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Quantum Computing for Finance: State of the Art and Future Prospects

Daniel J. Egger, Claudio Gambella, Jakub Marecek, Scott McFaddin, Martin Mevissen, Rudy Raymond, Andrea Simonetto, Stefan Woerner, Elena Yndurain

arXiv:2006.14510v3quant-phq-fin.ST

TL;DR

Financial services contain computationally challenging pricing, risk, optimization, and machine-learning problems for which classical methods can be costly. The paper surveys quantum approaches, maps them to financial applications, and demonstrates selected algorithms on IBM Quantum back-ends. It reports a quadratic speed-up for amplitude-estimation-based risk calculations, while quantum error correction remains a central technical challenge.

  • Problem

    Financial services include problems demanding high precision or runtime, including simulation, optimization, and machine learning tasks that challenge classical computers.

  • Method

    The paper introduces quantum foundations, surveys algorithms by problem class, maps them to financial solutions, and demonstrates selected methods on IBM Quantum back-ends.

  • Results

    A quadratic speed-up is reported for estimating portfolio economic capital requirements with amplitude estimation, omitting the logarithmic cost of VaR bisection.

  • Takeaways & Limitations

    Quantum algorithms may offer faster sampling for simulation and better solutions in fewer steps for exponentially scaling optimization problems.

Abstract

from arXiv · show

This article outlines our point of view regarding the applicability, state-of-the-art, and potential of quantum computing for problems in finance. We provide an introduction to quantum computing as well as a survey on problem classes in finance that are computationally challenging classically and for which quantum computing algorithms are promising. In the main part, we describe in detail quantum algorithms for specific applications arising in financial services, such as those involving simulation, optimization, and machine learning problems. In addition, we include demonstrations of quantum algorithms on IBM Quantum back-ends and discuss the potential benefits of quantum algorithms for problems in financial services. We conclude with a summary of technical challenges and future prospects.

I. INTRODUCTION

Quantum computing is presented as a complement to classical systems for computationally challenging financial problems. The paper introduces quantum foundations, maps algorithms to financial applications, and discusses potential benefits alongside hardware and regulatory constraints.

  • Paper scope: The paper surveys quantum algorithms, their benefits, technical challenges, and financial applications across banking, financial markets, and insurance.It offers a practical guide linking problem types to financial solutions and business benefits.
  • Quantum computing motivation: Quantum computing may address financial problems that become extremely large or time-consuming on classical computers.The paper frames quantum systems as complementary rather than replacements for classical computers.
  • Hardware constraints: Near-term quantum devices remain noisy, with relatively high error rates and limited coherence times, making technical progress toward Quantum Advantage essential.The paper expects early advantages in physical-process modelling, optimization, and related difficult problem classes.
  • Quantum workflow: Quantum solutions generally load classical data, compute through qubit manipulation, and measure a probabilistic output distribution.Data loading can be substantial, while computation often approximates an optimum and measurement is repeated across shots.
  • Financial-services context: Financial institutions face regulatory and risk-management pressures, including Basel III capital, liquidity, and portfolio-management requirements.LCR diversification rules and asset limits complicate risk assessment and portfolio management.

III. SIMULATION

The simulation section addresses computationally intensive pricing and risk estimation, where classical Monte Carlo requires many samples. It presents quantum amplitude estimation as a route to quadratic sampling improvement, while noting substantial circuit-depth requirements.

  • Simulation applications: Financial simulation supports pricing, risk estimation, and scenario analysis, but complex dependencies and path-dependence make analytical models insufficient.Monte Carlo simulations are therefore widely used for risk metrics, instrument pricing, and stress-test scenarios.
  • Classical Monte Carlo: Reducing a Monte Carlo confidence interval by one order of magnitude requires 100 times more computational cost.The confidence interval scales as O(1/sqrt(M)), where M is the number of samples.
  • Amplitude estimation: Quantum amplitude estimation estimates a parameter with convergence rate O(1/M), providing a quadratic speed-up over classical Monte Carlo.The estimator targets the probability of measuring |1⟩ in the final qubit.
  • Amplitude estimation: Amplitude estimation maps measured phase information to an estimator ã = sin^2(yπ/M) after controlled applications of Q and an inverse quantum Fourier transform.The measured integer y identifies constructively interfering phases and is then converted classically.

B. ESTIMATING VALUE AT RISK WITH AE

The paper uses amplitude estimation to estimate a risk quantity by encoding a discretized random variable and its weighted function into an ancilla amplitude. For VaR, the CDF is encoded and searched over thresholds to identify the smallest acceptable loss level.

