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Quantum Risk Analysis

Stefan Woerner, Daniel J. Egger

arXiv:1806.06893v1quant-ph

TL;DR

The paper addresses the computational cost of Monte Carlo methods for estimating financial risk measures. It uses quantum amplitude estimation to calculate measures including VaR and CVaR, studies circuit-depth trade-offs, and demonstrates the approach on Treasury portfolios. The reported results show improved convergence over Monte Carlo methods, while current hardware limits realistic two-asset calculations.

  • Problem

    Monte Carlo methods are traditionally used for financial risk estimation, but realistic portfolio calculations require many samples and can be computationally intensive.

  • Method

    The paper uses quantum amplitude estimation on gate-based quantum computers to estimate expectation, variance, VaR, and CVaR, with implementations demonstrated on Treasury portfolios.

  • Results

    The demonstrated algorithm achieves improved convergence over classical Monte Carlo methods, including quadratic speed-up claims for VaR and CVaR estimation.

  • Takeaways & Limitations

    Quantum amplitude estimation provides a flexible framework for quantum risk analysis, with applications demonstrated on real hardware and simulated Treasury portfolios.

  • Takeaways & Limitations

    Current hardware limits circuit depth for the two-asset portfolio, and practical quantum advantage requires more qubits, lower errors, and more than a small number of samples.

Abstract

from arXiv · show

We present a quantum algorithm that analyzes risk more efficiently than Monte Carlo simulations traditionally used on classical computers. We employ quantum amplitude estimation to evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer. Additionally, we show how to implement this algorithm and how to trade off the convergence rate of the algorithm and the circuit depth. The shortest possible circuit depth - growing polynomially in the number of qubits representing the uncertainty - leads to a convergence rate of $O(M^{-2/3})$. This is already faster than classical Monte Carlo simulations which converge at a rate of $O(M^{-1/2})$. If we allow the circuit depth to grow faster, but still polynomially, the convergence rate quickly approaches the optimum of $O(M^{-1})$. Thus, for slowly increasing circuit depths our algorithm provides a near quadratic speed-up compared to Monte Carlo methods. We demonstrate our algorithm using two toy models. In the first model we use real hardware, such as the IBM Q Experience, to measure the financial risk in a Treasury-bill (T-bill) faced by a possible interest rate increase. In the second model, we simulate our algorithm to illustrate how a quantum computer can determine financial risk for a two-asset portfolio made up of Government debt with different maturity dates. Both models confirm the improved convergence rate over Monte Carlo methods. Using simulations, we also evaluate the impact of cross-talk and energy relaxation errors.

I. INTRODUCTION

The paper motivates quantum risk analysis by the computational cost of Monte Carlo estimation and develops amplitude-estimation methods for expectation, variance, VaR, and CVaR. It also outlines quantum-circuit implementation and applications to Treasury portfolios.

  • Motivation: Monte Carlo simulations are widely used to determine VaR and CVaR but require many portfolio realizations for representative loss distributions.VaR is a loss-distribution quantile, while CVaR is the expected loss beyond VaR and is more sensitive to tail events.
  • Applications: The paper applies the approach to US Treasury portfolios, including a real-hardware single-period T-bill model and a simulated two-asset portfolio.The introduction presents amplitude estimation for expected T-bill value on a real quantum computer and simulation of a more comprehensive portfolio.
  • Quantum approach: Amplitude estimation converges as O(M^-1), providing a quadratic speed-up over classical Monte Carlo convergence.The algorithm estimates an unknown parameter through quantum amplitude estimation and uses M = 2^m applications of the relevant operator.
  • Quantum approach: Quantum states encode discrete random-variable distributions, whose expected values and variances are obtained by mapping functions such as x/(N −1) and x^2/(N −1)^2 onto an ancilla amplitude.The measured amplitude estimates the corresponding expectation, enabling variance calculation from E[X^2] − E[X]^2.
  • Quantum risk measures: VaR is estimated by applying threshold functions and using bisection to find the smallest level whose cumulative probability reaches 1 − α.The resulting accuracy scales as O(M^-1), again yielding a quadratic speed-up over classical Monte Carlo methods.
  • Quantum risk measures: CVaR is estimated by applying a function restricted to outcomes below the VaR threshold and normalizing the resulting probability.The paper states that replacing the threshold probability by an estimate still preserves a quadratic speed-up over classical Monte Carlo methods.

