Novel Quantum Circuit Designs of Random Injection and Payoff Computation for Financial Risk Assessment

Yeong-Jar Chang; Ying-Wei Tseng; Yu-Ting Kao

REVIEW 5 major objections 6 minor 15 references

Novel Quantum Circuit Designs of Random Injection and Payoff Computation for Financial Risk Assessment

T0 review · 5 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that randomness and payoff computation can be embedded in a single quantum circuit, removing classical preprocessing while preserving QAE's quadratic speedup.

desk verdict A concrete but trivial threshold-payoff circuit wrapped in unsupported QAE and quantum-supremacy claims — desk reject. read the letter →

arxiv 2507.23310 v1 pith:3SM42WJI submitted 2025-07-31 quant-ph

Yu-Ting Kao , Yeong-Jar Chang , Ying-Wei Tseng This is my paper · ORCID

classification quant-ph MSC 81P68 PACS 03.67.Lx

keywords quantumfinancerandomnumberinjectionpayoffcomputationAmplitudeEstimationmixed-signalcircuitfinancialriskassessmentparallelcomputing

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 sets out to solve a two-sided problem in quantum finance: how to inject randomness into massively parallel quantum circuits, and how to compute thresholded payoffs without first classifying the data classically. The authors propose two circuit components—random injection, which XORs each thread's result with a uniform random variable, and payoff computation, which uses a mixed-signal switch to route only prices above the strike to a weighted-average register. They argue that the expected value of random injection is exactly the exhaustive grid-sampling average, and that estimating it with Quantum Amplitude Estimation turns Monte Carlo's $O(m^{-1/2})$ sampling error into $O(m^{-1})$. If these claims hold, the result is a fully quantum financial risk pipeline that keeps the quadratic speedup and scales to $2^n$ parallel threads.

What carries the argument

The argument rests on four objects. Random Injection is the map $g(x)=f(x)\oplus (r==3)$, where $r\in\{0,1,2,3\}$ uniform, which flips the sign of a quarter of the parallel threads. Grid Sampling is the deterministic version that expands the thread count by four and takes the exhaustive average; the paper's equivalence theorem says the two expectations coincide. The Mixed-Signal Switch is a digital control ($AB=11$) combined with analog quantum operations that routes values above the strike threshold to the payoff register, replacing classical piecewise-linear interval classification. Quantum Amplitude Estimation is the measurement scheme that produces a deterministic average over all amplitudes, claimed to yield $O(m^{-1})$ convergence instead of the $O(m^{-1/2})$ of Monte Carlo sampling. A fitted scaling factor $m=1.57$ from a prior calibration study [4] connects the circuit's rotation angle to the Taylor-series payoff expression $x=1/2+i\theta$.

What would settle it

Recompute the payoff error with the scaling factor set to a value other than $1.57$, or derive $x=1/2+i\theta$ without dividing only the right-hand side; if the error rises back toward the original 3.912%, the 0.0134% accuracy is a calibration artifact rather than a property of the circuit.

Watch

Extended reading notes

Core claim

The central claim is that a single quantum circuit can do both jobs that previously forced classical help: generating randomness and computing a payoff. Random injection is defined by $g(x)=f(x)\oplus (r==3)$ with $r$ uniform on {0,1,2,3}; the paper derives $E[g(x)] = \frac{3}{4}f(x)+\frac{1}{4}(f(x)\oplus 1)$, which is identical to averaging over a deterministic grid of four $r$ values. The payoff component uses the two most significant bits of a 5-bit price as a control: when $AB=11$ (price at least 24, or strike 900 in the encoded example), the low-bit mean is routed to the payoff register, and otherwise the payoff is zero. Together with QAE, the architecture turns a random-sampling problem into a deterministic expectation estimation, giving a claimed $O(m^{-1})$ convergence and scaling to $2^n$ threads.

Load-bearing premise

The payoff accuracy claims rest on the fitted constant $m=1.57$, taken from an earlier study and inserted into the payoff formula; the step that divides only the right-hand side by 1.57 to obtain $x=1/2+i\theta$ is not algebraically justified, so if that constant is not universal the accuracy result is not derived from the circuit.

