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A Survey of Quantum Computing for Finance

Dylan Herman, Cody Googin, Xiaoyuan Liu, Alexey Galda, Ilya Safro, Yue Sun, Marco Pistoia, Yuri Alexeev

arXiv:2201.02773v4quant-phq-fin.CP

TL;DR

Finance offers computationally challenging problems that may suit quantum algorithms, motivating this survey of quantum computing for financial applications. The paper synthesizes stochastic modeling, optimization, and machine learning approaches, reviews their financial use cases and hardware feasibility, and reports potential speedups alongside substantial current hardware constraints.

  • Problem

    Financial applications such as risk modeling and derivative pricing involve computationally challenging stochastic problems for which numerical methods are required.

  • Method

    The survey reviews quantum algorithms for finance across stochastic modeling, optimization, and machine learning, including their applicability to financial use cases and quantum hardware.

  • Results

    Quantum Monte Carlo integration provides a quadratic speedup for Monte Carlo integration, while selected derivative-pricing approaches report exponential speedup in problem dimension.

  • Takeaways & Limitations

    The survey provides a reference for identifying financial problems that may benefit from quantum computing and for advancing research on their quantum solutions.

  • Takeaways & Limitations

    Current NISQ implementations arguably provide no decisive advantage over classical algorithms, while larger applications require more capable hardware and may face limited qubit counts.

Abstract

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Quantum computers are expected to surpass the computational capabilities of classical computers during this decade and have transformative impact on numerous industry sectors, particularly finance. In fact, finance is estimated to be the first industry sector to benefit from quantum computing, not only in the medium and long terms, but even in the short term. This survey paper presents 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, describing how these solutions, adapted to work on a quantum computer, can potentially help to solve financial problems, such as derivative pricing, risk modeling, portfolio optimization, natural language processing, and fraud detection, more efficiently and accurately. We also discuss the feasibility of these algorithms on near-term quantum computers with various hardware implementations and demonstrate how they relate to a wide range of use cases in finance. We hope this article will not only serve as a reference for academic researchers and industry practitioners but also inspire new ideas for future research.

1 Introduction

This introduction frames finance as a promising early domain for quantum computing and surveys how quantum algorithms may address financial problems. It covers quantum hardware, algorithms, and applications across stochastic modeling, optimization, and machine learning.

  • Finance is estimated to be the first industry sector to benefit from quantum computing, partly because many financial problems may suit near-term quantum algorithms.
  • Quantum computing may reduce the time and memory required for computational tasks, but quantum speedup requires algorithms designed to exploit quantum-mechanical properties.
  • The survey emphasizes that industry must identify which problems benefit from quantum computing and determine the extent of those benefits.
  • The survey reviews computationally intensive financial problems, quantum-computation fundamentals, and quantum-enhanced stochastic modeling, optimization, and machine learning.
  • Scope relative to prior surveys: It takes a broader view than prior surveys by covering multiple quantum technologies, financial optimization applications, and machine-learning approaches beyond narrower hardware or application focuses.

2 Applicability of Quantum Computing to Finance

The paper maps quantum-computing techniques to finance problems driven by stochasticity, regulation, personalization, security, and large-scale data. It organizes applications around stochastic modeling, optimization, and machine learning.

  • Financial risk metrics such as VaR and CVaR often require numerical methods because stochastic inputs generally prevent closed-form analytical solutions.
  • The survey groups financial applications and quantum algorithms into stochastic modeling, optimization, and machine learning.
  • Stochastic modeling addresses financial-asset prices and associated risks, while optimization covers portfolio and other combinatorial or convex problems.
  • Quantum machine learning applications discussed include anomaly detection, natural language modeling, asset pricing, and implied-volatility calculation.
  • Table 1 summarizes the classical and quantum techniques associated with the financial use cases covered by the survey.

3 Quantum Computing Concepts

This section introduces quantum-computing foundations, hardware diversity, computational models, and the practical constraints affecting current devices. It distinguishes theoretical potential from the present limitations of NISQ hardware.

