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Quantum computing for finance: overview and prospects
Roman Orus, Samuel Mugel, Enrique Lizaso
TL;DR
Financial computation includes optimization, machine learning, and Monte Carlo tasks that quantum methods may accelerate. The paper reviews quantum optimization and annealing, quantum machine learning, and quantum amplitude estimation, reporting a near-quadratic reduction in Monte Carlo samples while emphasizing hardware and software limitations. It concludes that quantum computers may eventually play a key role in quantitative finance.
Problem
Financial problems such as portfolio optimization, arbitrage, credit scoring, and Monte Carlo simulation can demand difficult optimization or substantial computational resources.
Method
The paper surveys quantum optimization and annealing, quantum machine learning, and quantum amplitude estimation for financial applications.
Results
Quantum amplitude estimation enables Monte Carlo estimation with O(σ/ϵ) samples up to polylogarithmic factors and 99% success probability, nearly quadratically fewer than classical sampling.
Takeaways & Limitations
The reviewed approaches suggest potential applications to portfolio optimization, arbitrage, credit scoring, and financial Monte Carlo methods.
Takeaways & Limitations
The paper notes that practical quantum advantage requires overcoming formidable hardware challenges and developing algorithms for noisy intermediate-scale quantum processors.
Abstract
from arXiv · showhide
We discuss how quantum computation can be applied to financial problems, providing an overview of current approaches and potential prospects. We review quantum optimization algorithms, and expose how quantum annealers can be used to optimize portfolios, find arbitrage opportunities, and perform credit scoring. We also discuss deep-learning in finance, and suggestions to improve these methods through quantum machine learning. Finally, we consider quantum amplitude estimation, and how it can result in a quantum speed-up for Monte Carlo sampling. This has direct applications to many current financial methods, including pricing of derivatives and risk analysis. Perspectives are also discussed.
I. INTRODUCTION
Quantum computing is presented as a potentially disruptive source of computational speedups for finance. The paper surveys financial applications spanning optimization, machine learning, and quantum-enhanced Monte Carlo methods, while cautioning that predictive reliability in qualitatively new events remains unproven.
- Quantum computation promises highly efficient algorithms, including exponential speedups for some technologically important problems, although only small processors are currently available.
- Financial applications discussed include stock-market prediction, portfolio optimization, and fraud detection.
- The paper provides an overview of finance fields that could benefit from quantum-computational speedups, potentially producing substantial savings for governments, institutions, and individuals.
- Accepted financial models can make erroneous predictions in qualitatively new situations, and whether quantum computing can predict such events remains unproven.
- It reviews quantum optimization, quantum machine learning, and quantum sampling approaches for financial problems.
II. BACKGROUND
The background frames finance as the study of uncertain asset behavior and develops optimization, machine-learning, and Monte Carlo approaches for financial analysis. These methods are useful but can require substantial computation, while Monte Carlo accuracy also depends on modeling assumptions.
- A. Some core problems in finance: Finance concerns uncertain future asset behavior, with risk measuring deviations between actual and expected returns and volatility describing return variation.
- A. Some core problems in finance: Portfolio construction seeks either maximum return at a given risk or minimum risk at a given return.
- A. Some core problems in finance: Options generally require numerical simulation methods such as Monte Carlo to value their complex, asset-dependent payoffs.
- B. Relevant approaches in quantitative finance: The paper focuses on optimization, machine learning, and Monte Carlo as three computational approaches to financial problems.
- B. Relevant approaches in quantitative finance: Monte Carlo methods require many runs for accurate estimates, and constant drift and volatility assumptions reduce short-term prediction accuracy.
- B. Relevant approaches in quantitative finance: These approaches can require colossal computational power, with the burden worsening as the amount of gathered data increases.
C. Some basics on quantum computing
Quantum computing represents information with qubits and exploits superposition, entanglement, interference, and reversible evolution. Quantum algorithms encode inputs, process superposed states, amplify desired outcomes, and measure a classical result.