Editorial extensions

If this is right

Random injection and grid sampling produce identical expectation values, so Monte Carlo-type random factors can be replaced by deterministic QAE averaging in threaded quantum risk circuits.
The payoff circuit computes thresholded averages entirely inside the quantum circuit, removing the classical preprocessing and piecewise-linear segmentation bottleneck.
The 8-thread demonstration extends to $2^n$ threads, meaning one million random factors could simultaneously influence one million parallel computations.
QAE reaches the same accuracy as $m^2$ classical Monte Carlo samples with only $m$ quantum operations, preserving the quadratic speedup.
The two components can feed value-at-risk and conditional value-at-risk estimates in a fully quantum pipeline.

Reading between the lines

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

The equivalence theorem reduces random injection to a simple identity, so the practically novel block is the mixed-signal switch; future tests should stress that switch with different thresholds and probability distributions.
If the $m=1.57$ factor is a fitted calibration rather than a universal constant, the 0.0134% payoff error may not generalize to other strikes, PDFs, or bit widths.
The claimed $O(m^{-1})$ convergence assumes QAE acts as an exact deterministic averaging oracle; on real hardware, circuit depth and noise will add errors that this analysis does not model.
The scaling to $2^n$ threads and the quantum-supremacy framing are extrapolations from the 8-thread demonstration rather than demonstrated capabilities.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

This one is a reject, and I think your assessment is largely right, though I'd put the QAE gap even more centrally than the m=1.57 problem. Let me give credit where it's due: the paper does describe an explicit, concrete Qiskit circuit for thresholded payoff computation that filters out prices below a strike and averages the rest without classical preprocessing. For a fixed threshold, the circuit appears to work as a toy demonstration — the histogram and payoff numbers are consistent with what you'd expect from a controlled operation. The mixed-signal switch idea is not fundamentally new, but the implementation is spelled out, which is more than many papers in this area do.

The soft spots are load-bearing. The random-injection 'equivalence' is just linearity of expectation: flipping a bit with probability 1/4 is definitionally 3/4 f + 1/4 (f XOR 1). That's not a theorem; it's a trivial identity. The payoff calibration is circular: m=1.57 is fitted from previous data to minimize error, then the reduced error is presented as validation. The algebra 'dividing only the right-hand side by 1.57' is simply invalid, and it's not a minor typo — it's the step that manufactures the agreement with the Taylor expansion.

The bigger problem is QAE. The abstract and conclusion claim a quadratic speedup via QAE, but no QAE circuit, no amplitude oracle, and no QAE experiment appears anywhere. The experiments use 1,600 direct shots. The O(m^-1) convergence claim is asserted without a derivation or even an explicit encoding of the expectation into an amplitude. Without QAE, the remaining contribution is a simple controlled gate and a thresholded average, neither of which supports a quantum-advantage narrative. And the 'million threads' supremacy claim misunderstands what quantum advantage means: thread count is a classical parallelism metric.

No code or data are shipped, so the already thin empirical claims are hard to verify independently.

Who is this for? Maybe someone looking for a minimal example of a threshold circuit, but the paper's framing vastly overreaches. It is not a serious candidate for peer review as is. A desk reject is appropriate; if the authors want to turn this into a solid short note, they'd need to drop the QAE and supremacy claims, supply code, and show the payoff circuit on a real task where the thresholding is nontrivial.

I'd recommend desk reject without review.

Best, [You]

Referee Report

5 major / 6 minor

Summary. The paper proposes two quantum circuit components for financial risk assessment: a random-injection circuit that XORs a two-qubit random state into a parallel computation, and a mixed-signal payoff circuit that filters and averages values above a fixed threshold of 24. It presents an equivalence between random injection and grid sampling, asserts compatibility with Quantum Amplitude Estimation (QAE) and an O(m^-1) convergence rate, and reports Qiskit experiments using 8 threads and 1,600 shots. The payoff validation introduces a scaling factor m=1.57, taken from a prior study, and reports a reduction in payoff error from 3.912% to 0.0134% after calibration.

Significance. The paper contains a correct but elementary identity connecting random injection and grid sampling, and it reports concrete Qiskit circuit diagrams and histograms; these parts are reproducible at the level of ordinary circuit simulation. If the QAE integration and payoff calibration were valid, the architecture would be of interest for quantum finance. However, the central speedup claim is not accompanied by any QAE circuit, oracle, or experiment, and the payoff accuracy is produced by a fitted constant that is then used to claim validation. The numerical agreement in Section 3.2 is therefore not evidence for the circuit's correctness, and the 'quantum supremacy' framing is unsupported by any resource estimate. The paper's main value proposition—a fully quantum random-injection plus payoff pipeline with quadratic speedup—is not established.

