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Credit Risk Analysis using Quantum Computers
Daniel J. Egger, Ricardo Gacía Gutiérrez, Jordi Cahué Mestre, Stefan Woerner
TL;DR
Credit-risk analysis requires estimating economic capital from loss distributions, while rare-event Monte Carlo simulations can be computationally costly. The paper develops a quantum amplitude-estimation algorithm for Value at Risk and analyzes its scaling, reporting a quadratic speedup and an estimated 30-minute runtime for a one-million-asset portfolio under assumed future hardware conditions.
Problem
Estimating economic capital requires Value at Risk calculations for loss distributions, and rare-event Monte Carlo simulations require many costly samples.
Method
The paper uses quantum amplitude estimation to estimate Value at Risk and combines it with expected-loss calculations for credit-risk models.
Results
A quadratic speedup over classical Monte Carlo simulations is reported, with an estimated runtime of 30 minutes for Value at Risk in a one-million-asset portfolio.
Takeaways & Limitations
The analysis indicates substantial potential for quantum computing in credit-risk analysis, including realistic problem sizes and more complex uncertainty models.
Takeaways & Limitations
The scaling and runtime estimates rely on assumptions about the performance of future quantum hardware.
Abstract
from arXiv · showhide
We present and analyze a quantum algorithm to estimate credit risk more efficiently than Monte Carlo simulations can do on classical computers. More precisely, we estimate the economic capital requirement, i.e. the difference between the Value at Risk and the expected value of a given loss distribution. The economic capital requirement is an important risk metric because it summarizes the amount of capital required to remain solvent at a given confidence level. We implement this problem for a realistic loss distribution and analyze its scaling to a realistic problem size. In particular, we provide estimates of the total number of required qubits, the expected circuit depth, and how this translates into an expected runtime under reasonable assumptions on future fault-tolerant quantum hardware.
I. INTRODUCTION
Economic capital and credit-risk metrics are important because firms use them to determine capital needed for solvency, but Monte Carlo estimation of credit-risk tails is computationally costly. The paper motivates quantum amplitude-estimation approaches and examines their implementation and scaling.
- Economic Capital determines the risk capital financial service firms require to remain solvent under adverse yet realistic conditions.
- Credit risk is the monetary loss resulting from a counterparty failing to meet a financial obligation.
- Monte Carlo simulations are commonly used to calculate Value at Risk and Economic Capital Requirement across many scenarios.
- Rare-event credit-risk estimation requires many Monte Carlo samples, making it computationally costly.Importance sampling lowers constant factors but does not change the asymptotic convergence rate.
- The paper discusses gate-based quantum algorithms, small-instance simulations, and scaling to realistic problem sizes.
II. CREDIT RISK ANALYSIS
The paper defines Economic Capital Requirement from portfolio loss distributions using expected loss and Value at Risk, then considers independent and conditionally independent default models. It applies quantum amplitude estimation to estimate the resulting risk measure with a quadratic speedup over classical Monte Carlo simulations.
- Economic Capital Requirement summarizes capital needed to remain solvent at a specified confidence level and time horizon.The paper considers ECR related only to default risk.
- The portfolio model combines single-asset default probabilities, loss given default, and correlations among default events.
- Value at Risk is the smallest total loss whose probability is at least the confidence level α.Common industry confidence levels for ECR are around 99.9%.
- The paper considers an independent-loss model with Bernoulli defaults and a more realistic conditional-independence model driven by a latent standard-normal variable.
- Aggregating similar assets into multivalued random variables is proposed to scale the model to larger portfolios.
- Quantum amplitude estimation is used to estimate ECR, yielding a quadratic speedup over classical Monte Carlo simulations.
III. QUANTUM ALGORITHM
The algorithm uses QAE to estimate the total-loss CDF and combines it with bisection search to obtain VaR, while encoding uncertainty, losses, and comparisons in quantum registers. Correlated defaults are loaded through a latent-factor model, and circuit-depth reductions trade additional ancillas for parallelism.
