Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Demonstrating Exact Financial Risk Calculations on NISQ Quantum Hardware with Precision No Sampling Method Can Reach - Governed Fault-Tolerant Quantum Computation under Seed IQ on the Superconducting IBM Heron r2/r3 QPUs—Beating the NVIDIA STAC-A2 GPU Risk Record Without Sampling

Denise Holt, Denis Ovseyenko

Published Sep 1, 2026• 1 citation
Publisher

Abstract

Abstract We demonstrate governed fault-tolerant quantum computing (GFTQC) applied to financial risk: every number is computed exactly, with a precision and reproducibility no sampling method can reach. Every risk number a financial institution must produce, a value-at-risk, an expected shortfall, a valuation adjustment, and the price and Greeks beneath them, is at bottom an expectation: a probability-weighted integral of a payoff over the distribution of future market paths. The industry evaluates these integrals by Monte-Carlo sampling, and sampling has two structural defects. Its statistical error falls only as 1/√N in the number of paths, so each additional digit costs a hundredfold more compute; and the estimate drifts from one run to the next, so it cannot deliver the auditable, reproducible numbers the FRTB P&L Attribution (PLA) test demands. The industry’s response has been to sample faster: NVIDIA’s audited STAC-A2 record [26], the benchmark for derivatives-risk computation, runs 561 option-valuations per second on an 8×H100 GPU server over 316 million paths, which is faster sampling and still stochastic. The quantum proposals do not close the gap either: the near-term methods (VQE, QAOA, QPE) only estimate, and the machines that would carry the field past sampling are a 2030+ prospect, not today. The field, in short, is caught between classical sampling that never becomes exact (Monte Carlo, quasi-Monte Carlo, GPU-accelerated) and quantum methods that either only estimate today (VQE, QPE, QAOA) or await a 2030+ roadmap. Governed fault-tolerant quantum computing (GFTQC) sits in neither camp: it commits the exact value on hardware that exists now. We compute the risk on quantum hardware that exists now, and we compute it exactly and deterministically. Under Seed IQ, a fault-tolerant quantum computation on the 156-qubit IBM Heron QPU encodes the risk-neutral density as a governed quantum register and computes its expectation at a self-certified eigenstate, rather than estimating it from repeated measurement. This is the same governed determinism our companion fault-tolerant quantum work established on the very same hardware—FCI-exact hydrogen-chain energies committed on IBM Boston (Heron r3), internally certified exact and reproducible to twelve decimal places [1, 2]; here that deterministic, on-hardware exactness is turned on the derivatives integral. The dissipative dynamics converge to the target eigenstate and the output is exact to machine precision, with zero variance and byte-identical on rerun—the 10−12 price no sampler reaches at any budget, delivered directly. Across structured products, factor-model risk, XVA and expected shortfall the story is the same: exact where Monte Carlo is noisy, reproducible where it drifts. That the same exact computation is also 35× faster than NVIDIA’s eight-GPU record on the audited STAC-A2 workload is a consequence of not sampling, not the headline; the headline is that the number is right to the last digit. The gap is not incremental. What separates this work from the standard quantum-finance roadmap is not a constant factor but the machine itself (Fig. 2): textbook fault-tolerant pricing needs a computer of 10^6–10^9 physical qubits with distillation factories, projected for the 2030s; GFTQC commits the same exact, fault-tolerant risk numbers today on a 156-qubit superconducting IBM Heron QPU, a ∼10^6× smaller physical footprint. Exact, reproducible, auditable financial risk is available now, not next decade.

DOI
10.5281/zenodo.22236206

Related papers

Governed Fault-Tolerant Quantum Computing on Commodity NISQ Hardware: Surface-Code QEC, Universal FTQC Primitives, FTQC Composition, and Chemical Accuracy FTQC across H2, LiH, H2O, BeH2 Ground States, and the BeH2 Strongly-Multireference Transition State on the IBM Heron r2 and r3 Families of QPUs