major comments (5)

[Section 3.2, 'Applying a linear transformation...'] The algebraic step that equates the calibrated expression with the Taylor-series form is invalid. From x=0.785+1.57 iθ, dividing both sides by 1.57 gives x/1.57 = 1/2 + iθ, not x=1/2+iθ as written. Consequently, cases (3) and (4) in Fig. 6 compare different classical inputs, and the claimed reduction from 3.912% to 0.0134% is not a valid test of the payoff circuit against the original payoff function.
[Section 2.1, 'QAE has three key features...' and Section 4] The QAE speedup is asserted but never constructed. No QAE circuit, amplitude oracle, or encoding of the expected value into an amplitude is shown, and the experiments in Section 3 use 1,600 direct measurement shots rather than amplitude estimation. The claimed convergence rate O(m^-1) is stated without a derivation or a definition of m in relation to QAE phase estimation. The sentence in Section 2.1 that random injection 'with QAE measurements' yields O(m^-1) is therefore unsupported, and the paper's central claim of quadratic speedup is not demonstrated.
[Section 3.2, Eq. (1)] The use of m=1.57 is circular. The factor is adopted from [4] because it minimizes the error, it is then inserted into the payoff formula (1), and the resulting low error is reported as validation. The agreement between the calibrated case and the Taylor-series case is a consequence of applying the same fitted scaling in both, not an independent confirmation. Without a derivation of m from the circuit or a test on data not used in the calibration, the 0.0134% error cannot be attributed to the quantum payoff circuit.
[Section 2.1, Methods 1 and 2] The equivalence between random injection and grid sampling is correct but only shows that an expectation over the four equiprobable values of r equals the average over four deterministic grid points. It does not construct an A-operator for QAE, it does not show that the random-injection circuit is compatible with QAE, and it does not by itself imply any improved convergence rate. The subsequent claim in the same section that this integration yields O(m^-1) rather than O(m^-1/2) is a non sequitur unless an actual QAE estimation procedure is specified.
[Section 3.2, Fig. 6 and Eq. (1)] The relation between Eq. (1) and the quantum circuit is unspecified. The role of the symbol i in x=0.785+1.57iθ is not defined; if it is the imaginary unit, the expression is not a valid rotation angle, and if it is an index, the notation is never explained. The text also does not explain how the probabilities P[i+24] are extracted from the circuit or how the input variables map to gate parameters. Without this mapping, the payoff experiment cannot be reproduced from the manuscript.

minor comments (6)

[Section 2.1] The strings '2266' and '2268' should be typeset with explicit superscripts (e.g., 2^266 and 2^268); as printed, the thread counts are ambiguous.
[Abstract] The abstract states that the current implementation uses '23 threads', which conflicts with the '8 threads' used elsewhere and appears to be a typographical error for '2^3 threads'.
[Section 3.2, Eqs. (1) and (2)] The constants m and k are introduced as 'constant' but are never defined or assigned values; m is later fixed to 1.57, while k never appears in the numerical discussion, making the payoff formulas unreproducible.
[Section 3.1] The 9.26% error is referenced as 'Fig.2', but Fig. 2 shows circuit diagrams rather than the histogram or error data; the reference should point to the appropriate figure.
[Introduction] The phrase '1,024 times (denoted by m2)' uses an undefined symbol; it should be written as 'm^2' or as a stated number of classical samples.
[References] Reference [4] is listed as 'SNUG Taiwan' without author names, making it difficult to verify the source of the m=1.57 calibration value.

Circularity Check

2 steps flagged · score 7.0 of 10

Payoff validation reduces to a fitted rescaling constant and a one-sided variable transformation.

fitted input called prediction [Section 3.2, Average calculation result (payoff calibration)]
"However, a prior study [4] demonstrated that the minimal error occurs when the scaling factor m=1.57, leading to the refined expression: x = 0.785 + 1.57 × i × θ ... the analog calibration method proposed in [4] and the Taylor series-based approach in [1] both reduce the approximation error of PF (payoff) from 3.912% to 0.0134%, demonstrating their mathematical equivalence in this context."