  • Quantum encoding: A discretized distribution of X is represented as an n-qubit state, with outcome i occurring with probability p_i.The outcomes are mapped to i ∈ {0, ..., N−1}, where N = 2^n.
  • Quantum encoding: Applying an ancilla operation for f(i) makes the probability of measuring |1⟩ equal to the weighted expectation ∑_i p_i f(i).This probability is the amplitude estimated by the quantum algorithm.
  • VaR estimation: For VaR, f(i) is chosen as a threshold indicator so the ancilla probability becomes P[X ≤ l].The threshold l determines which discretized losses contribute to the measured probability.
  • VaR estimation: A binary search finds the smallest l_α satisfying P[X ≤ l_α] ≥ 1 − α, and this threshold is the estimated VaR.The procedure uses amplitude estimation to evaluate the CDF at candidate thresholds.

C. CREDIT RISK

The credit-risk application models correlated loan defaults, computes total portfolio loss, and uses amplitude estimation with bisection search to obtain VaR and economic capital. An illustrative two-asset experiment recovers the 95% VaR at a loss of 2$, while a larger resource estimate demonstrates substantial circuit requirements.

  • Credit-risk model: Economic capital requirement is defined as portfolio VaR minus expected total loss, protecting the balance sheet against unexpected losses.The portfolio contains K loans with total loss L.
  • Credit-risk model: Defaults are modeled as Bernoulli variables with losses L_k = λ_kX_k, correlated through a latent standard-normal factor Z under conditional independence.Conditional on Z, defaults are independent, but their probabilities depend on the factor realization.
  • Quantum circuit: The operator A is built as A = CSU: U loads uncertainty, S computes total loss, and C marks whether loss is at most x for CDF evaluation.Separate registers encode Z, the asset defaults, the summed losses, and the comparison result.
  • Quantum circuit: The weighted-sum operator stores λ_1x_1 + ··· + λ_Kx_K in an n_S-qubit register, using divide-and-conquer addition across assets.The register size is chosen to represent all possible integer loss sums.
  • Illustrative experiment: In the two-asset example, two qubits encode four portfolio states, and the sum register requires n_S = 2 qubits.The selected asset parameters generate the loss distribution shown in Figure 6.
  • Illustrative experiment: 16 quantum samples identify the 95% VaR at a loss of 2$ on the first bisection iteration.The simulation uses m = 4 evaluation qubits and a total of 12 qubits.
  • Resource estimate: For K = 220 assets, the estimated fault-tolerant implementation requires approximately 37 million T/Toffoli gates and a runtime estimate of 30 minutes under stated assumptions.The estimate uses n_Z = 10, n_S = 30, and m = 10.

D. DISTRIBUTION LOADING

Efficient distribution loading is necessary for amplitude estimation to retain potential quantum advantage, but arbitrary state preparation can be exponentially costly. The paper also frames optimization benefits as promising yet still under active research, with examples simplified by omitted practical constraints.

  • Distribution loading: Replacing Monte Carlo with amplitude estimation requires efficiently loading model distributions into the quantum computer.Otherwise, data-loading cost can diminish the potential quantum advantage.
  • Distribution loading: Efficiently integrable distributions such as log-concave distributions can be loaded with controlled rotations.These loading techniques provide a constructive route for some structured distributions.
  • Distribution loading: Loading arbitrary quantum states requires exponentially many gates, making arbitrary distributions inefficient to encode directly as quantum gates.The paper identifies qGANs as a possible polynomial-gate alternative for distributions of special form.
  • Optimization scope: The optimization discussion introduces idealized next-period problems that ignore transaction costs and do not enforce Basel III constraints.The resulting quantum algorithms therefore address simplified problem formulations.
  • Optimization scope: Different optimization classes require different quantum algorithms, while benefits in solution quality or computational time remain an active research area.The paper expects tangible benefits in some problem classes as technology advances.

A. PROBLEM CLASSES: CONVEX PROBLEMS

The paper surveys quantum algorithms for convex optimization, especially semidefinite programming, and contrasts their apparent scaling benefits with parameter dependencies and lower bounds that can limit practical speedups.

  • Convex optimization includes linear, quadratic, and semidefinite programming with continuous decision variables.
  • Semidefinite programming generalizes linear programming, second-order cone programming, and convex quadratically constrained quadratic programming.
  • Reported quantum algorithms include scalings of O(√mn) and O(√m poly(log(m), log n)).
  • These upper bounds assume primal and dual solution diameter parameters P and D remain dimension-independent, which may be difficult in practice.
  • For continuous Markowitz portfolio optimization, dimension-dependent P and D yield quantum runtime lower bounds, suggesting practical speedups may be limited.
  • Quantum SDP algorithms quantize multiplicative-weight-update methods by replacing selected classical steps with quantum subroutines.
  • Some Markowitz instances have quantum complexity equal to classical complexity, so potential benefits depend on the input data.

C. PROBLEM CLASSES: COMBINATORIAL PROBLEMS

Combinatorial optimization uses discrete decision variables and is generally non-convex and not polynomial-time solvable classically. Variational quantum methods such as VQE and QAOA use quantum evaluations within classically optimized parameter loops.

