Quantum computing for finance

Introduction

• Financial institutions use stochastic modeling, optimization, and machine learning for forecasting and optimization tasks such as pricing, risk estimation, portfolio selection, trading, and hedging.Source: Introduction · p. 2
• Quantum Monte Carlo integration has shown promise for a quadratic speedup in convergence compared with its classical counterpart.Source: Introduction · p. 2
• End-to-end quantum advantages remain challenging because state preparation, readout, pre- and post-processing, and current noisy hardware can limit practical performance.Current NISQ devices have low fidelity and few qubits, while classical computation remains important for circuit optimization, initial conditions, and error mitigation.Source: Introduction · p. 2
• The Review covers quantum algorithms for financial problems in stochastic modeling, optimization, and machine learning.It also places these algorithms alongside classical techniques used in each area.Source: Introduction · pp. 2–3
• The Review discusses whether quantum algorithms could be useful, their implementation challenges, and potential research directions.Source: Introduction · pp. 2–3

Stochastic modeling

• Stochastic processes model market quantities such as stock prices, interest rates, and volatilities to support investment decisions involving returns and risks.Source: Stochastic modeling · p. 3

Monte Carlo pricing and risk analysis

• Classical Monte Carlo integration estimates derivative prices from simulated payoffs, requiring O(σ2/ε2) samples for estimation error ε.Source: Quantum methods for Monte Carlo-based… · p. 4
• Quantum Monte Carlo integration can estimate the same price with O(σ/ε) quantum samples at constant success probability.QMCI uses quantum amplitude estimation to estimate the probability of observing the payoff qubit in the |1⟩ state.Source: Quantum methods for Monte Carlo-based… · p. 4
• Practical QMCI advantage depends on efficient unitary state preparation, but distribution loading can lose quadratic speedup, while path-increment loading uses qubits linear in time steps.Quantum arithmetic can also be expensive, leaving alternative payoff-distribution loading procedures unresolved.Source: Quantum methods for Monte Carlo-based… · p. 4

Greeks computation

• Greeks measure derivative-price sensitivities and support systematic hedging, while classical bump-and-reprice and path-wise derivative methods compute them under stated continuity or smoothness conditions.Source: Quantum methods for Monte Carlo-based… · p. 5

Monte Carlo pricing and risk analysis

• QMCI can also target risk metrics involving expected quantities, including value-at-risk and credit risk.Source: Quantum methods for Monte Carlo-based… · p. 5

Differential-equation methods

• Quantum approaches also combine finite-difference discretization with quantum linear-system algorithms, variational quantum simulation, or quantum neural networks for differential-equation-based pricing and risk analysis.Source: Quantum methods for… · pp. 5–6
• For heuristic variational and neural approaches, whether any quantum advantage exists remains unclear without further experimentation.Source: Quantum methods for… · p. 6

Scope

• Financial optimization spans continuous, discrete, and mixed-variable problems, including highly constrained linear or quadratic programs.Quantum heuristics and algorithms are reviewed across these categories.Source: Optimization · p. 6

Portfolio optimization

• Portfolio optimization selects asset allocations to trade off expected return and financial risk under a feasible set of constraints.The review considers continuous-valued allocations and may allow a non-convex feasible set.Source: Quantum methods for continuous… · p. 7

Continuous optimization

• Structured convex programs such as LPs, SOCPs, and SDPs have efficient classical algorithms, leaving only a small practical margin for quantum speedup.Quantum worst-case reductions can arise from fast linear-algebra subroutines, but practical comparisons are complicated by QBLAS caveats.Source: Quantum methods for continuous… · pp. 6–7
• QBLASs incur polynomial dependence on matrix conditioning, sampling costs for retrieving quantum data, and restrictions on efficient superposition access to classical inputs.Efficient input access without quantum memory is available only in certain sparse-matrix cases.Source: Quantum methods for continuous… · p. 7

Polynomial-time methods

• Quantum MMW achieves optimal general dependence on the number of constraints and variables, but its financial applications remain limited in study.The cited quantum complexity is ˜O(s√nγ5), measured in gates and matrix-element-oracle calls.Source: Quantum methods for continuous… · p. 7
• Quantum interior-point methods may solve only approximations of the original optimization problem, and resource estimates for simple portfolio optimization show no advantage.Quantum sampling negates favorable classical error dependence and yields approximate Newton-system solutions.Source: Quantum methods for continuous… · pp. 7–8