III. QUANTUM CIRCUITS

The section maps amplitude-estimation algorithms onto quantum circuits for expectations, variance, VaR, and CVaR, using polynomial approximations to reduce implementation cost. Circuit depth and convergence rate are traded off, with faster-than-classical rates approaching the optimal rate as depth increases.

  • Circuit construction: Quantum circuits map amplitude estimation to expectation, variance, VaR, and CVaR estimation for random distributions.VaR uses a threshold operator and bisection search; CVaR controls the expectation circuit with the VaR threshold ancilla.
  • Circuit complexity: Preparing the uncertainty distribution may require O(2^n) gates in general, although polynomial complexity is possible for distributions such as log-concave ones.The method assumes an operator R that prepares the distribution state.
  • Circuit construction: Polynomially many gates can construct the operator for a polynomial function using multi-controlled Y-rotations.The approach avoids ancillas for polynomial approximations, although the resulting convergence rate is lower than the ideal rate.
  • Circuit complexity: O(n^2) gates suffice in the smallest linear-function scenario when the number of uncertainty qubits n grows.This case uses linear f and p, corresponding to s = 1 and u = 0.
  • Convergence trade-off: O(M^-2/3) convergence is already faster than classical Monte Carlo, while increasing polynomial circuit depth approaches O(M^-1).The convergence trade-off follows from selecting the scaling and polynomial-approximation parameters.
  • Risk measures: VaR estimation uses an O(n)-gate threshold operator and a bisection search requiring at most n steps.The threshold operator marks x ≤ l, allowing amplitude estimation to approximate the cumulative probability.

IV. T-BILL ON A SINGLE PERIOD BINOMIAL TREE

The paper applies amplitude estimation to a one-period binomial model for a T-bill exposed to a possible interest-rate increase. The simple model can serve as a building block for more complex multi-period, path-dependent debt models.

  • Model: The T-bill model values a zero-coupon bond when the interest rate may rise by δr in the next time step.The bond value is represented as a probability-weighted combination of low-rate and high-rate outcomes.
  • Model: A one-year T-bill example uses a 0.25%-point rate increase with 70% probability and no change with 30% probability.The probabilities correspond to a possible Federal Open Market Committee announcement scenario.
  • Quantum implementation: Amplitude estimation maps the T-bill value to [0, 1], with the low and high values corresponding to $0 and $1.The implementation uses a single uncertainty qubit and an objective qubit.
  • Quantum implementation: The one-dimensional amplitude-estimation operator is Q = Ry(2θp), enabling efficient construction of its controlled powers.The circuit estimates p = E[X] = 30%.
  • Extensions: A single-period binomial tree can be extended straightforwardly to multi-period multinomial trees with path-dependent assets.The authors describe the single-period model as the smallest building block for scenarios of arbitrary complexity.

Results from real quantum hardware

Real-hardware experiments apply amplitude estimation with increasing numbers of evaluation qubits to the T-bill problem. The observed estimator approaches the true value while resolution improves as the number of evaluation qubits increases.

  • Circuit structure: The m = 3 circuit includes controlled Q^2^j operations, a qubit swap, and an inverse QFT.The swap compensates for limited processor connectivity; omitted U2 and U3 rotation parameters are formally defined elsewhere.
  • Hardware setup: m = 1, 2, 3, 4 evaluation qubits correspond to M = 2, 4, 8, 16 samples, respectively.The circuits require at most five qubits and were run on the IBM Q 5 Yorktown processor.
  • Observed results: The most frequent estimator approaches the real value, and algorithmic resolution increases with m.Each circuit was repeated 8192 times to obtain a reliable estimate despite probabilistic success and hardware errors.

