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Quantum computing for finance


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 modelling, 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.

Key points

  • Quantum algorithms for stochastic modelling, optimization and machine learning are applicable to various financial problems.

  • Quantum Monte Carlo integration and gradient estimation can provide quadratic speedup over classical methods, but more work is required to reduce the amount of quantum resources for early fault-tolerant feasibility and achieving an actual speedup.

  • Financial optimization problems can be continuous (convex or non-convex), discrete or mixed, and thus quantum algorithms for these problems can be applied.

  • The advantages and challenges of quantum machine learning for classical problems are also apparent in finance.

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The authors appreciate support from the Chicago Quantum Exchange. Y.A. acknowledges support from the Office of Science, US Department of Energy, under contract DE-AC02-06CH11357 at Argonne National Laboratory. D.H., Y.S. and M.P. appreciate the insightful discussions they had with their colleagues from the Global Technology Applied Research centre at JPMC.

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A financial product that pays the holder a high return if the value of the underlying asset passes an upside barrier.

Black–Scholes model

A mathematical model for the dynamics of a financial market containing derivative investment instruments.


A method to estimate the sensitivity of the price of a financial derivative with respect to an underlying parameter by evaluating the price at different values of the parameter and taking the difference.

Financial derivative

A financial contract that derives its value from the performance of an underlying entity.

Hamilton–Jacobi–Bellman equation

An equation that gives a necessary and sufficient condition for optimality of a control with respect to a loss function.

Martingale measure

A probability measure such that the conditional expectation of a random variable in a sequence given the value of a random variable prior in the sequence is equal to the value of this prior random variable on which the expectation is conditioned.


A financial contract that gives the holder the right, but not the obligation, to buy or sell an underlying asset at an agreed-upon price and time frame.

Target accrual redemption forward

A financial product that allows the holder to achieve a target rate (interest rate, exchange rate and so on) or rate range on a pre-defined schedule (for example, monthly) up to a limit on the maximum payout and under certain conditions on the extreme values of the rate observed in the market (spot rate). It achieves this goal by paying the holder a positive amount if the spot rate is higher than a target value and negative if lower, until the maximum amount of accrual has been reached or the spot rate hits certain upper and/or lower barriers.

Vapnik–Chervonenkis (VC) dimension

A measure of the capacity of a set of functions that can be learnt by a statistical binary classification algorithm, defined as the cardinality of the largest set of data points that the algorithm can always learn a perfect classifier for an arbitrary labelling.

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Herman, D., Googin, C., Liu, X. et al. Quantum computing for finance. Nat Rev Phys 5, 450–465 (2023).

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