Non-convex optimization

• Convex relaxations can provide nearly identical solution quality to exact solvers for some tax-aware and sparse portfolio problems at lower computational cost.Quantum methods have also shown speedups for certain continuous non-convex landscapes, though the evidence concerns selected problems.Source: Quantum methods for continuous… · p. 8

Discrete optimization

• Quantum-walk search offers almost quadratic tree-search speedups, but excluding practical heuristics and subproblem costs leaves time-to-solution uncertain.Extensions incorporate depth-first and broader practical heuristics, yet the overall resource burden remains unresolved.Source: Quantum methods for discrete… · p. 9

• Financial machine learning supports forecasting, anomaly detection, classification, and other applications using historical data and sophisticated models.Source: Machine learning · p. 10
• Quantum machine-learning methods either accelerate classical techniques with QBLASs or use quantum-native models that are harder to simulate classically.QBLAS-based acceleration typically requires error correction, whereas quantum-native methods may be nearer term.Source: Machine learning · p. 10
• Quantum-native methods may be intractable to simulate classically, but their advantage on classical problems is currently unclear and may appear unlikely.Further algorithmic research and real-world benchmarking are needed.Source: Machine learning · pp. 10–11
• Quantum algorithms have been proposed for regression, classification, clustering, boosting, generative learning, and Bayesian inference in finance.Source: Quantum methods for regression · p. 11, Quantum methods for classification · p. 11, Quantum methods for boosting · p. 11

Supervised learning

• Quantum classifiers can achieve a quadratic improvement for training with constant margin, while quantum Gaussian-process regression can obtain up to exponential speedup under certain kernel conditions.Source: Quantum methods for classification · p. 11, Quantum methods for regression · p. 11

Generative learning

• QCBM at least matches restricted Boltzmann-machine performance and demonstrates superior performance as the model scales in numerical distribution-learning simulations.Source: Quantum methods for generative learning · p. 12

• Quantum-inspired algorithms could benefit high-dimensional financial problems when high precision is not required, despite refuting many originally claimed exponential speedups.Source: Dequantized algorithms · p. 14

• Finance contains computationally challenging problems where quantum algorithms may provide real-world speedups, but end-to-end advantage on practical problems remains unclear.Wall-clock speed and model accuracy can directly affect financial profit and loss.Source: Outlook · p. 14

Hardware challenges

• State preparation, reversible arithmetic, and multi-qubit interactions are hardware requirements that can increase circuit complexity and qubit counts.Native multicontrolled or multi-qubit gates and efficient quantum floating-point arithmetic could reduce these burdens.Source: Outlook · p. 14
• Existing quantum memory technologies have fundamental limitations that make low-cost, scalable quantum memory without active error correction highly challenging.Source: Outlook · p. 14
• Quantum hardware operates more slowly than classical computers, limiting practical usability; variational machine-learning algorithms also require many high-quality circuit evaluations for gradient estimation.Source: Outlook · p. 14
• Quantum error correction introduces significant overhead that can potentially negate certain quantum speedups for finance.Improving error correction and quantum architecture is described as critical for commercial applications.Source: Outlook · p. 15

Research outlook

• Further algorithmic research and benchmarking of increasingly complex, especially heuristic, algorithms are needed as quantum hardware advances.Source: Outlook · p. 15

• An autocallable is a financial product that pays the holder a high return if the underlying asset passes an upside threshold.Source: Glossary · p. 28
• The Black–Scholes model is a mathematical model for the dynamics of a financial market containing derivative investment.Source: Glossary · p. 28
• Bump-and-reprice estimates a derivative price sensitivity by evaluating prices at different underlying-parameter values and taking their difference.Source: Glossary · p. 28
• A financial derivative is a contract whose value derives from the performance of an underlying entity.Source: Glossary · p. 28
• The Hamilton–Jacobi–Bellman equation gives a necessary and sufficient condition for control optimality with respect to a loss function.Source: Glossary · p. 28
• An option gives its holder the right, but not the obligation, to buy or sell an underlying asset at an agreed price and time.Source: Glossary · p. 28
• A target accrual redemption forward pays according to whether a spot rate is above or below a target, subject to schedules, limits, and barriers.Source: Glossary · p. 28
• The Vapnik–Chervonenkis dimension measures the capacity of a binary-classification function set by the largest always-perfectly-classifiable arbitrarily labelled data set.Source: Glossary · p. 28

• The paper is informational and is not a product of JPMorgan Chase & Co.’s Research Department or its affiliates.Source: Disclaimer · p. 29
• JPMorgan Chase & Co. and its affiliates make no explicit or implied warranty and accept no liability regarding the paper.Source: Disclaimer · p. 29