Algorithms · Amplitude estimation
Value at Risk by quantum amplitude estimation
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.
Attested & literatureUpdated
A small worked example, computed in your browser. It is not a result from this record's papers.
Canonical amplitude estimation
3 counting qubits estimating a = sin²(θ) for the one-qubit state A|0⟩ = RY(π/4)|0⟩, θ = π/8.
Step 1 of 4 · RY(π/4)
Probability of each outcome after this step
- 000085%
- 100015%
Prepares the target qubit as A|0⟩ = cos(θ)|0⟩ + sin(θ)|1⟩; the quantity being estimated is a = sin²(θ), the probability that measuring this qubit alone would give |1⟩.
Probability spreads from 1 outcome to 2.
Spreads the 3 counting qubits into an equal superposition of all 8 values.
Probability spreads from 2 outcomes to 16.
Applies controlled powers of the Grover operator Q = A Z A⁻¹ Z, one power per counting qubit: phase estimation of Q, using A|0⟩ in place of a single eigenstate.
Probability moves between the same 16 outcomes. Down: 0110 10.7% → 1.8%, 0010 10.7% → 1.8%. This is the step that entangles the qubits: from here on no single qubit has a state of its own, only the register as a whole does.
An inverse QFT on the counting register reads the phase into a 3-bit estimate.
Probability gathers from 16 outcomes onto 4. 0001 goes from 1.8% to 25.0%.
Readout The counting register reads 001 or 111, each equally likely: these are θ/π = 1/8 and its mirror 1 − 1/8 = 7/8, both exact in 3 bits by construction. Either one gives a = sin²(π/8) = (2 − √2) / 4 exactly, the standard amplitude-estimation formula a = sin²(πλ) applied to the smaller of the two readings.
Open in Studio- Record typeLiterature reference
- Reference implementationvalue_at_risk
- Reported costA 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.
- Papers1
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.
TakesNothingWhat joins here
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
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..
TakesNothingWhat joins here
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
- Record type
- Literature reference
- Reference implementation
- value_at_risk
- 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.
- Complexity 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 as O(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 demonstration this record covers states no complexity bound of its own. Those are the only sources read for this field.
- Circuit
- Not supplied
This record is not a concrete circuit.
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.
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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 as O(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 demonstration this record covers states no complexity bound of its own. Those are the only sources read for this field.
PRIMARY SOURCE: Stefan Woerner, Daniel J. Egger (2018), Quantum Risk Analysis — https://arxiv.org/abs/1806.06893
This is a literature reference record, not an executable circuit.A reference record, not runnable source. Leona cannot execute it, so it cannot be saved to your Library as a circuit.
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References
- Quantum Risk Analysis
Stefan Woerner, Daniel J. Egger · 2018
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.