The value m=1.57 is introduced as the result of a prior empirical minimization ('minimal error occurs'), not derived from the payoff function or the circuit. The same fitted factor is then inserted into the payoff formulas, producing the reported error drop to 0.0134%, and that same drop is cited as validation of the circuit and as proof that calibration matches the Taylor-series approach. The low-error 'prediction' is therefore statistically forced by the fitted input rather than independently confirmed.
self definitional [Section 3.2, Average calculation result (transformation to Taylor series)]
"Applying a linear transformation by dividing only the right-hand side by 1.57 yields: x = 1/2 + i×θ. This result exactly matches the expression derived from the Taylor series in [1], thereby providing theoretical consistency across different formulations."

Starting from x = 0.785 + 1.57 iθ, the paper divides only the right-hand side by 1.57 and declares x = 1/2 + iθ. This is not a valid linear transformation; it is a redefinition of the variable that makes the calibrated expression coincide with the Taylor-series expression by construction. The claimed 'theoretical consistency' is an artifact of this one-sided rescaling, not an independent derivation.

full rationale

The central payoff-accuracy validation is circular. In Section 3.2, the scaling factor m=1.57 is taken from prior work [4] as the value that minimizes the approximation error; it is then inserted into the payoff formulas, and the resulting 0.0134% error is presented as empirical validation of the circuit and as demonstration that the calibration equals the Taylor-series approach. The error reduction is a property of the fitted constant, not an independent confirmation. The accompanying step of 'dividing only the right-hand side by 1.57' to obtain x=1/2+iθ is algebraically invalid and forces the claimed match with the Taylor series by definition. The Random Injection / Grid Sampling equivalence in Section 2.1 is a true but trivial identity: both methods define the same observation function, so the equality is entailed by the definitions; it does not by itself establish that the random-injection unitary can serve as a QAE oracle. The asserted QAE O(m^-1) convergence and quantum-supremacy prospects are not demonstrated by the experiments, which use 1,600 direct shots and no amplitude-estimation oracle, but that is an evidence gap, not a circularity, so it is not counted in the score. Overall, the payoff prediction reduces to its fitted input, giving a circularity score of 7.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central validation relies on two unevidenced assumptions: the uniform-random XOR model standing in for financial randomness, and the calibrated factor m=1.57 that manufactures the quoted accuracy. QAE resources are assumed but not implemented, and the unspecified constant k blocks independent reproduction.

free parameters (2)

m (payoff scaling factor) = 1.57
Introduced via analog calibration from prior study [4]; used to transform x=0.785+1.57 i*theta into x=1/2+i*theta. The reported error reduction from 3.912 percent to 0.0134 percent depends on this fitted value.
k (payoff offset constant) = not stated
Appears in the PF and LM formulas in Eqs. (1) and (2) as an unspecified constant; without its value the payoff computation cannot be independently reproduced.

assumptions (4)

domain assumption QAE estimates the expectation of the random-injection function with O(m^-1) convergence
Stated in Section 2.1 and the Conclusion, but QAE is never run; the authors assume the standard QAE result from [6] transfers to their circuit without proof or implementation.
domain assumption The two-qubit uniform random variable r yields a faithful model of financial randomness
Section 2.1 defines g=f XOR (r==3); this is a single Bernoulli perturbation, not a return distribution or market scenario.
ad hoc to paper m=1.57 is a valid universal calibration constant for payoff approximation
Taken from SNUG Taiwan [4] to minimize error; no derivation shows it holds for arbitrary payoff functions or strike prices.
domain assumption The probability density can be loaded into the 5-qubit state at negligible cost
Section 3.2 simulates prices directly; no state preparation circuit or cost analysis is given.

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Cite this review

Pith. "Pith review of Novel Quantum Circuit Designs of Random Injection and Payoff Computation for Financial Risk Assessment." pith.science (2026). https://pith.science/paper/3SM42WJI

@misc{pith2026250723310,
  author       = {Pith},
  title        = {Pith review of: Novel Quantum Circuit Designs of Random Injection and Payoff Computation for Financial Risk Assessment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SM42WJI}},
  note         = {Machine review of arXiv:2507.23310}
}

read the original abstract

Quantum entanglement enables exponential computational states, while superposition provides inherent parallelism. Consequently, quantum circuits are theoretically capable of supporting large scale parallel computation. However, applying them to financial analysis particularly in the areas of random number generation and payoff computation remains a significant challenge. Experts generally believe that quantum computing relies on matrix operations, which are deterministic in nature without randomness. This inherent determinism makes it particularly challenging to design quantum circuits that require random number injection. JP Morgan[1] introduced the piecewise linear (PWL) approach for modeling payoff computations but did not disclose a quantum circuit capable of identifying values exceeding the strike price, suggesting a possible reliance on classical pre processing for interval classification. This paper presents an integrated quantum circuit with two key components: one for random number injection, applicable to risk assessment, and the other for direct payoff computation, relevant to financial pricing. These components are compatible with a scalable framework that leverages large scale parallelism and Quantum Amplitude Estimation (QAE) to achieve quadratic speedup. The circuit was implemented on IBM Qiskit and evaluated using 8 parallel threads and 1600 measurement shots. Results confirmed both the presence of randomness and the correctness of payoff computation. While the current implementation uses 8 threads, the design scales to 2 to the power of n threads, for arbitrarily large n, offering a potential path toward demonstrating quantum supremacy.