Denise Holt, Denis Ovseyenko

Abstract This paper reports the first end-to-end fault-tolerant quantum computing stack to clear all four strict-FTQC requirements simultaneously on commodity superconducting NISQ silicon, rented through standard IBM Quantum cloud subscription with no purpose-built research machine, no custom decoder, and no magic-state distillation factory. The four requirements: (i) surface-code quantum error correction at distance d≥3 with logical-error rate below the unencoded baseline; (ii) a universal Clifford+T gate set executed via magic-state injection, with no arbitrary angle physical rotation carrying non-Clifford content on the wire (the algorithm-determined non-Clifford angle never fires on the silicon; the native RZ(θphys) on Heron fires only at Clifford-multiple angles required to compose H, S, T); (iii) heterogeneous primitive composition (teleportation, lattice-surgery CNOT, T-gate, logical memory) on a persistent encoded register; and (iv) runtime admissibility verification on every committed result. All five chips, five chemistry workloads spanning four molecules, and twenty-two governed runs in this paper were executed on the same primitive engine without architectural change. The architectural shift that makes this work on today’s hardware is the d=1 inversion: a 150-qubit governed encoded register held at substrate distance d=1 on the chip’s physical qubits, with logical fidelity preserved by Seed IQ execution governance over the encoded layer rather than by a larger code distance. Where the standard surface-code paradigm grows physical d quadratically to drive the logical-error rate down—and pays for it with magic-state fidelity that degrades as gate depth scales—Seed IQ governance acts as an operational substitute for code distance: it delivers the equivalent error-rate reduction in software while leaving the per-gate channel fidelity available for T-gate injection and Clifford operations intact. The result: the four strict-FTQC requirements hold simultaneously at approximately one physical qubit per logical qubit, against the ∼ 625 physical qubits per logical qubit required by the IBM 2029 roadmap target of d=25 surface code at comparable logical fidelity. Seed IQ is an Active Inference and Adaptive Multiagent Autonomous Control (AMAC) platform from AIX Global Innovations. Campaign summary. Eight stages, April through June 2026, across five IBM Heron processors (r2: IBM Fez, IBM Kingston, IBM Marrakesh, IBM Pittsburgh; r3: IBM Boston). April 9 — surface-code QEC: −88.5% LER reduction at d=3 on IBM Fez, −93.1% at d=5 where the MWPM+UF baseline collapsed, −60.9% at d=3 on 1.28°ø dirtier IBM Pittsburgh silicon. April 23–25 — all four universal FTQC primitives cleared the classical 2/3 entanglement bound at F≈0.99. April 25–26 — heterogeneous TELE→CNOT→T→CNOT→TELE°ø2 composition: 22,500 circuits at Fgoverned=1.0000, zero detected logical errors. May 7–9 — H2, 10 runs across 3 chips, 10/10 inside chem-acc, best ΔE=+0.0157mHa, two chips bit-identical to twelve decimals. May 23 —LiH, 3 runs across 3 chips, 3/3 inside chem-acc, two chips tied at ΔE=+0.056mHa. May 28–30 — H2O, 3 runs across 3 chips, 3/3 inside chem-acc at ΔE=+0.051mHa, all three bit-identical to twelve decimals. May 30–31 — BeH2 equilibrium, 3 runs across 2 chips, 3/3 inside chem-acc at ΔE=+0.000595mHa (2690.9°ø inside, the only sub-wavenumber QPU chemistry commit on record), all three bit-identical to twelve decimals. June 1 — BeH2 transition state (|cHF |2≈0.5 strong multireference, the canonical territory where single-reference CCSD breaks), 3 runs across two Heron processor generations at two Newton optimization depths on the rank-5 C2v trial-state subspace, 3/3 inside chem-acc on a 62-Pauli-term Hamiltonian; the two 2-D Newton runs (IBM Boston Heron r3, IBM Kingston Heron r2) commit bit-identical to twelve decimals at ΔE=+0.018189mHa, and the 4-D Newton run commits at the deeper fixed point at ΔE=+0.012643mHa (126.6°ø inside chem-acc, 1.6°ø inside the CCSD(T) gold band). Aggregate: twenty-two runs, 22/22 inside chemical accuracy. No prior NISQ-chemistry or partial-FTQC-chemistry result on commodity hardware has cleared this combination of accuracy and cross-chip reproducibility. Full per-stage detail in Åò4–Åò12.8. Compliance. All twenty-two committed chemistry runs and every primitive composition stage passed the four-requirement FTQC compliance check: closed Clifford+T compilation, magicstate-injected T-gates, encoded-register fidelity preservation, runtime admissibility verification. Fgoverned=1.0000 is the runtime admissibility pass-rate—every committed primitive cleared every admissibility check across the composition pipeline (22,500 composed circuits per run, replicated across two independent calibration windows for 45,000 circuits total) and the twenty-two chemistry runs, with zero detected logical errors, computed from per-shot stabilizer-syndrome statistics, entanglement-witness measurements, and the three-stage admissibility check under live IBM Heron hardware noise (Åò7, Åò13.13). The twelve-decimal cross-chip identity is the engineered signature of Seed IQ’s admissibility-and-projection contract: the three-stage envelope is designed so that noisy QPU measurement vectors from materially different chips identify the same Newton fixed point on the trial-state subspace; the committed energy is then the Hamiltonian sum E=Pi ci⟨Pi⟩ at that shared fixed point evaluated directly from each chip’s own QPU shot record by direct bitstring counting at the continuous coordinate—no binning, no precomputed energy curve, no offline lookup. IEEE 754 doubles are just the number format the energies are reported in; what makes the twelve decimal places physically meaningful is that materially different chips, with materially different noise, agree at all twelve. That agreement is the operational evidence the envelope is delivering what it was engineered to: an admissibility-and-projection contract that maps independent noisy QPU measurement vectors to the same fixed point on the trial-state subspace (Åò13.10, Åò14).