- Quantum advantage: QAE provides a quadratic speedup because its estimation error scales inversely with the number of quantum samples, compared with classical Monte Carlo scaling as O(1/√M).The QAE success probability is 81%, and repeating the procedure with median aggregation can increase reliability.
- Value-at-Risk estimation: QAE estimates VaRα[L] by evaluating P[L ≤ x] and using bisection to find the smallest xα with probability at least α.The expected total loss E[L] is computed classically, so the quantum procedure focuses on VaR estimation.
- CDF operator: A = CSU maps uncertainty to total loss and flips an objective qubit when the loss is at most x, thereby encoding the CDF.U loads uncertainty, S computes the weighted loss sum, and C performs the comparison.
- Uncertainty loading: Correlated defaults are modeled by a standard-normal latent variable Z, conditioned-independent default events Xk, and Z-controlled rotations preparing their probabilities.The latent distribution is truncated and discretized before controlling rotations on the Xk registers.
- Circuit scaling: O(nZK) circuit depth for uncertainty loading can be reduced to O(log K) by adding O(K) ancilla qubits and parallelizing operations.The number of ancillas can be reduced when optimizing overall performance under qubit and algorithm-complexity constraints.
- Loss computation: The weighted-sum operator stores λ1x1 + ··· + λKxK using nS = ⌊log2(λ1 + ··· + λK)⌋ + 1 qubits for integer losses.The comparison operator then flips the target qubit for loss realizations no greater than x.
IV. RESULTS
The paper demonstrates the algorithm on a two-asset illustrative model simulated with classical computers. The experiment uses a compact quantum representation and applies QAE followed by bisection to determine the 95% VaR.
- Illustrative example: K = 2 assets, nZ = 2 latent-factor qubits, and nS = 2 loss qubits yield a seven-qubit operator A including the objective qubit.The example parameters include each asset’s loss given default, default probability, and sensitivity.
- Simulation setup: 12 qubits are required for the simulated experiment, including one QAE ancilla and four evaluation qubits representing 16 quantum samples.The bisection search requires at most two steps because nS = 2.
- VaR search: α = 95% and m = 4 define the illustrated CDF target and bisection-search configuration for estimating VaR.The figure shows the total-loss CDF, the target level, and the evolving upper, lower, and estimated bounds against the exact value.
V. SCALING TO REAL WORLD PROBLEM
The paper estimates resource requirements for a fault-tolerant quantum credit-risk algorithm at realistic portfolio sizes, focusing on circuit depth, qubits, and runtime. For a one-million-asset portfolio, the analysis estimates an overall runtime of about one hour, reducible to 30 minutes through parallelization.
- Resource analysis: The analysis models circuit depth in T/Toffoli-gates because T-gates are expected to dominate fault-tolerant quantum runtime.The circuit uses a Clifford + T decomposition, and Toffoli gates are treated as equivalent through ancilla-assisted constructions.
- Resource analysis: The total qubit requirement scales as O(K), because both asset representation and required ancillas scale linearly with the number of assets.The paper does not further elaborate the exact qubit count because runtime is the primary focus.
- Realistic-size estimate: For K = 2^20 assets, nZ = 10, nS = 30, and m = 10, the circuit depth for A is about 600 T/Toffoli-gates, while overall QAE depth is approximately 37 million gates.The assumptions include 1,024 discretized values for Z and 0.06%-points accuracy for α = 99.9%.
- Hardware assumptions: Limited processor connectivity may approximately double CNOT-gate counts, but this should not significantly affect runtime because T-gates dominate.This conclusion relies on an empirical factor-of-two increase observed for comparable circuits.
- Runtime estimate: Assuming error-corrected T/Toffoli-gates take 10^-4 seconds, the estimated runtime is around one hour, or 30 minutes when QPE removal enables parallelization on two quantum devices.The estimate concerns VaR for a one-million-asset portfolio.