- A qubit is a two-dimensional quantum system that encodes classical bits 0 and 1 in basis states.
- Superposition allows quantum systems to occupy multiple states simultaneously, supporting massively parallel computation.
- Entanglement describes states that cannot be factorized into separate individual-qubit states and can make some systems difficult for classical computers.
- Quantum algorithms encode input data, create superposition, apply an algorithm across states, amplify the correct state, and measure qubits.
- Measurement is random, so algorithms are engineered such that the most probable measured answer encodes the classical solution.
- Table III summarizes QML subroutine speedups, using O(√N) for square-root speedup and O(log N) for exponential speedup.
D. The case for quantum computing in finance
The paper surveys quantum algorithms that may accelerate financial computation, including search, optimization, Fourier-transform, and linear-system subroutines. Their potential relevance follows from the computational demands of financial machine learning and optimization.
- Quantum algorithms may offer substantial speedups because operation counts can grow more slowly with input size than the best known classical alternatives.
- Grover’s algorithm searches an unordered database in O(√N) steps, compared with O(N) for the best classical algorithm.
- QAOA can find a good optimization solution in polynomial time for problems requiring exponential classical time.
- The QFT has complexity O((log N)^2), compared with O(N log N) for the classical FFT, and may support artificial-intelligence methods.
- The HHL algorithm solves linear systems with an exponential improvement relative to the best classical alternative and is used by many QML methods.
E. Currently available quantum hardware
Quantum finance relies on distinct hardware families and optimization paradigms, but practical advantage remains constrained by decoherence, error correction, data encoding, and immature NISQ software. Quantum optimization has nevertheless produced proof-of-principle financial demonstrations.
- Hardware families: Quantum hardware comprises gate-model processors and quantum annealers, with annealers designed to find local minima in combinatorial optimization problems.The passage identifies Google’s 72-qubit processor and D-Wave machines with over 2000 superconducting qubits as examples of the two families.
- Hardware challenges: Decoherence limits quantum operations by destroying quantum behavior, while fault-tolerant operation can require many thousands of physical qubits per logical qubit.One study found that an apparent speedup disappeared after accounting for the classical processing needed for error correction.
- Hardware challenges: NISQ algorithms target faulty processors that operate despite decoherence, but their practical use is limited by the lack of an established algorithm library.Developing algorithms for near-term quantum computers is presented as a major software challenge.
- Hardware challenges: Quantum machine learning also faces a hardware boundary because efficient quantum encoding and long-term storage of large classical datasets require unavailable qRAM.The paper identifies qRAM as one of the largest current hardware challenges.
- Quantum optimization: Adiabatic quantum computation encodes a cost function in a problem Hamiltonian, slowly deforms an easily prepared initial Hamiltonian, and measures the resulting ground state.The adiabatic theorem links sufficiently slow evolution and non-degenerate energy levels to a high probability of obtaining the problem Hamiltonian’s ground state.
- Quantum optimization: Quantum annealing implements this optimization process through quantum tunneling, but only approximately satisfies the conditions required for adiabatic computation.The system may fail to evolve fully adiabatically or begin in the true initial ground state, so the resulting solution is close to optimal rather than guaranteed optimal.
- Financial applications: Quantum optimization case studies in finance are proof-of-principle results that indicate quantum annealers may acquire practical value for financial problems.The paper frames these examples as near-term prospects rather than established superiority over classical methods.
A. Optimal trading trajectory
Dynamic portfolio optimization seeks a best trading trajectory while accounting for transaction costs and market impact. A discrete multiperiod formulation was implemented on D-Wave processors, where small instances matched classical hardware performance.
- Problem formulation: Dynamic portfolio optimization aims to find an optimal trajectory through portfolio space while accounting for transaction costs and market impact.The problem is described as a discrete multiperiod optimization task suitable for quantum annealers.