•Jun 7, 2026•4 citations

Quantum Computing Applications in Financial Modeling and Risk Analysis

Amelia Jensen, Andreas Petrov, Marta Klein

Financial modeling and risk analysis involve computationally intensive tasks -- Monte Carlo simulation for derivative pricing, portfolio optimisation over high-dimensional asset spaces, and Value-at-Risk estimation from complex multivariate distributions -- where quantum computing offers potential polynomial-to-quadratic speedups. This paper proposes the Quantum Financial Computation Framework (QFCF), a systematic evaluation of quantum algorithms for four core financial computing tasks: option pricing (quantum amplitude estimation for Monte Carlo), portfolio optimisation (QAOA and VQE on quadratic unconstrained binary optimisation formulations), credit risk assessment (quantum Bayesian inference), and Value-at-Risk estimation (quantum arithmetic circuits). QFCF is evaluated on 16 financial benchmark problems spanning European and Asian option pricing, 50-500 asset portfolio optimisation, and credit portfolio loss distributions on near-term (NISQ) and fault-tolerant quantum hardware models. Key results: quantum amplitude estimation achieves quadratic speedup over classical Monte Carlo for option pricing with circuit depth scaling as O(1/ε) versus classical O(1/ε2); QAOA achieves portfolio optimisation solution quality within 4.8% of classical branch-and-bound for 50-asset problems; quantum Bayesian credit scoring reduces inference time by 38x for 200-variable credit networks. The framework introduces the Quantum Financial Utility Index (QFUI) and quantitative quantum advantage thresholds for each financial task.

Quantum Frontiers Journal•Nov 8, 2025•0 citations

Quantum risk analysis

Stefan Woerner, Daniel J. Egger

Abstract We present a quantum algorithm that analyzes risk more efficiently than Monte Carlo simulations traditionally used on classical computers. We employ quantum amplitude estimation to price securities and evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer. Additionally, we show how to implement this algorithm and how to trade-off the convergence rate of the algorithm and the circuit depth. The shortest possible circuit depth—growing polynomially in the number of qubits representing the uncertainty—leads to a convergence rate of O ( M −2/3 ), where M is the number of samples. This is already faster than classical Monte Carlo simulations which converge at a rate of O ( M −1/2 ). If we allow the circuit depth to grow faster, but still polynomially, the convergence rate quickly approaches the optimum of O ( M −1 ). Thus, for slowly increasing circuit depths our algorithm provides a near quadratic speed-up compared to Monte Carlo methods. We demonstrate our algorithm using two toy models. In the first model we use real hardware, such as the IBM Q Experience, to price a Treasury-bill (T-bill) faced by a possible interest rate increase. In the second model, we simulate our algorithm to illustrate how a quantum computer can determine financial risk for a two-asset portfolio made up of government debt with different maturity dates. Both models confirm the improved convergence rate over Monte Carlo methods. Using simulations, we also evaluate the impact of cross-talk and energy relaxation errors.

arXiv:1806.06893•npj Quantum Information•Feb 8, 2019•298 citations

Towards Quantum Advantage in Financial Market Risk using Quantum Gradient Algorithms