  • Quantum computers process information using quantum-mechanical systems, with qubits replacing classical binary bits as the fundamental information unit.
  • Multiqubit systems use tensor-product state spaces and can exhibit entanglement, producing correlations that cannot be described through independent subsystem states.
  • Quantum measurements are probabilistic, and expected observable values can be computed from the system state and observable operator.
  • Current limitations: Near-term algorithms target NISQ devices, whereas theoretically advantageous algorithms require many high-fidelity logical qubits; none of the discussed NISQ implementations arguably provides decisive classical advantage yet.
  • Gate-speed trade-offs: Quantum hardware must balance gate speed, spatial resources, and fidelity because faster gates can increase error rates, especially in repeated-execution algorithms.

4 Foundational Quantum Algorithms

This section introduces foundational quantum algorithms and complexity concepts used to assess potential computational speedups. It covers search, amplitude amplification, phase estimation, linear-system, variational, and adiabatic optimization methods, together with their practical requirements and limitations.

  • Complexity and speedups: Quantum algorithms seek computational speedups by reducing complexity, although many near-term algorithms rely on heuristics without proven asymptotic reductions.Most NISQ algorithms fall into the heuristic category.
  • Search and amplification: Grover’s algorithm provides an asymptotic quadratic speedup for unstructured search and is optimal in quantum query complexity.The algorithm amplifies the probability of measuring a marked element.
  • Search and amplification: Quantum amplitude amplification boosts a marked state’s measurement probability to Ω(1) and achieves a quadratic speedup over classical sampling.Classically, O(1/sin^2(θ_a)) samples are required in expectation.
  • Linear-system algorithms: HHL solves quantum linear systems with an exponential speedup in system size N for well-conditioned matrices over known classical algorithms.Its complexity depends on matrix sparsity, condition number, system size, and error tolerance.
  • Variational methods: Variational quantum algorithms are classical–quantum hybrids that update circuit parameters using measurement-derived cost functions, but their training is NP-hard and heuristic.VQAs apply to eigenvalue, optimization, linear, and nonlinear problems.
  • Adiabatic optimization: Quantum annealing targets QUBO problems using quantum tunneling, with demonstrated usefulness for tall, narrow cost-function peaks and potential near-term applicability.Its overall benefit remains an ongoing research topic, while current annealer scale supports possible near-term use.

5 Stochastic Modeling

The survey reviews stochastic modeling methods used in finance and their quantum adaptations, focusing on Monte Carlo integration and differential-equation approaches. It describes potential speedups for pricing, risk, and related financial computations while noting model and implementation constraints.

  • Financial stochastic models represent quantities such as stock prices, interest rates, and volatilities as random variables whose evolution supports investment and risk decisions.
  • Most complex financial stochastic differential equations require numerical methods because analytical solutions exist only for limited cases.
  • The survey reviews quantum solutions for Monte Carlo integration and differential-equation methods, then connects them to financial applications.
  • Monte Carlo Integration: Quantum Monte Carlo integration uses quantum amplitude estimation to achieve a quadratic reduction in queries for a target expectation at the same error.The expectation is computed from a quantum encoding of sample paths and function values.
  • Differential Equations: Quantum differential-equation methods transform discretized stochastic problems into linear systems or variational objectives that can be solved with quantum algorithms.For linear-system approaches, the potential complexity is O(poly(d, log(1/ϵ))) in dimension d and error tolerance ϵ.
  • Financial Applications: Applications include path-dependent derivative pricing, multi-asset Black–Scholes equations, Greeks, counterparty credit valuation adjustment, and quantum inner-product estimation.Reported approaches include quadratic speedups for computing Greeks and for estimating inner products used in portfolio valuation.

6 Optimization

This section surveys quantum algorithms for financial optimization, emphasizing portfolio optimization, arbitrage, and other problems with potential relevance on near-term quantum hardware. It also introduces hybrid approaches for large and mixed-integer problems.

  • Portfolio optimization and arbitrage are identified as the most promising commercially relevant applications of optimization on NISQ hardware.
  • Quantum algorithms are applied to NP-hard combinatorial optimization problems, particularly those with quadratic cost functions.
  • Large-scale optimization can use hybrid approaches that decompose large problems into smaller subtasks solvable on near-term quantum devices.
  • Mixed-integer programming can potentially be handled by combining quantum optimization algorithms with classical-quantum hybrid methods.