- Scaling opportunities: Aggregating similar assets can potentially reduce the estimated 30-minute runtime toward near real-time while also lowering the required number of qubits.The same aggregation strategy is described as applicable to both classical and quantum simulations.
- Other objectives: Estimating CVaR requires only one additional, slightly more expensive QAE application and does not require the VaR bisection search.The paper therefore argues that CVaR can be estimated without much additional overhead.
VI. CONCLUSION
The paper develops and analyzes a quantum algorithm for estimating economic capital requirement with a quadratic speedup, demonstrating it in simulation and evaluating realistic-size scaling. Its projections depend on future quantum hardware, while further algorithmic work could reduce qubit and gate requirements.
- Contribution: The paper develops and analyzes a quantum algorithm to estimate ECR with a quadratic speedup.It demonstrates the algorithm using simulation and analyzes scaling and expected runtime for realistic problem sizes.
- Scope: The analysis argues that the results also extend to more complex uncertainty models and objectives such as CVaR without much additional overhead.The conclusion presents this as part of the algorithm’s potential scope.
- Future work: Further algorithmic research could reduce both the required number of qubits and the circuit depth.
- Limitations: The runtime and scaling analysis assumes performance characteristics for future quantum hardware that are not yet available at the required scale.The paper emphasizes transparency so estimates can be adjusted as hardware and algorithmic understanding improve.
Appendix A: Expected Total Loss
The appendix gives classical expressions for expected total loss under independent and conditionally independent default models, including numerical integration for the latter.
- The expected total loss for independent defaults follows from the linearity of expectation and independence of the default variables.
- Under conditional independence, the expected loss is computed using the conditional default structure.
- The standard-normal density term in the conditional model can be efficiently approximated through classical numerical integration.
Appendix B: Amplitude Estimation
The appendix explains how QAE estimates a measurement probability using quantum phase estimation and repeated applications of its operator, yielding a quadratic Monte Carlo speedup.
- QAE estimates a, the probability of measuring |1⟩ in the last qubit, using operator Q and quantum phase estimation.
- QAE uses m evaluation qubits and M = 2^m −1 applications of Q.
- QAE provides a quadratic speedup over classical Monte Carlo simulations because classical estimation error follows an O(1/√M) convergence rate.
Appendix C: Uncertainty Model
The uncertainty-model implementation uses controlled rotations, while duplicating uncertainty qubits trades ancillas and CNOT operations for substantially lower circuit depth.
- Each X_k qubit requires one uncontrolled Y-rotation and n_Z controlled Y-rotations for state preparation.
- For ϵ = 2^-10 ≈ 10^-3, uncontrolled and controlled Y-rotations have T-depths of 26 and 28, respectively.
- A straightforward implementation of U uses K uncontrolled and n_ZK controlled rotations, with depth K controlled Y-rotations when n_Z rotations run in parallel.
- Duplicating the Z register w−1 times requires n_Z(w−1) ancillas and 2n_Zw CNOT gates, while reducing preparation CNOT-depth to 2 log_2(w).
Appendix D: Weighted Sum Operator
The weighted-sum operator computes total loss by recursively combining asset pairs with adders, using parallel intermediate computations across log_2(K) iterations.
- A divide-and-conquer procedure first sums asset pairs, then repeatedly combines partial sums until the total loss is computed.
- The construction begins with a weighted-sum operator and uses adder circuits to iteratively combine intermediate results.
- The average loss-per-asset representation uses at most n_S − log_2(K) qubits before one qubit is added per iteration.
- After log_2(K) iterations, the total loss is computed, with intermediate results assumed to be evaluated in parallel.
- An n-qubit adder can achieve Toffoli-depth ⌊log_2(n)⌋ + ⌊log_2(n/3)⌋ + 7 using a linear number of ancilla qubits.