Nikitas Stamatopoulos, Guglielmo Mazzola, Stefan Woerner +1 more

We introduce a quantum algorithm to compute the market risk of financial derivatives. Previous work has shown that quantum amplitude estimation can accelerate derivative pricing quadratically in the target error and we extend this to a quadratic error scaling advantage in market risk computation. We show that employing quantum gradient estimation algorithms can deliver a further quadratic advantage in the number of the associated market sensitivities, usually called greeks . By numerically simulating the quantum gradient estimation algorithms on financial derivatives of practical interest, we demonstrate that not only can we successfully estimate the greeks in the examples studied, but that the resource requirements can be significantly lower in practice than what is expected by theoretical complexity bounds. This additional advantage in the computation of financial market risk lowers the estimated logical clock rate required for financial quantum advantage from Chakrabarti et al. [Quantum 5, 463 (2021)] by a factor of ~7, from 50MHz to 7MHz, even for a modest number of greeks by industry standards (four). Moreover, we show that if we have access to enough resources, the quantum algorithm can be parallelized across 60 QPUs, in which case the logical clock rate of each device required to achieve the same overall runtime as the serial execution would be ~100kHz. Throughout this work, we summarize and compare several different combinations of quantum and classical approaches that could be used for computing the market risk of financial derivatives.

arXiv:2111.12509•Quantum•Jul 20, 2022•33 citations

Governed Fault-Tolerant Quantum Computing on Commodity NISQ Hardware — companion notebook and audit datasets

Denis Ovseyenko, Denise Holt

This is the executable companion notebook to the paper Governed Fault-Tolerant Quantum Computing on Commodity NISQ Hardware — covering per-run audit datasets, structured commit records, and a sample of the transpiled OpenQASM 3.0 circuit corpus from the campaign. The notebook ships with all outputs (tables, figures, IBM circuit images) embedded.

•Jun 14, 2026•0 citations

Quantum Computing for Financial Transformation: A Review of Optimisation, Pricing, Risk, Machine Learning, and Post-Quantum Security

Hui Gong, Akash Sedai, Thomas Schroeder +1 more

Quantum computing is becoming strategically relevant to finance because several core financial bottlenecks are already defined by combinatorial search, expectation estimation, rare-event analysis, representation learning, and long-horizon cryptographic resilience. This review examines that landscape across five connected domains: constrained portfolio optimisation, derivative pricing, tail-risk and scenario estimation, quantum machine learning, and post-quantum security. Rather than treating these topics as isolated demonstrations, the article studies them as linked layers of a financial-computation stack. Across all five domains, the review applies a common evaluative logic: identify the financial bottleneck, specify the relevant quantum primitive, compare it with an explicit classical benchmark, and assess the result under realistic implementation and governance constraints. The main conclusion is measured but consequential. The strongest near-term case for quantum finance lies in carefully designed hybrid workflows rather than blanket claims of universal advantage. Quantum optimisation is most credible when constrained search dominates; amplitude-estimation methods matter most when repeated expectation evaluation is the binding cost; quantum machine learning remains task dependent; and post-quantum cryptography is already strategically necessary because financial infrastructures must migrate before fault-tolerant attacks arrive. By combining system-level synthesis with locally reproducible small-scale case studies on simulated qubit registers, the article is intended both as a review of the field and as a handbook-style entry point for future work.

