computed by applying the paper's own formulas to a specific engineered example (3-qubit register, values {6,7}, six self-additions), and the paper is transparent about this being an 'ad hoc' / 'engineered case.' This is a question of empirical breadth, not circularity. Self-citations [7,8,9,10] refer to prior work on quantum GANs, data encoding, and quantum integration—different topics that are not load-bearing for the CEFV definition, algorithms, or correctness proofs. The b_lead optimization via dual annealing (§2.4) is a parameter search for algorithm performance, not a fitted input presented as a first-principles prediction. No step in the derivation chain reduces to its inputs by construction. Score 1 reflects the presence of minor self-citations that are not load-bearing for the central claims.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper introduces no new physical entities, particles, or forces. The CEFV is a data structure / encoding scheme, not a postulated object. The free parameters (a, b, b_lead, n_max) are design choices within the encoding, not fitted constants. The axioms are standard quantum mechanics plus the external SBPEval result.

free parameters (3)
  • b_lead (output scaling factor) = chosen per-operation via dual annealing optimization or heuristically as b_1 or b_2
    The output scaling factor b_lead is a free parameter chosen by the circuit designer at each arithmetic step. The paper studies its effect (Fig. 2) but does not derive it from first principles; it is optimized numerically.
  • n_max (output register size) = set by available hardware qubits
    The maximum output register size is a free parameter constrained by hardware, not derived. The algorithm adapts to it.
  • a, b, ε± (CEFV parameters per variable) = a and b set by data encoding; ε± computed from encoding approximation
    The offset a and scaling b are chosen when encoding a random variable into the quantum register. They are free in the sense that the paper does not prescribe a unique choice, only studies the effect of different choices.
axioms (4)
  • standard math Semi-boolean polynomial evaluation (SBPEval) produces a unitary circuit evaluating integer-coefficient polynomials on qubit basis states with stated gate complexity (Prop. S2.1, from Ref. [17]).
    The entire arithmetic framework builds on SBPEval as a black-box primitive. This is an external result from Seidel et al. [17], not re-proven here.
  • standard math Quantum parallelism: a unitary operating correctly on basis states extends by linearity to superpositions (§2.1).
    Standard quantum mechanics axiom, invoked to extend basis-state correctness to random-variable operations.
  • domain assumption The target random variable's domain is compatible with a uniformly spaced grid (affine encoding x = a + bz).
    The CEFV representation assumes the data can be approximated on a uniformly spaced grid. Variables with highly non-uniform distributions or multi-scale structure may not benefit as much.
  • domain assumption Classical processing in the algorithm scales polynomially (Remark S2.16).
    The paper asserts but does not prove that the classical overhead (weight computation, M selection) is polynomial. If it were exponential, it would negate the quantum advantage.

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 5 internal anchors

  1. [1]

    Rieffel and W

    E. Rieffel and W. Polak,Quantum Computing: A Gentle Introduction. Scientific and Engineering Computation, MIT Press, 2011

    work page 2011

  2. [2]

    Quantum Computation and Quantum Information: 10th Anniver- sary Edition,

    M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information: 10th Anniver- sary Edition,” Dec. 2010

    work page 2010

  3. [3]

    Optimizing Quantum Circuits for Arithmetic,

    T. H¨ aner, M. Roetteler, and K. M. Svore, “Optimizing Quantum Circuits for Arithmetic,” May

    work page

  4. [4]

    arXiv:1805.12445 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    Quantum algorithm for stochastic optimal stopping problems with applications in finance

    J. F. Doriguello, A. Luongo, J. Bao, P. Rebentrost, and M. Santha, “Quantum algorithm for stochas- tic optimal stopping problems with applications in finance,” May 2022. arXiv:2111.15332 [quant-ph, q-fin]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  6. [6]

    Quantum Risk Analysis

    S. Woerner and D. J. Egger, “Quantum Risk Analysis,”npj Quantum Information, vol. 5, p. 15, Dec. 2019. arXiv: 1806.06893

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  7. [7]

    The quantum amplitude estimation algorithms on near-term devices: A practical guide,

    M. Maronese, M. Incudini, L. Asproni, and E. Prati, “The quantum amplitude estimation algorithms on near-term devices: A practical guide,”Quantum Reports, vol. 6, no. 1, pp. 1–13, 2024

    work page 2024

  8. [8]

    Optimal tuning of quantum generative adversarial networks for multi- variate distribution loading,

    G. Agliardi and E. Prati, “Optimal tuning of quantum generative adversarial networks for multi- variate distribution loading,”Quantum Reports, vol. 4, no. 1, pp. 75–105, 2022

    work page 2022

  9. [9]