- Optimization model: The proposed quantum-annealing approach optimizes portfolio return under holdings constraints across all time steps.The formulation includes forecast returns, covariance, risk aversion, and transaction-cost contributions.
- Optimization model: The portfolio must maintain total holdings equal to K at every time step, while each asset’s maximum allowed holdings sum to at most K′.These constraints define the feasible trading trajectories used by the optimization.
- Experimental result: 512- and 1152-qubit D-Wave chips solved small instances with performance similar to classical hardware and a high success rate.Fine-tuning the D-Wave machines produced important improvements in success rates.
B. Optimal arbitrage opportunities
Optimal arbitrage can be formulated as a graph-cycle optimization problem and recast as QUBO for quantum annealers. A five-asset implementation matched exhaustive classical optimization, while later extensions incorporated risk and repeated purchases.
- Optimal arbitrage is NP-Hard, although classical algorithms can efficiently detect whether any profitable cycle exists.
- Quantum annealing models arbitrage by finding the most profitable cycle in a directed graph whose nodes are assets and edges carry conversion rates.Transaction costs are assumed to be included in the edge weights.
- Boolean variables xij encode whether each directed link belongs to the selected arbitrage cycle.The objective includes cycle cost, flow constraints, and restrictions preventing repeated traversal of an asset.
- The graph problem becomes a quadratic unconstrained binary optimization problem amenable to quantum annealers.
- The D-Wave 2X produced the same optimal solutions as an exhaustive classical solver on a five-asset example.The study was later extended to include risk variables and cases where an asset can be bought multiple times.
C. Optimal feature selection in credit scoring
Credit scoring requires selecting informative features when some data are irrelevant, unavailable, weakly correlated, or too costly to process jointly. The selection objective can be formulated as QUBO and optimized with a quantum annealer as a proof of principle.
- Credit scoring identifies borrowers as high- or low-risk using attributes such as income, age, financial history, and collateral.
- Feature selection is needed when applicant data are irrelevant, weakly correlated, incomplete, or too computationally expensive to use in full.
- The method represents applicant attributes in matrix U and past credit decisions in vector V, then minimizes a feature-selection cost function.
- Binary variable xi indicates feature inclusion, while correlations with credit outcomes and among features define the objective terms.Parameter α controls the relative weight between feature influence and feature independence.
- The resulting objective was implemented as a proof of principle using the 1QBit SDK, showing that future quantum annealers can select credit-analysis features.
A. Data classification
Data classification supports credit scoring, pattern recognition, and fraud detection, but high-dimensional projections can make classification computationally demanding. Quantum proposals encode data as quantum states or accelerate related linear-algebra operations, while readout and state storage remain important constraints.
- Classification assigns new customer vectors to loan-risk classes and also supports pattern recognition and outlier detection for fraud detection.
- Large training sets and many attributes require numerous high-dimensional projections, limiting confidence in assigning classes to new vectors.
- Quantum classification proposals encode data points as quantum states and estimate classical distances using repeated swap tests.
- Quantum support vector machines seek separating hyperplanes for labeled classes and have attracted attention for potentially favorable scaling.
- Pattern recognition on quantum computers remains an early field, although experimental demonstrations have been reported.
- Quantum regression methods target the expensive matrix inversion used to fit least-squares models, with proposals claiming exponential speedups for suitable data matrices.For sparse matrices, optimal fit parameters can be encoded in quantum-state amplitudes; state tomography may itself be exponentially expensive.
C. Principal component analysis
Principal component analysis extracts dominant trends from financial correlation matrices for forecasting and portfolio risk analysis. Quantum-PCA proposals address the prohibitive cost of classical diagonalization, while quantum methods also target neural-network training and richer financial patterns.
- PCA analyzes stock-price changes through a covariance matrix whose principal components represent important trends for predicting future movements.
- Classical diagonalization costs O(N^2) for an N × N matrix, making PCA impractical when financial datasets contain millions of stocks.