Figures

Figures reproduced from arXiv: 2507.23310 by the authors.

**Figure 3.** The quantum circuit automatically filters and averages the PDF [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

**Figure 4.** Quantum Circuit Architecture for Financial [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages

[4]

random injection

The execution flow involves input data of 2 266 threads, injecting random number r during computation, and computing average after completion. The expectation value calculation shows: 𝐸[𝑔(𝑥)] = 𝐸[𝑓(𝑥) 𝑋𝑂𝑅 (𝑟 == 3)] = 1 4 × [𝑓(𝑥) 𝑋𝑂𝑅 0 + 𝑓(𝑥) 𝑋𝑂𝑅 0 + 𝑓(𝑥) 𝑋𝑂𝑅 0 + 𝑓(𝑥) 𝑋𝑂𝑅 1] = 3 4 × 𝑓(𝑥) + 1 4 × ( 𝑓(𝑥) 𝑋𝑂𝑅 1) Method 2: Grid Sampling defines the observation...

work page
[1]

It also introduced the PWL approach for modeling payoff computations

INTRODUCTION JP Morgan [1] proposed a quantum circuit for mean computation, though loading arbitrary probability density function (PDF) into quantum computers remains a challenging problem. It also introduced the PWL approach for modeling payoff computations. Building on this foundation, Wilkens [2] introduc ed the use of classical preprocessing to partit...

work page
[2]

Random Injection Figure 1 illustrates the original 8 threads operating in massive parallel processing

METHODOLOGY 2.1. Random Injection Figure 1 illustrates the original 8 threads operating in massive parallel processing. After incorporating random numbers, the system maintains 8 threads in massive parallel processing while using random number generation to create 4 types of combinations, r₀r₁=(00, 01, 10, 11), such that 1 4 probability (11) will cause E ...

work page
[3]

proposed analog parallel parameter multiplication and addition circuits that significantly red uce the circuit complexity than digital parallel designs. More recently, SNUG Taiwan [4] developed mixed- signal quantum circuits for option pricing, combining the simplicity of analog design with the flexibility and synthesizability of digital circuits. This ap...

work page
[5]

Demonstration of Certified Randomness Generation Using a 56 -Qubit Quantum System,

Quantinuum, "Demonstration of Certified Randomness Generation Using a 56 -Qubit Quantum System," Nature Physics, vol. 21, no. 5, pp. 123-130, 2025

work page 2025
[6]

Random Injection Histogram Analysis Verification It compares the quantum circuit results with the expected E sign - change probability of 0.25

EXPERIMENTAL RESULTS 3.1. Random Injection Histogram Analysis Verification It compares the quantum circuit results with the expected E sign - change probability of 0.25. Under the condition of 1600 shots, the average error is 9.26%(Fig.2), with the error attributed to randomness effects. This research confirms that a single circuit can generate 2n groups ...

work page
[7]

By utilizing the deterministic computation capabilities of quantum circuits through Quantum Amplitude Estimation (QAE), we successfully approximate stochastic operations

CONCLUSION This paper presents two innovative quantum circuits, Random injection and Payoff computation, specifically designed for financial risk simulation applications. By utilizing the deterministic computation capabilities of quantum circuits through Quantum Amplitude Estimation (QAE), we successfully approximate stochastic operations. Our invention c...

work page
[8]

& Alexeev, Y

Herman, D., Googin, C., Liu, X., Galda, A., Safro, I., Sun, Y., ... & Alexeev, Y. (2022). A survey of quantum computing for finance. arXiv preprint arXiv:2201.02773

arXiv 2022

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Wilkens, S., & Moorhouse, J. (2023). Quantum computing for financial risk measurement. Quantum Information Processing, 22(1), 51

