Evaluate risk measures of a financial position — Value at Risk and Conditional Value at Risk among them — where the classical route is a Monte Carlo simulation over sampled realisations of the uncertainty.
value at riskrisk analysisamplitude estimationmonte carlofinance
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Evaluate risk measures of a financial position — Value at Risk and Conditional Value at Risk among them — where the classical route is a Monte Carlo simulation over sampled realisations of the uncertainty. Woerner and Egger present a quantum algorithm that, they state, analyzes risk more efficiently than the Monte Carlo simulations traditionally used on classical computers. The algorithm employs quantum amplitude estimation to evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer, and the paper additionally shows how to implement it and how to trade off the convergence rate against the circuit depth. That trade-off is the substance of the result: at the shortest possible circuit depth — one the authors describe as growing polynomially in the number of qubits representing the uncertainty — the convergence rate is O(M⁻²ᐟ³), which the paper notes is already faster than classical Monte Carlo simulations converging at O(M⁻¹ᐟ²); allowing the depth to grow faster, but still polynomially, brings the rate toward the optimum of O(M⁻¹). The authors demonstrate the algorithm on two toy models, using real hardware such as the IBM Q Experience to measure the financial risk in a Treasury-bill faced by a possible interest rate increase in the first, and simulating the algorithm for a two-asset portfolio of government debt with different maturity dates in the second. They report that both models confirm the improved convergence rate over Monte Carlo methods, and that they also evaluate the impact of cross-talk and energy relaxation errors using simulations.
Circuit & simulation
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algorithm
transformmeasure
readout
measure
What this takes and returns
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How it works
Woerner and Egger present a quantum algorithm that, they state, analyzes risk more efficiently than the Monte Carlo simulations traditionally used on classical computers. The algorithm employs quantum amplitude estimation to evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer, and the paper additionally shows how to implement it and how to trade off the convergence rate against the circuit depth. That trade-off is the substance of the result: at the shortest possible circuit depth — one the authors describe as growing polynomially in the number of qubits representing the uncertainty — the convergence rate is O(M⁻²ᐟ³), which the paper notes is already faster than classical Monte Carlo simulations converging at O(M⁻¹ᐟ²); allowing the depth to grow faster, but still polynomially, brings the rate toward the optimum of O(M⁻¹). The authors demonstrate the algorithm on two toy models, using real hardware such as the IBM Q Experience to measure the financial risk in a Treasury-bill faced by a possible interest rate increase in the first, and simulating the algorithm for a two-asset portfolio of government debt with different maturity dates in the second. They report that both models confirm the improved convergence rate over Monte Carlo methods, and that they also evaluate the impact of cross-talk and energy relaxation errors using simulations. The Classiq library carries this subject under applications · finance. Reported cost: A convergence rate of O(M⁻²ᐟ³) at the shortest possible circuit depth, which the abstract describes as growing polynomially in the number of qubits representing the uncertainty, against O(M⁻¹ᐟ²) for classical Monte Carlo; allowing the depth to grow faster but still polynomially brings the rate toward the optimum of O(M⁻¹), which the abstract characterises as a near quadratic speed-up compared to Monte Carlo methods for slowly increasing circuit depths. The abstract states the result only in those terms: it gives no gate count, no qubit count, no dependence on the number of risk factors, and no constant factor..
Implementation
Unsupported
risk-analysis-amplitude-estimation.txt
ALGORITHM: Value at Risk by quantum amplitude estimation
PROBLEM: Evaluate risk measures of a financial position — Value at Risk and Conditional Value at Risk among them — where the classical route is a Monte Carlo simulation over sampled realisations of the uncertainty.
IDEA: Woerner and Egger present a quantum algorithm that, they state, analyzes risk more efficiently than the Monte Carlo simulations traditionally used on classical computers. The algorithm employs quantum amplitude estimation to evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer, and the paper additionally shows how to implement it and how to trade off the convergence rate against the circuit depth. That trade-off is the substance of the result: at the shortest possible circuit depth — one the authors describe as growing polynomially in the number of qubits representing the uncertainty — the convergence rate is O(M⁻²ᐟ³), which the paper notes is already faster than classical Monte Carlo simulations converging at O(M⁻¹ᐟ²); allowing the depth to grow faster, but still polynomially, brings the rate toward the optimum of O(M⁻¹). The authors demonstrate the algorithm on two toy models, using real hardware such as the IBM Q Experience to measure the financial risk in a Treasury-bill faced by a possible interest rate increase in the first, and simulating the algorithm for a two-asset portfolio of government debt with different maturity dates in the second. They report that both models confirm the improved convergence rate over Monte Carlo methods, and that they also evaluate the impact of cross-talk and energy relaxation errors using simulations.
REPORTED COST: A convergence rate of O(M⁻²ᐟ³) at the shortest possible circuit depth, which the abstract describes as growing polynomially in the number of qubits representing the uncertainty, against O(M⁻¹ᐟ²) for classical Monte Carlo; allowing the depth to grow faster but still polynomially brings the rate toward the optimum of O(M⁻¹), which the abstract characterises as a near quadratic speed-up compared to Monte Carlo methods for slowly increasing circuit depths. The abstract states the result only in those terms: it gives no gate count, no qubit count, no dependence on the number of risk factors, and no constant factor.
BASIS: abstract of arXiv:1806.06893 (TeX rendered into Unicode: the abstract writes the rates in inline math mode as $O(M^{-2/3})$, $O(M^{-1/2})$ and $O(M^{-1})$, written above asO(M⁻²ᐟ³), O(M⁻¹ᐟ²) and O(M⁻¹)): "The shortest possible circuit depth - growing polynomially in the number of qubits representing the uncertainty - leads to a convergence rate of $O(M^{-2/3})$." and "This is already faster than classical Monte Carlo simulations which converge at a rate of $O(M^{-1/2})$." and "If we allow the circuit depth to grow faster, but still polynomially, the convergence rate quickly approaches the optimum of $O(M^{-1})$." The comparison is the abstract's own: "Thus, for slowly increasing circuit depths our algorithm provides a near quadratic speed-up compared to Monte Carlo methods." The same abstract fixes what is being computed — "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." — and quotes no gate count, no qubit count and no constant anywhere. The Classiq index entry this record covers, applications/finance/value_at_risk, gives a directory path and a file list and states no bound. Those are the only sources read for this field.
DEMONSTRATED BY: the Classiq library entry applications/finance/value_at_risk
PRIMARY SOURCE: Stefan Woerner, Daniel J. Egger (2018), Quantum Risk Analysis — https://arxiv.org/abs/1806.06893
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Quantum vs classical
Classical baseline
Compare Amplitude estimation with the strongest classical method for the same instance, input budget, and output metric.
Quantum claim
This reference exposes a quantum circuit pattern; it does not imply an application-level speedup without a matched benchmark.
How to compare
Report input loading, circuit depth, repetitions, classical preprocessing, post-processing, and wall-clock time together.
Declared gaps
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Literature & references
Quantum Risk Analysis2018 · Stefan Woerner, Daniel J. Egger
Primary source: it presents the algorithm, employs quantum amplitude estimation to evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer, states the convergence rates at the shortest possible circuit depth and at faster-growing polynomial depths, and reports the two toy models — the T-bill on real hardware such as the IBM Q Experience, and the simulated two-asset government-debt portfolio. Consult it for how the uncertainty is loaded, how the risk measure is read out of the amplitude, what the circuit depths actually are, and for the cross-talk and energy relaxation study; the abstract states none of those.