    Optimized quantum generative adversarial networks for distribution loading,

    G. Agliardi and E. Prati, “Optimized quantum generative adversarial networks for distribution loading,” in2022 IEEE International Conference on Quantum Computing and Engineering (QCE), pp. 824–827, IEEE, 2022

    work page 2022

  10. [10]

    Quantum data encoding as a distinct abstraction layer in the design of quantum circuits,

    G. Agliardi and E. Prati, “Quantum data encoding as a distinct abstraction layer in the design of quantum circuits,”Quantum Science and Technology, vol. 10, no. 2, p. 025008, 2025

    work page 2025

  11. [11]

    Quantum integration of elementary particle pro- cesses,

    G. Agliardi, M. Grossi, M. Pellen, and E. Prati, “Quantum integration of elementary particle pro- cesses,”Physics Letters B, p. 137228, 2022

    work page 2022

  12. [12]

    Quantum networks for elementary arithmetic operations,

    V. Vedral, A. Barenco, and A. Ekert, “Quantum networks for elementary arithmetic operations,” Physical Review A, vol. 54, pp. 147–153, July 1996. Publisher: American Physical Society

    work page 1996

  13. [13]

    Addition on a Quantum Computer,

    T. G. Draper, “Addition on a Quantum Computer,”arXiv: Quantum Physics, Aug. 2000

    work page 2000

  14. [14]

    A new quantum ripple-carry addition circuit

    S. A. Cuccaro, T. G. Draper, S. A. Kutin, and D. P. Moulton, “A new quantum ripple-carry addition circuit,” Oct. 2004. arXiv:quant-ph/0410184. 35

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  15. [15]

    Design of efficient reversible logic-based binary and BCD adder circuits,

    H. Thapliyal and N. Ranganathan, “Design of efficient reversible logic-based binary and BCD adder circuits,”ACM Journal on Emerging Technologies in Computing Systems, vol. 9, pp. 17:1–17:31, Oct. 2013

    work page 2013

  16. [16]

    A Resource-Efficient Design for a Reversible Floating Point Adder in Quantum Computing,

    T. D. Nguyen and R. Van Meter, “A Resource-Efficient Design for a Reversible Floating Point Adder in Quantum Computing,”ACM Journal on Emerging Technologies in Computing Systems, vol. 11, pp. 13:1–13:18, Nov. 2014

    work page 2014

  17. [17]

    Quantum arithmetic with the quantum Fourier trans- form,

    L. Ruiz-Perez and J. C. Garcia-Escartin, “Quantum arithmetic with the quantum Fourier trans- form,”Quantum Information Processing, vol. 16, p. 152, June 2017

    work page 2017

  18. [18]

    Efficient Floating Point Arithmetic for Quantum Computers,

    R. Seidel, N. Tcholtchev, S. Bock, C. K.-U. Becker, and M. Hauswirth, “Efficient Floating Point Arithmetic for Quantum Computers,”IEEE Access, vol. 10, pp. 72400–72415, 2022. Conference Name: IEEE Access

    work page 2022

  19. [19]

    Quantum Circuits for Floating-Point Arith- metic,

    T. Haener, M. Soeken, M. Roetteler, and K. M. Svore, “Quantum Circuits for Floating-Point Arith- metic,” inReversible Computation(J. Kari and I. Ulidowski, eds.), Lecture Notes in Computer Science, (Cham), pp. 162–174, Springer International Publishing, 2018

    work page 2018

  20. [20]

    Quantum Alphatron: quantum advantage for learning with kernels and noise

    S. Yang, N. Guo, M. Santha, and P. Rebentrost, “Quantum Alphatron: quantum advantage for learning with kernels and noise,” Dec. 2022. arXiv:2108.11670 [quant-ph] version: 2

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  21. [21]

    SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen,...

    work page 2020

  22. [22]

    Silq — overview of key concepts

    “Silq — overview of key concepts.”https://silq.ethz.ch/overview. [Online; accessed 2024-01- 04]

    work page 2024

  23. [23]

    What is qrisp? — documentation

    “What is qrisp? — documentation.”https://www.qrisp.eu/index.html. [Online; accessed 2023- 06-21]

    work page 2023

  24. [24]

    Elementary gates for quantum computation,

    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,”Physical review A, vol. 52, no. 5, p. 3457, 1995. 36

    work page 1995

  25. [25]

    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, p. 022311, Feb. 2016. Publisher: American Physical Society

    work page 2016

  26. [26]

    Qiskit: An open-source framework for quantum computing

    G. Aleksandrowicz, T. Alexander, P. Barkoutsos, L. Bello, Y. Ben-Haim, D. Bucher, F. J. Cabrera- Hern´ andez, J. Carballo-Franquis, A. Chen, C.-F. Chen,et al., “Qiskit: An open-source framework for quantum computing.”https://www.qiskit.org. 37

    work page