- Quantum-PCA approximates principal components of a correlation matrix with computational and query complexity O((log N)^2).The paper presents this as broadening PCA’s applicability to previously infeasible risk-estimation and profit-maximization settings.
- Neural networks are used for market prediction, credit-risk analysis, and financial time-series prediction, motivating quantum approaches to accelerate them.
- Quantum annealers can train neural-network models while allowing the trained algorithm to run on a classical computer.The paper expects this training approach to be less prone to local minima than several standard methods.
- A D-Wave proof of principle demonstrated efficient training of a Boltzmann machine, while quantum-PCA methods were proposed to accelerate iterative gradient descent.
- Fully quantum neural-network proposals include quantum perceptrons and quantum hidden Markov models, the latter being relevant to financial forecasting.
V. QUANTUM AMPLITUDE ESTIMATION AND MONTE CARLO
Quantum amplitude estimation reduces the sampling burden of Monte Carlo estimation, offering a quadratic speedup with applications to financial simulation and risk-related calculations.
- Monte Carlo motivation: Monte Carlo estimates system properties by statistically sampling realizations, making it useful for complex financial systems.In finance, applications include portfolio evaluation, personal finance planning, risk evaluation, and derivatives pricing.
- Monte Carlo motivation: k = O(σ2/ϵ2) samples are sufficient for approximately 99% success when estimating a distribution mean under the classical sampling bound.The convergence rate worsens for wide distributions or very small errors, potentially making simulations extremely large.
- Quantum amplitude estimation: Quantum amplitude amplification increases a desired probability to almost one in O(1/√p) operations, quadratically faster than classical search.This result motivated quantum amplitude estimation for expectation-value calculations.
- Quantum amplitude estimation: Quantum amplitude estimation combines amplitude-amplification operations with a quantum Fourier transform to estimate a state amplitude.The paper directs readers to the cited references for implementation details and notes a success probability of at least 8/π2.
- Quantum Monte Carlo: O(σ/ϵ) samples are required by Montanaro’s quantum Monte Carlo algorithm to estimate a mean with 99% success probability, up to polylogarithmic factors.The speedup comes from quantum amplitude estimation.
- Financial applications: The paper reviews two proposals applying quantum-accelerated Monte Carlo to financial problems, with potential speedups from quantum sampling and estimation.When combined, the two sampling speedups concatenate.
A. Pricing of financial derivatives
Quantum-accelerated Monte Carlo is presented as a route to faster financial estimation, especially for derivative pricing and risk analysis, while practical deployment remains limited by hardware requirements and scope.
- Pricing of financial derivatives: Derivative pricing determines a fair price from the future, potentially stochastic, price trajectory of an underlying asset.Classical approaches include simplified models such as Black-Scholes-Merton and Monte Carlo sampling.
- Pricing of financial derivatives: Quantum-accelerated Monte Carlo is proposed to obtain a quadratic speedup in pricing financial derivatives.The method constructs a quantum operator matching the derivative’s probability distribution and estimates its expectation value.
- Pricing of financial derivatives: The proposed derivative-pricing method is discussed for European call options and Asian options.These applications use the same expectation-estimation strategy based on a matched probability distribution.
- Risk analysis: VaR measures a portfolio-loss distribution, while CVaR measures expected loss beyond VaR.Both quantities are commonly estimated through Monte Carlo sampling in quantitative finance.
- Risk analysis: Quantum amplitude estimation was used to determine VaR and CVaR with excellent accuracy and a quadratic speedup relative to classical methods.The approach used a tailored oracle and was implemented for small examples on the IBM Q Experience.
- Perspectives and scope: Before a universal quantum processor can surpass present-day supercomputers, experimental progress and substantially higher qubit quality are required.Faulty quantum computers may nevertheless find applications before fault-tolerant quantum computing is achieved.
- Perspectives and scope: The review excludes several finance-related quantum applications, including blockchain, cryptocurrencies, quantum finance, quantum money, quantum cryptography, and quantum simulators.These topics are identified as areas that could be discussed in future work.