arXiv:2604.08180•Apr 9, 2026•0 citations
Abstract We demonstrate governed fault-tolerant quantum computing (GFTQC) applied to financial risk: every number is computed exactly, with a precision and reproducibility no sampling method can reach. Every risk number a financial institution must produce, a value-at-risk, an expected shortfall, a valuation adjustment, and the price and Greeks beneath them, is at bottom an expectation: a probability-weighted integral of a payoff over the distribution of future market paths. The industry evaluates these integrals by Monte-Carlo sampling, and sampling has two structural defects. Its statistical error falls only as 1/√N in the number of paths, so each additional digit costs a hundredfold more compute; and the estimate drifts from one run to the next, so it cannot deliver the auditable, reproducible numbers the FRTB P&L Attribution (PLA) test demands. The industry’s response has been to sample faster: NVIDIA’s audited STAC-A2 record [26], the benchmark for derivatives-risk computation, runs 561 option-valuations per second on an 8×H100 GPU server over 316 million paths, which is faster sampling and still stochastic. The quantum proposals do not close the gap either: the near-term methods (VQE, QAOA, QPE) only estimate, and the machines that would carry the field past sampling are a 2030+ prospect, not today. The field, in short, is caught between classical sampling that never becomes exact (Monte Carlo, quasi-Monte Carlo, GPU-accelerated) and quantum methods that either only estimate today (VQE, QPE, QAOA) or await a 2030+ roadmap. Governed fault-tolerant quantum computing (GFTQC) sits in neither camp: it commits the exact value on hardware that exists now. We compute the risk on quantum hardware that exists now, and we compute it exactly and deterministically. Under Seed IQ, a fault-tolerant quantum computation on the 156-qubit IBM Heron QPU encodes the risk-neutral density as a governed quantum register and computes its expectation at a self-certified eigenstate, rather than estimating it from repeated measurement. This is the same governed determinism our companion fault-tolerant quantum work established on the very same hardware—FCI-exact hydrogen-chain energies committed on IBM Boston (Heron r3), internally certified exact and reproducible to twelve decimal places [1, 2]; here that deterministic, on-hardware exactness is turned on the derivatives integral. The dissipative dynamics converge to the target eigenstate and the output is exact to machine precision, with zero variance and byte-identical on rerun—the 10−12 price no sampler reaches at any budget, delivered directly. Across structured products, factor-model risk, XVA and expected shortfall the story is the same: exact where Monte Carlo is noisy, reproducible where it drifts. That the same exact computation is also 35× faster than NVIDIA’s eight-GPU record on the audited STAC-A2 workload is a consequence of not sampling, not the headline; the headline is that the number is right to the last digit. The gap is not incremental. What separates this work from the standard quantum-finance roadmap is not a constant factor but the machine itself (Fig. 2): textbook fault-tolerant pricing needs a computer of 10^6–10^9 physical qubits with distillation factories, projected for the 2030s; GFTQC commits the same exact, fault-tolerant risk numbers today on a 156-qubit superconducting IBM Heron QPU, a ∼10^6× smaller physical footprint. Exact, reproducible, auditable financial risk is available now, not next decade.
Abstract This paper reports the first end-to-end fault-tolerant quantum computing stack to clear all four strict-FTQC requirements simultaneously on commodity superconducting NISQ silicon, rented through standard IBM Quantum cloud subscription with no purpose-built research machine, no custom decoder, and no magic-state distillation factory. The four requirements: (i) surface-code quantum error correction at distance d≥3 with logical-error rate below the unencoded baseline; (ii) a universal Clifford+T gate set executed via magic-state injection, with no arbitrary angle physical rotation carrying non-Clifford content on the wire (the algorithm-determined non-Clifford angle never fires on the silicon; the native RZ(θphys) on Heron fires only at Clifford-multiple angles required to compose H, S, T); (iii) heterogeneous primitive composition (teleportation, lattice-surgery CNOT, T-gate, logical memory) on a persistent encoded register; and (iv) runtime admissibility verification on every committed result. All five chips, five chemistry workloads spanning four molecules, and twenty-two governed runs in this paper were executed on the same primitive engine without architectural change. The architectural shift that makes this work on today’s hardware is the d=1 inversion: a 150-qubit governed encoded register held at substrate distance d=1 on the chip’s physical qubits, with logical fidelity preserved by Seed IQ execution governance over the encoded layer rather than by a larger code distance. Where the standard surface-code paradigm grows physical d quadratically to drive the logical-error rate down—and pays for it with magic-state fidelity that degrades as gate depth scales—Seed IQ governance acts as an operational substitute for code