  • Combinatorial optimization problems have discrete decision variables and are generally non-convex and not polynomial-time solvable classically.
  • Variational algorithms are attractive for near-term quantum machines because they have non-trivial guarantees and are not efficiently simulatable classically.
  • A variational approach parameterizes quantum states with relatively few parameters and uses classical optimization to select values minimizing or maximizing a utility function.
  • QUBOs can be mapped to Ising Hamiltonians by converting binary variables to spin variables and adding quadratic penalties for equality constraints.
  • VQE: VQE alternates quantum expectation estimation with classical parameter updates to minimize ⟨ψ(θ)|H|ψ(θ)⟩.
  • QAOA: QAOA generalizes VQE by using parameterized rotation and mixing Hamiltonians in a similar quantum-classical optimization loop.

E. COMBINATORIAL APPLICATION 1: ACTIVE INVESTMENT MANAGEMENT, PORTFOLIO OPTIMIZATION

The paper demonstrates VQE and QAOA for active portfolio selection and diversification-related combinatorial problems. Simulations recover budget-feasible and efficient-frontier solutions, while a small diversification instance shows nearly equal classical and quantum benefits.

  • Active portfolio optimization: The active portfolio example selects B = 3 assets from n = 6 and equally allocates capital under simplified equal-price and full-budget assumptions.
  • Active portfolio optimization: VQE and QAOA produce states whose three most probable samples select three assets and are either optimal or near-optimal.
  • Active portfolio optimization: VQE solutions closely follow the efficient frontier across risk-return trade-off values q without a budget constraint.
  • Portfolio diversification: Passive diversification selects q representative assets from q clusters to approximate a larger index with limited budget and transaction costs.
  • Portfolio diversification: The diversification model uses binary variables to select fund stocks and assign each stock to its most similar selected representative.
  • Portfolio diversification: A three-asset, two-cluster instance produces a 12-qubit Hamiltonian solved classically with CPLEX and quantumly with depth-7 fully entangled VQE.
  • Portfolio diversification: 2.001 versus 2.000: the classical diversification benefit is slightly higher than the quantum VQE benefit on the small instance.

G. MULTI-BLOCK ADMM HEURISTIC FOR MIXED-BINARY OPTIMIZATION

The 3-ADMM-H heuristic decomposes mixed-binary optimization into a quantum QUBO block and classical convex blocks. Its implementation alternates these updates, while convergence and optimality depend on stated convexity assumptions.

  • Mixed-binary optimization combines binary and continuous decisions subject to equality and inequality constraints and includes finance applications such as combinatorial auctions.
  • The heuristic assumes a quadratic binary objective together with convex constraint and coupling functions and convex feasible sets.
  • The algorithm performs first, second, and third block updates, a dual-variable update, merit evaluation, and returns the iterate with minimum ηk.
  • ADMM has residual, objective, and dual-variable convergence properties when its convexity assumptions hold, while comprehensive 3-ADMM-H conditions are outside the paper’s scope.
  • 3-ADMM-H decomposes certain mixed-binary problems into a quantum QUBO subproblem and a classically solved continuous convex constrained subproblem.
  • The quantum QUBO block can use variational algorithms such as VQE or QAOA, while the convex block uses classical optimization solvers.

H. COMBINATORIAL APPLICATION 3: AUCTIONS

The auction problem maximizes revenue while respecting item-availability constraints, and 3-ADMM-H uses a quantum solver for its demanding QUBO subproblem. In simulation, the heuristic reached a finite-iteration termination despite an inexact quantum solver.

  • Auction formulation: Combinatorial auction winner selection maximizes revenue from bids on item subsets subject to available-unit constraints.Each bid is represented by a binary decision indicating whether it wins.
  • Quantum optimization approach: Because inequality constraints prevent direct QUBO reformulation, the auction problem is solved with the 3-ADMM-H heuristic.The reported approach uses 3-ADMM-H rather than applying VQE directly to the original formulation.
  • Experimental setup: The simulated instance contains 3 items, 16 bids, and 6 available units per item, with the number of bids determining the required qubits.Profits and bid compositions were randomly generated within the stated instance parameters.
  • Results: The classical CPLEX optimum accepts B0 = {0}, B1 = {1}, and B4 = {1, 2}, yielding a profit of 24.The solution was obtained by solving the integer program with IBM ILOG CPLEX.
  • Results: With VQE as the QUBO solver, 3-ADMM-H converges to the same solution in 43 iterations despite the quantum solver’s inexactness.The authors state that convergence guarantees do not apply to inexact QUBO solvers, but the residual curves show finite termination and tolerance to inexact computation.
  • Quantum optimization approach: The algorithm’s intended advantage over classical solvers such as CPLEX is using quantum algorithms to tackle QUBO subproblems.The QUBO subproblem is identified as the most computationally demanding component of the decomposition.

