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Quantum computing for finance
Dylan Herman, Cody Googin, Xiaoyuan Liu, Yue Sun, Alexey Galda, Ilya Safro, Marco Pistoia, Yuri Alexeev
TL;DR
Financial institutions face computationally challenging forecasting and optimization problems, motivating investigation of quantum methods for finance. This Review surveys quantum approaches across stochastic modeling, optimization, and machine learning, compares them with classical techniques, and identifies implementation challenges. It reports potential speedups and application results, while emphasizing that end-to-end quantum advantage remains constrained by resource, hardware, and algorithmic caveats.
Problem
Financial institutions routinely face computationally challenging forecasting and optimization problems, including pricing, risk estimation, portfolio selection, trading, and hedging.
Method
The Review surveys quantum algorithms for financial applications across stochastic modeling, optimization, and machine learning, comparing them with classical techniques and discussing implementation challenges.
Results
The Review identifies potential quantum speedups and financial applications, including QMCI for expected-value estimation, quantum methods for risk metrics, and quantum generative models for distribution learning.
Takeaways & Limitations
Quantum computing for finance offers promising algorithmic approaches, but practical value depends on resolving bottlenecks in state preparation, readout, processing, and hardware.
Takeaways & Limitations
End-to-end quantum advantages remain difficult for commercially relevant problems because quantum algorithms face resource, conditioning, sampling, state-preparation, and hardware constraints.
Abstract
from arXiv · showhide
Quantum computers are expected to surpass the computational capabilities of classical computers and have a transformative impact on numerous industry sectors. We present a comprehensive summary of the state of the art of quantum computing for financial applications, with particular emphasis on stochastic modeling, optimization, and machine learning. This Review is aimed at physicists, so it outlines the classical techniques used by the financial industry and discusses the potential advantages and limitations of quantum techniques. Finally, we look at the challenges that physicists could help tackle.
Introduction
This Review surveys quantum algorithms for financial applications across stochastic modeling, optimization, and machine learning, while comparing them with classical techniques and identifying implementation challenges and research directions.
- 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.
- Introduction: Quantum Monte Carlo integration has shown promise for a quadratic speedup in convergence compared with its classical counterpart.
- Introduction: 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.
- Introduction: 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.
- Introduction: The Review discusses whether quantum algorithms could be useful, their implementation challenges, and potential research directions.
Stochastic modeling
Stochastic modeling underlies financial pricing and risk analysis, where quantum methods target Monte Carlo, differential-equation, and gradient-estimation workloads. Quantum Monte Carlo methods offer improved theoretical error scaling, but state preparation, arithmetic, sampling, and hardware constraints limit demonstrated practical advantage.
- Stochastic modeling: Stochastic processes model market quantities such as stock prices, interest rates, and volatilities to support investment decisions involving returns and risks.
- 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.
- Monte Carlo pricing and risk analysis: 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.
- Monte Carlo pricing and risk analysis: 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.
- 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.
- Monte Carlo pricing and risk analysis: QMCI can also target risk metrics involving expected quantities, including value-at-risk and credit risk.
- 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.
- Differential-equation methods: For heuristic variational and neural approaches, whether any quantum advantage exists remains unclear without further experimentation.
Optimization
Quantum algorithms for financial optimization target continuous, discrete, and mixed highly constrained problems, but their practical advantage remains uncertain. The review compares quantum approaches with efficient classical methods and highlights resource, conditioning, sampling, and data-access barriers.
- 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.
- 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.
- 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.
- Continuous optimization: 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.
- 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.
- Polynomial-time methods: 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.
- 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.
- 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.
Machine learning
The Review surveys quantum machine-learning approaches for financial tasks, spanning accelerated classical methods and quantum-native models. It reports promising applications and speedups, while emphasizing that practical advantage remains uncertain and further research is needed.
- Financial machine learning supports forecasting, anomaly detection, classification, and other applications using historical data and sophisticated models.
- 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.
- 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.
- Quantum algorithms have been proposed for regression, classification, clustering, boosting, generative learning, and Bayesian inference in finance.
- 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.
- Generative learning: QCBM at least matches restricted Boltzmann-machine performance and demonstrates superior performance as the model scales in numerical distribution-learning simulations.
- Quantum-inspired algorithms could benefit high-dimensional financial problems when high precision is not required, despite refuting many originally claimed exponential speedups.
Outlook
The outlook identifies substantial opportunities for quantum computing in finance but stresses that practical, end-to-end advantage depends on unresolved algorithmic, hardware, memory, speed, and error-correction challenges.
- 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.
- 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.
- Hardware challenges: Existing quantum memory technologies have fundamental limitations that make low-cost, scalable quantum memory without active error correction highly challenging.
- Hardware challenges: 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.
- Hardware challenges: 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.
- Research outlook: Further algorithmic research and benchmarking of increasingly complex, especially heuristic, algorithms are needed as quantum hardware advances.
Glossary
The glossary defines financial products, models, methods, equations, probability concepts, and a statistical capacity measure used in the Review.
- An autocallable is a financial product that pays the holder a high return if the underlying asset passes an upside threshold.
- The Black–Scholes model is a mathematical model for the dynamics of a financial market containing derivative investment.
- Bump-and-reprice estimates a derivative price sensitivity by evaluating prices at different underlying-parameter values and taking their difference.
- A financial derivative is a contract whose value derives from the performance of an underlying entity.
- The Hamilton–Jacobi–Bellman equation gives a necessary and sufficient condition for control optimality with respect to a loss function.
- An option gives its holder the right, but not the obligation, to buy or sell an underlying asset at an agreed price and time.
- A target accrual redemption forward pays according to whether a spot rate is above or below a target, subject to schedules, limits, and barriers.
- The Vapnik–Chervonenkis dimension measures the capacity of a binary-classification function set by the largest always-perfectly-classifiable arbitrarily labelled data set.
Disclaimer
The paper was prepared for informational purposes with contributions from JPMorgan Chase & Co.’s Global Technology Applied Research center, not its Research Department.
- The paper is informational and is not a product of JPMorgan Chase & Co.’s Research Department or its affiliates.
- JPMorgan Chase & Co. and its affiliates make no explicit or implied warranty and accept no liability regarding the paper.