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Carrera Vaz quez, A., & Woerner, S. (2021). Efficient state preparation for quantum amplitude estimation. Physical Review Applied, 15(3), 034027

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Mixed-Signal Quantum Circuit Design for Option Pricing Using Design Compiler

SNUG Taiwan, " Mixed-Signal Quantum Circuit Design for Option Pricing Using Design Compiler ", Proceedings of the 2025 SNUG Taiwan Conference, Taipei, Taiwan, 2025

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Woerner, S., & Egger, D. J. (2019). Quantum risk analysis. npj Quantum Information, 5(1), 15

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Matsakos, T., & Nield, S. (2024). Quantum Monte Carlo simulations for financial risk analytics: scenario generation for equity, rate, and credit risk factors. Quantum, 8, 1306

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Montanaro, A. (2015). Quantum speedup of Monte Carlo methods. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2181), 20150301

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These PDFs can come from direct inputs or large -scale parallel computations. Traditionally, the PWL method uses classical computers to filter results greater than or equal to 24 and then sends the data to quantum computers for averaging, that may suffer from classical bottlen...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [4]

random injection

The execution flow involves input data of 2 266 threads, injecting random number r during computation, and computing average after completion. The expectation value calculation shows: 𝐸[𝑔(𝑥)] = 𝐸[𝑓(𝑥) 𝑋𝑂𝑅 (𝑟 == 3)] = 1 4 × [𝑓(𝑥) 𝑋𝑂𝑅 0 + 𝑓(𝑥) 𝑋𝑂𝑅 0 + 𝑓(𝑥) 𝑋𝑂𝑅 0 + 𝑓(𝑥) 𝑋𝑂𝑅 1] = 3 4 × 𝑓(𝑥) + 1 4 × ( 𝑓(𝑥) 𝑋𝑂𝑅 1) Method 2: Grid Sampling defines the observation...

work page

[2] [1]

It also introduced the PWL approach for modeling payoff computations

INTRODUCTION JP Morgan [1] proposed a quantum circuit for mean computation, though loading arbitrary probability density function (PDF) into quantum computers remains a challenging problem. It also introduced the PWL approach for modeling payoff computations. Building on this foundation, Wilkens [2] introduc ed the use of classical preprocessing to partit...

work page

[3] [2]

Random Injection Figure 1 illustrates the original 8 threads operating in massive parallel processing

METHODOLOGY 2.1. Random Injection Figure 1 illustrates the original 8 threads operating in massive parallel processing. After incorporating random numbers, the system maintains 8 threads in massive parallel processing while using random number generation to create 4 types of combinations, r₀r₁=(00, 01, 10, 11), such that 1 4 probability (11) will cause E ...

work page

[4] [3]

proposed analog parallel parameter multiplication and addition circuits that significantly red uce the circuit complexity than digital parallel designs. More recently, SNUG Taiwan [4] developed mixed- signal quantum circuits for option pricing, combining the simplicity of analog design with the flexibility and synthesizability of digital circuits. This ap...

work page

[5] [5]

Demonstration of Certified Randomness Generation Using a 56 -Qubit Quantum System,

Quantinuum, "Demonstration of Certified Randomness Generation Using a 56 -Qubit Quantum System," Nature Physics, vol. 21, no. 5, pp. 123-130, 2025

work page 2025

[6] [6]

Random Injection Histogram Analysis Verification It compares the quantum circuit results with the expected E sign - change probability of 0.25

EXPERIMENTAL RESULTS 3.1. Random Injection Histogram Analysis Verification It compares the quantum circuit results with the expected E sign - change probability of 0.25. Under the condition of 1600 shots, the average error is 9.26%(Fig.2), with the error attributed to randomness effects. This research confirms that a single circuit can generate 2n groups ...

work page

[7] [7]

By utilizing the deterministic computation capabilities of quantum circuits through Quantum Amplitude Estimation (QAE), we successfully approximate stochastic operations

CONCLUSION This paper presents two innovative quantum circuits, Random injection and Payoff computation, specifically designed for financial risk simulation applications. By utilizing the deterministic computation capabilities of quantum circuits through Quantum Amplitude Estimation (QAE), we successfully approximate stochastic operations. Our invention c...

work page

[8] [8]

& Alexeev, Y

Herman, D., Googin, C., Liu, X., Galda, A., Safro, I., Sun, Y., ... & Alexeev, Y. (2022). A survey of quantum computing for finance. arXiv preprint arXiv:2201.02773

arXiv 2022