V. MACHINE LEARNING

The paper surveys quantum machine-learning applications in finance, focusing on binary classification for tasks such as fraud detection and credit-risk scoring. It describes quantum-enhanced feature maps and variational classifiers, reporting that QRAC-based VQC achieves better training losses and accuracy comparable to an RBF-kernel SVC.

  • Financial applications: Financial machine learning supports prediction, classification, and pattern discovery, including credit scoring, fraud detection, and customer-life-cycle applications.The paper connects these tasks to customer identification, product targeting, transaction monitoring, and retention.
  • Problem setting: The paper focuses on supervised binary classification using labeled training data, with the assumption that future data share the same underlying distribution.The classifier is trained on S and evaluated on test data T and unseen future labels.
  • Quantum feature spaces: Quantum-enhanced classification embeds classical data into a higher-dimensional quantum feature space through a feature map Φ(x).The embedding maps x ∈ R^d into a space with n > d dimensions and may be realized as a quantum state.
  • Variational quantum classifier: VQC training minimizes empirical risk, and classification uses the learned parameters to evaluate the sign of the resulting classifier function.Because measurements are probabilistic, both training and classification require repeated shots.
  • Variational quantum classifier: A VQC applies UΦ(x) to encode data, then trains a variational circuit W(θ) whose measurements produce the classifier output.The parameter set θ is optimized by minimizing empirical risk using variational methods such as COBYLA or SPSA.
  • Results: VQCwQRAC achieves better training losses than standard VQC, with accuracy comparable to an RBF-kernel support-vector classifier.The comparison concerns a synthetic credit-card transaction dataset evaluated through 5-fold cross-validation.

VI. TECHNICAL CHALLENGES IN QUANTUM COMPUTING

The paper frames technical challenges as constraints that must be addressed when solving computationally challenging problems on quantum computers.

  • Technical challenges: The section introduces the technical challenges involved in solving computationally challenging problems on quantum computers.It motivates the subsequent discussion of constraints on quantum computing.

A. LOADING DATA

Quantum data loading differs fundamentally from classical loading because state preparation can itself dominate computation. Preparing arbitrary states with one- and two-qubit gates has exponential complexity in the number of qubits, although structured or approximate loading can mitigate the issue.

  • Classical–quantum contrast: Classical systems assume routine movement between storage and memory, whereas current quantum architectures generally lack quantum hard drives and RAM.Quantum computations are described as reversible and without ordinary branching.
  • State-preparation complexity: With k qubits, quantum state preparation requires at least 4^k gates for almost all arbitrary states when restricted to one- and two-qubit gates.The passage characterizes this as a generic lower bound rather than merely a worst-case result.
  • State-preparation complexity: Although a 4^k-dimensional state can be loaded with O(4^k) circuit complexity, this is exponential in the number of qubits k.Thus state preparation can be a central part of quantum-solution design rather than a negligible input step.
  • Mitigations: Potential workarounds include larger gates, separate state-preparation systems such as qRAM circuitry, and exploiting structure in the data.The paper notes that these alternatives pose control challenges or depend on exploitable data properties.
  • Mitigations: Approximate loading can replace exact state preparation when preparing a sufficiently close quantum state is adequate for the application.The paper presents approximate schemes as one way to work around loading complexity.

B. QUANTUM ERROR CORRECTION

Quantum error correction is a key boundary between noisy and fault-tolerant quantum computing, but its qubit overhead and error-threshold trade-off remain major technical challenges. The paper contrasts error-corrected amplitude-estimation applications with near-term variational optimization and machine-learning applications.

  • Quantum error correction separates noisy quantum devices from universal fault-tolerant quantum computers.
  • QEC schemes can correct residual errors below a physical-qubit error threshold, but require overhead in physical qubits per protected logical qubit.For one commonly discussed QEC class, a 0.1% depolarizing-error probability is sometimes associated with more than 1,000 physical qubits per protected qubit.
  • Amplitude estimation can replace Monte Carlo simulation, but its controlled Q2j operators make the resulting circuits too deep without error correction.The paper therefore anticipates that amplitude-estimation applications will require fault-tolerant quantum computers.
  • Variational quantum circuits such as VQE and QAOA may support optimization and machine-learning applications on near-term noisy quantum computers.Unlike amplitude estimation, these heuristic algorithms do not provide a theoretical guarantee and require further research to determine when they outperform classical methods.
  • Quantum algorithms target finance problems involving exponentially scaling optimization, complex-data machine learning, and time-limited simulation.The paper links these classes to potential improvements in solution quality, pattern accuracy, or sampling efficiency, and demonstrates selected applications on IBM Quantum back-ends.
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