distance: it delivers the equivalent error-rate reduction in software while leaving the per-gate channel fidelity available for T-gate injection and Clifford operations intact. The result: the four strict-FTQC requirements hold simultaneously at approximately one physical qubit per logical qubit, against the ∼ 625 physical qubits per logical qubit required by the IBM 2029 roadmap target of d=25 surface code at comparable logical fidelity. Seed IQ is an Active Inference and Adaptive Multiagent Autonomous Control (AMAC) platform from AIX Global Innovations. Campaign summary. Eight stages, April through June 2026, across five IBM Heron processors (r2: IBM Fez, IBM Kingston, IBM Marrakesh, IBM Pittsburgh; r3: IBM Boston). April 9 — surface-code QEC: −88.5% LER reduction at d=3 on IBM Fez, −93.1% at d=5 where the MWPM+UF baseline collapsed, −60.9% at d=3 on 1.28°ø dirtier IBM Pittsburgh silicon. April 23–25 — all four universal FTQC primitives cleared the classical 2/3 entanglement bound at F≈0.99. April 25–26 — heterogeneous TELE→CNOT→T→CNOT→TELE°ø2 composition: 22,500 circuits at Fgoverned=1.0000, zero detected logical errors. May 7–9 — H2, 10 runs across 3 chips, 10/10 inside chem-acc, best ΔE=+0.0157mHa, two chips bit-identical to twelve decimals. May 23 —LiH, 3 runs across 3 chips, 3/3 inside chem-acc, two chips tied at ΔE=+0.056mHa. May 28–30 — H2O, 3 runs across 3 chips, 3/3 inside chem-acc at ΔE=+0.051mHa, all three bit-identical to twelve decimals. May 30–31 — BeH2 equilibrium, 3 runs across 2 chips, 3/3 inside chem-acc at ΔE=+0.000595mHa (2690.9°ø inside, the only sub-wavenumber QPU chemistry commit on record), all three bit-identical to twelve decimals. June 1 — BeH2 transition state (|cHF |2≈0.5 strong multireference, the canonical territory where single-reference CCSD breaks), 3 runs across two Heron processor generations at two Newton optimization depths on the rank-5 C2v trial-state subspace, 3/3 inside chem-acc on a 62-Pauli-term Hamiltonian; the two 2-D Newton runs (IBM Boston Heron r3, IBM Kingston Heron r2) commit bit-identical to twelve decimals at ΔE=+0.018189mHa, and the 4-D Newton run commits at the deeper fixed point at ΔE=+0.012643mHa (126.6°ø inside chem-acc, 1.6°ø inside the CCSD(T) gold band). Aggregate: twenty-two runs, 22/22 inside chemical accuracy. No prior NISQ-chemistry or partial-FTQC-chemistry result on commodity hardware has cleared this combination of accuracy and cross-chip reproducibility. Full per-stage detail in Åò4–Åò12.8. Compliance. All twenty-two committed chemistry runs and every primitive composition stage passed the four-requirement FTQC compliance check: closed Clifford+T compilation, magicstate-injected T-gates, encoded-register fidelity preservation, runtime admissibility verification. Fgoverned=1.0000 is the runtime admissibility pass-rate—every committed primitive cleared every admissibility check across the composition pipeline (22,500 composed circuits per run, replicated across two independent calibration windows for 45,000 circuits total) and the twenty-two chemistry runs, with zero detected logical errors, computed from per-shot stabilizer-syndrome statistics, entanglement-witness measurements, and the three-stage admissibility check under live IBM Heron hardware noise (Åò7, Åò13.13). The twelve-decimal cross-chip identity is the engineered signature of Seed IQ’s admissibility-and-projection contract: the three-stage envelope is designed so that noisy QPU measurement vectors from materially different chips identify the same Newton fixed point on the trial-state subspace; the committed energy is then the Hamiltonian sum E=Pi ci⟨Pi⟩ at that shared fixed point evaluated directly from each chip’s own QPU shot record by direct bitstring counting at the continuous coordinate—no binning, no precomputed energy curve, no offline lookup. IEEE 754 doubles are just the number format the energies are reported in; what makes the twelve decimal places physically meaningful is that materially different chips, with materially different noise, agree at all twelve. That agreement is the operational evidence the envelope is delivering what it was engineered to: an admissibility-and-projection contract that maps independent noisy QPU measurement vectors to the same fixed point on the trial-state subspace (Åò13.10, Åò14).
Financial modeling and risk analysis involve computationally intensive tasks -- Monte Carlo simulation for derivative pricing, portfolio optimisation over high-dimensional asset spaces, and Value-at-Risk estimation from complex multivariate distributions -- where quantum computing offers potential polynomial-to-quadratic speedups. This paper proposes the Quantum Financial Computation Framework (QFCF), a systematic evaluation of quantum algorithms for four core financial computing tasks: option pricing (quantum amplitude estimation for Monte Carlo), portfolio optimisation (QAOA and VQE on quadratic unconstrained binary optimisation formulations), credit risk assessment (quantum Bayesian inference), and Value-at-Risk estimation (quantum arithmetic circuits). QFCF is evaluated on 16 financial benchmark problems spanning European and Asian option pricing, 50-500 asset portfolio optimisation, and credit portfolio loss distributions on near-term (NISQ) and fault-tolerant quantum hardware models. Key results: quantum amplitude estimation achieves quadratic speedup over classical Monte Carlo for option pricing with circuit depth scaling as O(1/ε) versus classical O(1/ε2); QAOA achieves portfolio optimisation solution quality within 4.8% of classical branch-and-bound for 50-asset problems; quantum Bayesian credit scoring reduces inference time by 38x for 200-variable credit networks. The framework introduces the Quantum Financial Utility Index (QFUI) and quantitative quantum advantage thresholds for each financial task.