V. TWO ASSET PORTFOLIO

The paper models daily risk in a two-asset Treasury portfolio using correlated interest-rate changes and principal components, then evaluates amplitude estimation on real hardware and compares its convergence with Monte Carlo.

  • V. TWO ASSET PORTFOLIO: The portfolio contains one-year US Treasury bills and two-year US Treasury notes, with interest-rate changes as the primary risk factors.The model uses Constant Maturity Treasury rates for the one-year and two-year yields.
  • V. TWO ASSET PORTFOLIO: Amplitude estimation on real hardware uses m = 1, ..., 4 and 8192 shots per setting, with estimates compared against a 30% target.The figure distinguishes the most frequent estimate from other outcomes and maps measured values classically to amplitudes.
  • V. TWO ASSET PORTFOLIO: For M ≥16 samples, the quantum algorithm's error falls below the Monte Carlo results despite starting with a larger estimation error.The comparison includes real experiments using up to five qubits and projections for six- and seven-qubit experiments.
  • V. TWO ASSET PORTFOLIO: 96% of the variance is explained by the first three principal components: shift, twist, and butterfly.The high correlation among Treasury-rate changes enables dimensionality reduction for more complex portfolios.
  • V. TWO ASSET PORTFOLIO: The demonstration retains shift and twist, whose correlation coefficient is −1%, and fits separate discrete distributions to them.The paper retains two components for a tractable simulation, although all three are expected on real quantum hardware.

A. Uncertainty representation in the quantum computer

The portfolio uncertainty is represented with principal-component variables encoded as discrete quantum states, using separate register sizes and affine mappings for shift and twist.

  • A. Uncertainty representation in the quantum computer: Three qubits encode shift and two qubits encode twist, allowing eight and four possible values, respectively.Shift receives more qubits because it explains a larger share of the variance.
  • A. Uncertainty representation in the quantum computer: The operator R prepares a five-qubit state representing the discrete distributions of shift and twist up to the affine mappings.The distributions are fitted separately because shift and twist are nearly uncorrelated in the demonstration.
  • A. Uncertainty representation in the quantum computer: Figure 7 overlays historical one-year-versus-two-year Treasury rates with the resulting shift and twist principal-component vectors.The shift vector is longer than the twist vector in the figure's representation.
  • A. Uncertainty representation in the quantum computer: Figure 8 pairs 8-bin shift and 4-bin twist histograms with fitted distributions and circuits that load them into the quantum computer.Histogram labels identify the quantum state associated with each probability.

B. Portfolio model on the quantum computer

The portfolio model maps discrete shift and twist variables into an approximated, normalized portfolio value, with circuit depth traded against model fidelity.

  • B. Portfolio model on the quantum computer: The portfolio value is rewritten in terms of the shift and twist registers rather than the original Treasury yields.The construction uses the portfolio equations together with the affine mappings of the uncertainty variables.
  • B. Portfolio model on the quantum computer: A first-order approximation around x = 3.5 and y = 1.5 represents the portfolio through its duration.Higher-order expansions such as convexity could be included at the cost of increased circuit depth.
  • B. Portfolio model on the quantum computer: The approximated portfolio value is normalized to f ∈ [0, 1] using its minimum at (7, 3) and maximum at (0, 0).The scaling supports construction of an operator F for a chosen parameter c ∈ (0, 1].

C. Results from simulations of an ideal quantum computer

The ideal-quantum-computer simulation estimates the portfolio's 95% VaR and examines sampling accuracy alongside the gate overhead caused by qubit connectivity.