6.1 Combinatorial Optimization

Combinatorial optimization is formulated through binary variables and quadratic objectives that can be mapped to Ising Hamiltonians. Quantum annealing and QAOA then provide distinct quantum approaches, although hardware connectivity and adiabaticity impose practical constraints.

  • Combinatorial optimization in this paper refers specifically to integer optimization problems containing only binary variables.
  • A QUBO problem is an unconstrained binary quadratic program that naturally fits many quantum algorithms.
  • Constraints can be incorporated into QUBO formulations by optimizing a Lagrangian whose dual variables penalize constraint violations.
  • The optimal QUBO solution is equivalent to the ground state of its corresponding Ising Hamiltonian.
  • Quantum Annealing: Quantum annealing natively solves QUBO formulations, while higher-order problems can be converted to QUBO at the cost of additional variables.
  • Quantum Annealing: Commercial quantum annealers have limited connectivity, requiring additional qubits for arbitrary QUBO embeddings, and provide no guarantees of adiabatic evolution.
  • QAOA: QAOA uses parameterized phase-separation and mixing operators to seek high-quality assignments for unconstrained combinatorial problems on gate-based computers.
  • QAOA: Recent QAOA demonstrations indicate that current gate-based hardware can handle unconstrained and constrained optimization problems.

6.2 Convex Optimization

The survey covers quantum algorithms for convex optimization, including linear, second-order cone, and semidefinite programming, with applications such as portfolio optimization.

  • Quantum algorithms have been developed for linear programming, second-order cone programming, and semidefinite programming.
  • These convex optimization formulations have financial applications including portfolio optimization.

6.3 Large-Scale Optimization

Near-term quantum devices are expected to have too few qubits for many large-scale financial problems. Decomposition-based hybrid methods address this constraint by dividing problems between classical and quantum computation.

  • Limited qubit counts are a major obstacle to applying near-term quantum devices to large-scale financial problems.
  • Decomposition-based hybrid approaches let classical routines split large problems into subproblems that quantum devices solve separately.
  • The classical computer combines the solutions returned by the quantum device.

6.4 Financial Applications

The survey examines quantum optimization across financial applications, especially portfolio construction, swap netting, and arbitrage. It covers combinatorial and convex formulations, quantum annealing, and quantum linear-system methods, while emphasizing hardware-scale constraints and conditional speedups.

  • Portfolio Optimization: Portfolio optimization selects asset quantities under objectives involving risk and expected return, with continuous, discrete, or mixed variables.The survey discusses both combinatorial and convex formulations of this problem.
  • Portfolio Optimization: Risk minimization and mean-variance portfolio problems can be encoded as QUBOs with covariance, return, budget, and penalty terms.Quantum annealing is used for the discussed formulations, including long, short, or neutral position decisions.
  • Portfolio Optimization: Quantum annealing has produced more experimental results with larger portfolio problems because annealers possess more qubits than existing gate-based devices.
  • Portfolio Optimization: Relaxing convex portfolio optimization to sampling or statistic estimation yields a quantum linear-system formulation with potentially exponential speedup in system dimension N.The speedup requires sparsity and well-conditioned matrices, and only partial solution information may be needed.
  • Other Financial Applications: Swap-netting optimization maximizes netted notional value while enforcing cancellation and compatibility constraints through a QUBO solved with quantum annealing.The procedure can be applied in parallel across subsets of potentially nettable swaps.
  • Other Financial Applications: Arbitrage detection seeks rapid identification of price discrepancies across markets, including currency arbitrage opportunities.

7 Machine Learning

This section surveys quantum machine-learning approaches for financial regression, classification, and related data-analysis tasks. It contrasts fault-tolerant algorithms seeking speedups with near-term models that explore quantum representations, while highlighting data-loading and trainability challenges.