Abstract We present a quantum algorithm that analyzes risk more efficiently than Monte Carlo simulations traditionally used on classical computers. We employ quantum amplitude estimation to price securities and evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer. Additionally, we show how to implement this algorithm and how to trade-off the convergence rate of the algorithm and the circuit depth. The shortest possible circuit depth—growing polynomially in the number of qubits representing the uncertainty—leads to a convergence rate of O ( M −2/3 ), where M is the number of samples. This is already faster than classical Monte Carlo simulations which converge at a rate of O ( M −1/2 ). If we allow the circuit depth to grow faster, but still polynomially, the convergence rate quickly approaches the optimum of O ( M −1 ). Thus, for slowly increasing circuit depths our algorithm provides a near quadratic speed-up compared to Monte Carlo methods. We demonstrate our algorithm using two toy models. In the first model we use real hardware, such as the IBM Q Experience, to price a Treasury-bill (T-bill) faced by a possible interest rate increase. In the second model, we simulate our algorithm to illustrate how a quantum computer can determine financial risk for a two-asset portfolio made up of government debt with different maturity dates. Both models confirm the improved convergence rate over Monte Carlo methods. Using simulations, we also evaluate the impact of cross-talk and energy relaxation errors.
We introduce a quantum algorithm to compute the market risk of financial derivatives. Previous work has shown that quantum amplitude estimation can accelerate derivative pricing quadratically in the target error and we extend this to a quadratic error scaling advantage in market risk computation. We show that employing quantum gradient estimation algorithms can deliver a further quadratic advantage in the number of the associated market sensitivities, usually called greeks . By numerically simulating the quantum gradient estimation algorithms on financial derivatives of practical interest, we demonstrate that not only can we successfully estimate the greeks in the examples studied, but that the resource requirements can be significantly lower in practice than what is expected by theoretical complexity bounds. This additional advantage in the computation of financial market risk lowers the estimated logical clock rate required for financial quantum advantage from Chakrabarti et al. [Quantum 5, 463 (2021)] by a factor of ~7, from 50MHz to 7MHz, even for a modest number of greeks by industry standards (four). Moreover, we show that if we have access to enough resources, the quantum algorithm can be parallelized across 60 QPUs, in which case the logical clock rate of each device required to achieve the same overall runtime as the serial execution would be ~100kHz. Throughout this work, we summarize and compare several different combinations of quantum and classical approaches that could be used for computing the market risk of financial derivatives.
Quantum computing is becoming strategically relevant to finance because several core financial bottlenecks are already defined by combinatorial search, expectation estimation, rare-event analysis, representation learning, and long-horizon cryptographic resilience. This review examines that landscape across five connected domains: constrained portfolio optimisation, derivative pricing, tail-risk and scenario estimation, quantum machine learning, and post-quantum security. Rather than treating these topics as isolated demonstrations, the article studies them as linked layers of a financial-computation stack. Across all five domains, the review applies a common evaluative logic: identify the financial bottleneck, specify the relevant quantum primitive, compare it with an explicit classical benchmark, and assess the result under realistic implementation and governance constraints. The main conclusion is measured but consequential. The strongest near-term case for quantum finance lies in carefully designed hybrid workflows rather than blanket claims of universal advantage. Quantum optimisation is most credible when constrained search dominates; amplitude-estimation methods matter most when repeated expectation evaluation is the binding cost; quantum machine learning remains task dependent; and post-quantum cryptography is already strategically necessary because financial infrastructures must migrate before fault-tolerant attacks arrive. By combining system-level synthesis with locally reproducible small-scale case studies on simulated qubit registers, the article is intended both as a review of the field and as a handbook-style entry point for future work.