  • C. Results from simulations of an ideal quantum computer: The simulation uses m sampling qubits, requiring m + 12 qubits for expected value and VaR and m + 13 for CVaR.It compares all-to-all connectivity with connectivity corresponding to the IBM Q 20 chip.
  • C. Results from simulations of an ideal quantum computer: The exact 95% VaR is $0.288, corresponding to 0.093 after the paper's mapping, and the quantum estimate rapidly approaches it as m increases.With m = 5 sample qubits, the difference between classical and quantum estimates is 9%.
  • C. Results from simulations of an ideal quantum computer: The VaR circuit's CNOT count scales as O(M), while the resulting estimation error scales as O(M −1).Adding a sample qubit approximately doubles the number of required CNOT gates.
  • C. Results from simulations of an ideal quantum computer: IBM Q 20 connectivity increases the CNOT count by a factor of 2.5 compared with all-to-all connectivity.This quantifies the gate overhead imposed by a realistic chip architecture.

D. Results from simulations of a noisy quantum computer

The noisy two-asset portfolio simulation identifies when relaxation and cross-talk still permit correct risk estimation, while highlighting substantial hardware and modeling constraints.

  • The error study models energy relaxation and two-qubit-gate cross-talk as the leading error sources.Single-qubit gates are assumed instantaneous and exempt from errors, while CNOT gates last 100 ns.
  • Relaxation rates rγ < 10−4 s−1 and cross-talk strength |α| < 1% preserve successful identification of the expected value 0.5.Success means measuring 0.5 with probability greater than 50%, despite the 4383-gate circuit.
  • The expected-value test uses two sample qubits because the symmetric portfolio distribution maps to an exactly estimable value of 0.5.The algorithm is considered successful when 0.5 is identified with probability above 50%.
  • Current-hardware estimates suggest the simulation may be possible near term if measurement and gate-calibration errors remain controlled.The stated hardware estimates are γ = 10−4 s−1 and α = −2%.
  • Practical quantum advantage remains constrained because realistic scenarios need more qubits, hardware errors must fall, and classical Monte Carlo can be massively parallelized.The paper states that more samples are needed for a practical quantum advantage.
  • For real-hardware VaR on the two-asset portfolio, qubit coherence times may need to increase by several orders of magnitude and cross-talk must be further suppressed.
  • Quantum phase-estimation improvements, more versatile gate sets, and error mitigation could shorten circuits or reduce hardware requirements.The paper identifies direct SWAP gates and error mitigation as possible contributors.
  • Loading autocorrelated random processes remains costly and requires further investigation; importance sampling may improve results or reduce circuit depth.

Appendix A: Q-Operator

The appendix defines the amplitude-estimation Q-operator and develops approximation-error bounds, including the conclusion that CVaR retains a quadratic speedup over classical Monte Carlo.

  • The Q-operator is constructed from reflections around the all-zero state and the prepared state A|0⟩.For n = 0, it reduces to Q = AZA†Z; for A = Ry(θ), Q = Ry(2θ).
  • For a Taylor approximation of order 2u + 1, the error bound is controlled by the next Taylor coefficient, corresponding to order 2u + 3.
  • The Taylor-series bound uses coefficient estimates obtained after rescaling y by c ∈ (0, 1] and evaluating extrema at y = ±1.
  • CVaR error analysis accounts for estimating both the numerator and denominator rather than applying the amplitude-estimation bound directly.The derivation uses first-order Taylor expansion in the estimation errors and omits higher-order terms.
  • The quantum estimation of CVaR achieves a quadratic speedup compared to classical Monte Carlo methods.

Appendix D: U2, U3 single qubit rotations

This appendix introduces the single-qubit rotations used in the circuit figures and documents the hardware topologies and comparison procedures used in the experiments.

  • The appendix defines the U2 and U3 single-qubit rotations used in Fig. 3.
  • The Monte Carlo comparison uses a 95%-confidence interval and an optimistic standard-error bound for the small-sample setting.
  • The quantum estimate maps the most probable integer outcome y to ã = sin2(yπ/M) and converts the resulting phase interval into an estimate interval.
  • The study uses Constant Maturity Treasury rates across ten maturities and more than 5′000 data points.The maturities range from 1/4 year to 30 years, using periods where all rates are available.
  • IBM Q 5 and IBM Q 20 topology figures encode connectivity through lines joining qubit pairs that support CNOT gates.
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