  • Overview: Financial machine learning supports stock-pattern detection, transaction-anomaly discovery, news classification, and portfolio optimization.
  • Overview: Fault-tolerant quantum machine-learning methods often require quantum data loading, which can take exponential time without qRAM.Near-term approaches instead target small NISQ devices and may seek hard-to-represent quantum correlations.
  • Regression: Quantum regression methods encode model parameters in quantum states and use HHL- or CKS-based procedures to estimate fit quality and learn parameters.
  • Regression: Gaussian-process regression can estimate mean predictions and variances through repeated measurements of HHL output states, with potentially exponential speedup over classical algorithms.
  • Classification: Quantum support vector machines use kernel-matrix inversion and were proposed with O(log MN) runtime for training and classification.The approach relies on qRAM-based state construction and effectively uses a low-rank kernel approximation through eigenvalue filtering.
  • Classification: Quantum nearest-neighbor classification encodes Euclidean distances and inner products in amplitudes, then uses amplitude amplification to estimate distances.

7.3 Clustering

The survey presents quantum clustering and generative-learning methods for discovering structure, reducing dimensionality, and modeling financial distributions. These approaches include q-means, spectral clustering, quantum generative models, and quantum neural-network variants, with several relying on qRAM or remaining exploratory.

  • Clustering: Clustering can support diversified portfolio construction and group stocks whose returns are highly correlated.
  • Quantum k-Means: Quantum q-means has running time polylogarithmic in the number of data points, while a NISQ version and quantum expectation-maximization variants have also been proposed.
  • Quantum Spectral Clustering: Spectral clustering faces O(N^3) classical running time, whereas quantum approaches use projected or graph-Laplacian states and can scale polylogarithmically with input-vector dimension.The cited quantum method depends polynomially on the number of clusters k and assumes efficient qRAM access for state preparation.
  • Generative Modeling: Quantum generative modeling is relevant to finance because many applications require samples from complex probability distributions.
  • Generative Modeling: Quantum circuit Born machines represent probability distributions with quantum pure states, and simulations suggest performance at least matching restricted Boltzmann machines for distribution learning.
  • Quantum Neural Networks: Quantum neural networks use diverse constructions, including quantum neurons, orthogonal feedforward networks, parameterized circuits, and quantum-enhanced feature maps.The surveyed proposals vary in their proximity to classical neural networks.

7.7 Quantum Reinforcement Learning

The section places quantum reinforcement learning within a broader set of financial quantum-machine-learning applications. It describes quantum representations of reinforcement-learning states and actions, alongside applications spanning pricing, portfolio allocation, language, anomaly detection, and asset pricing.

  • Reinforcement Learning: Reinforcement learning maximizes reward through actions in an environment and has applications including contingent-claim pricing, hedging, portfolio allocation, market making, and tax optimization.
  • Reinforcement Learning: The standard reinforcement-learning framework uses finite-state Markov decision processes with states, actions, transition probabilities, rewards, and an objective function.
  • Quantum Reinforcement Learning: A proposed quantum reinforcement-learning method represents states in superposition, samples actions by observation, updates amplitudes from rewards, and amplifies good actions with Grover iterations.
  • Natural Language Processing: Quantum machine-learning models may potentially enhance natural-language architectures, while quantum NLP frameworks represent grammar and semantics through structures that can be encoded as quantum circuits.
  • Anomaly Detection: Quantum anomaly-detection proposals include parameterized-circuit generators, amplitude-estimation density methods, quantum GANs, and quantum clustering variants.
  • Asset Pricing: Quantum machine-learning methods have been proposed for asset pricing and implied-volatility learning, but their applicability to asset pricing remains an area needing investigation.

8 Hardware Implementations of Use Cases

The survey reviews hardware experiments applying quantum algorithms to financial problems, including risk analysis, portfolio optimization, and quantum linear-system methods. Results illustrate potential algorithmic advantages alongside constraints from current devices, such as limited qubit resources, connectivity, noise, and decoherence.

  • Experiments used superconducting universal processors, trapped-ion processors, and superconducting quantum annealers to solve or simulate financial problems.The reviewed hardware experiments include quantum risk analysis, portfolio optimization, and linear-system-based portfolio methods.
  • Quantum Risk Analysis on a Transmon Device: Quantum amplitude estimation was tested on a Treasury-bill pricing problem and a simulated two-asset portfolio risk problem.The algorithm estimates quantities associated with a random variable encoded as a quantum state and computes expected values of functions of that variable.
  • Quantum Risk Analysis on a Transmon Device: Lower error rates than Monte Carlo were observed for the 4-qubit QAE version on IBMQ 5 Yorktown, with the estimated gap growing for 5- and 6-qubit instances.The comparison used 8,192 shots per evaluation-qubit setting.
  • Portfolio Optimization with Reverse Quantum Annealing: Reverse annealing was reported as over 100 times faster than forward annealing on average using the time to solution metric.The protocol begins from a user-provided classical state, explores a quantum superposition, and then completes the annealing schedule toward the problem ground state.
  • Portfolio Optimization on a Trapped-Ion Device: Quantum linear-system methods were applied to convex mean-variance portfolio optimization after Lagrange multipliers converted the problem into a linear system.The reviewed implementation used HHL-related methods and a controlled-SWAP test on small portfolios.
  • Portfolio Optimization on a Trapped-Ion Device: Near-term implementations of QPE and HHL remain constrained by ancillary-qubit, controlled-operation, circuit-depth, connectivity, and decoherence requirements.QCL-QPE reduces ancillary-qubit requirements through mid-circuit measurements, resets, and classically controlled gates; matrix scaling was developed to separate eigenvalues in the output distribution.

9 Conclusion and Outlook

The conclusion positions finance as an early potential beneficiary of quantum computing because many financial problems involve stochastic inputs and may tolerate approximate solutions. It emphasizes that current systems do not yet outperform classical computers on real-scale industry problems, while surveying possible applications and the conditions needed for future advantage.

  • Finance is considered an early potential beneficiary of quantum computing because many financial use cases can tolerate solution imprecision.The paper connects this tolerance to the stochastic nature of many financial problems.
  • Quantum computing requires redesigning financial algorithms around quantum hardware rather than directly porting classical programs.
  • Commercial viability depends on whether an application needs greater speed or accuracy and whether a quantum speedup can emerge with hardware development.
  • Today’s quantum computers do not yet solve real-life-scale industry problems more efficiently and accurately than classical computers.The paper nevertheless notes existing proofs of concept for quantum advantage and quantum supremacy.
  • Potential quantum solutions have been developed for portfolio optimization, derivatives pricing, risk modeling, fraud detection, and natural language processing.
  • The paper provides a comprehensive review of quantum algorithms for computationally challenging financial problems and aims to support future research.

Funding

The funding statement identifies support from the U.S. Department of Energy for the listed contributors and institutional work.

  • The work received support from the U.S. Department of Energy through WDTS and Argonne National Laboratory contract DE-AC02-06CH11357.

Disclaimer

The disclaimer states that the paper was prepared for information purposes by researchers from the identified institutions and is not a JPMorgan Chase research product. It disclaims representations, warranties, and liability by JPMorgan Chase and its affiliates.

  • The paper was prepared for information purposes by researchers from the identified institutions, including JPMorgan Chase’s FLARE group.
  • The paper is not a product of JPMorgan Chase’s Research Department or its affiliates.
  • JPMorgan Chase and its affiliates disclaim explicit or implied representations, warranties, and liability connected with the paper.

Acronyms

This section defines acronyms for quantum-computing methods, computational concepts, machine-learning techniques, and finance applications used in the paper.

  • Quantum computing: Quantum-computing methods include NISQ, AQC, QCL, VQE, QAOA, QNN, PQC, VQA, QPE, QAA, QAE, QLSP, QLS, QLSA, HHL, QSVT, QSVE, CKS, qRAM, ITE, VarQITE, MCI, and QMCI.These abbreviations span hardware contexts, quantum algorithms, optimization methods, linear-system techniques, and simulation or estimation procedures.
  • Machine learning and data analysis: Quantum machine-learning and data-analysis acronyms include QML, GAN, SVM, PCA, TDA, NLP, and RL.The terms cover learning models, language processing, reinforcement learning, and classical or topological data-analysis approaches.
  • Optimization and computing: Optimization and computational-complexity terms include IQP, QUBO, NP, CPU, and GPU.These acronyms refer to integer quadratic programming, binary optimization, complexity classification, and conventional processing units.
  • Finance: Finance-specific acronyms include CVaR, CDO, and ECR.They denote Conditional Value at Risk, Collateralized Debt Obligation, and Economic Capital Requirement.
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