Business risk, estimated on a quantum computer.
Build a network of risk events, compile it to a quantum circuit, and estimate the tail risk with quantum amplitude estimation — the quadratic speedup over Monte Carlo, running here on a simulator.
That tail estimate is the inner loop of a deeper result in our 2021 paper — the sensitivity analysis: a Grover search layered on top of the amplitude estimation, using it as an imperfect oracle, to find which risk drives the tail the most. Stacking Grover’s quadratic search over QAE’s quadratic estimation compounds into a quartic speedup. The demo here runs the inner estimate; the sensitivity layer is explained below.
Near-term use case Business risk is the most accessible target — a compact network of relatively few parameters. Credit and liquidity risk are natural next applications of the same construction, but call for many more data points to model faithfully.
Why this matters
An enterprise risk model is a network of events: a market shock can trigger a counterparty default, which can trigger a liquidity squeeze, each with its own probability and financial impact. The quantity that matters to a risk manager is the tail — the probability that total losses exceed a capital threshold.
Classically, that probability is estimated by Monte Carlo: draw many random scenarios and count. Halving the error costs four times the samples (error ~ 1/√N). Quantum amplitude estimation reaches the same accuracy with a quadratic speedup (error ~ 1/N) — the result behind our 2021 work on a risk model used at Deutsche Börse Group.
The demo below is a small, runnable version of exactly that construction.
Risk network
Node = risk item (intrinsic probability, impact). Arrow = a conditional trigger: if the source risk occurs, it raises the probability of the target.
Parameters
1 · Compiled circuit
The state-preparation circuit encodes the risk network: an Ry rotation per risk item sets its intrinsic probability, and a controlled Ry per arrow applies a trigger. With QAE on, the evaluation register and controlled Grover iterations are appended automatically.
2 · Tail-risk estimate
3 · Resources used
| State qubits (risk items) | — |
|---|---|
| QAE evaluation qubits | — |
| Total qubits | — |
| Gates | — |
| Circuit depth | — |
| Grover iterations | — |
| Simulator wall-time | — |
How it works
Five steps turn a risk network into a quantum estimate — and then rank what drives it. Open each for the detail.
1 Risk network → quantum state
One qubit per risk item. An Ry rotation puts each qubit into a superposition whose |1⟩ weight equals the risk's intrinsic probability:
Ry(θ)|0⟩ = √(1−p) |0⟩ + √p |1⟩, θ = 2·arcsin(√p)
A conditional trigger (an arrow in the graph) is a controlled Ry: when the source qubit is |1⟩, it adds rotation to the target, raising its probability. The whole network becomes one state-preparation operator A acting on |0…0⟩.
2 Marking the loss event
Each basis state of the risk register is one scenario; its total loss is the sum of the impacts of the risks that occurred. We mark every scenario whose loss reaches the threshold as a «good» state — a reflection Sχ that flips the sign of those amplitudes.
a = P(loss ≥ threshold) = Σ over good scenarios |amplitude|²
This a is exactly the tail probability we want to estimate.
3 Amplitude estimation & the quadratic speedup
The Grover/amplitude operator Q = −A S0 A† Sχ rotates by an angle 2θ with a = sin²θ. Quantum phase estimation reads that angle: an evaluation register of m qubits applies controlled powers Q, Q², Q⁴, … then an inverse QFT.
classical Monte Carlo: error ~ 1/√N (N samples) quantum amplitude est.: error ~ 1/N (N = 2ᵐ oracle calls)
Same budget, quadratically smaller error. More evaluation qubits m → finer estimate.
4 Reading the answer
The evaluation register is measured, giving an integer y. The tail probability is recovered as:
â = sin²(π y / 2ᵐ)
The demo compares this quantum estimate against the exact value (full enumeration) and against a classical Monte Carlo run given the same number of model evaluations.
5 Sensitivity analysis & the quartic speedup
A tail probability is one number. The question a risk manager actually acts on is which input drives it: of all the intrinsic probabilities and trigger strengths, which one, when it moves, moves the tail the most? That is the sensitivity analysis — and it is the headline result of our 2021 paper.
Classically you would perturb each parameter and re-run the whole Monte Carlo estimate, then search for the largest response — a search on top of an estimation. The quantum version stacks the same two ideas:
- Inner loop — amplitude estimation. Evaluating the tail probability for one parameter setting is the QAE of steps 3–4: a quadratic speedup over Monte Carlo.
- Outer loop — Grover search. Finding the most sensitive parameter is a maximum-finding search over parameter space. Grover search delivers a second quadratic speedup over scanning candidates one by one.
The subtlety is that the inner QAE is an imperfect oracle: it returns an estimate with bounded error, not an exact yes/no bit. Grover normally assumes a perfect oracle, so the construction uses a noise-robust amplitude-amplification variant that tolerates that inner uncertainty. Composing the two quadratic speedups gives an overall quartic speedup:
Monte Carlo + scan: error / cost ~ 1/√N QAE (estimate): ~ 1/N (quadratic) Grover × QAE (sensitivity): ~ 1/N² effective → quartic (N⁴ fewer evaluations)
This stacked Grover-over-QAE structure — not the single tail estimate — is the algorithm the paper’s title refers to.
What this needs at production scale
The toy above runs in your browser. The real model is larger — but still strikingly modest in qubits.
Our 2021 analysis of a Deutsche Börse Group risk model showed the full construction would run on fewer than 200 error-corrected qubits, with a provable quadratic speedup over the classical Monte Carlo it would replace. Layering Grover search over that amplitude estimation — the sensitivity analysis that names the paper — compounds it into a quartic speedup for ranking which risks drive the tail. That is squarely within the envelope of the fault-tolerant machines now on vendor roadmaps — see the convergence chart on the Shor demo.
Reference — M. C. Braun, T. Decker, N. Hegemann, S. F. Kerstan, C. Schäfer, A Quantum Algorithm for the Sensitivity Analysis of Business Risks (2021).
Honest caveats
- Additive trigger model. Conditional triggers add rotation angle on the target qubit. This is a faithful, compact encoding of a probabilistic risk network — the published model handles richer dependency structures.
- Grover operator as one block. The amplitude operator Q is computed as a unitary and applied as a single labelled block, then its controlled powers are built by matrix squaring. A gate-level decomposition exists but would obscure the structure at this scale.
- State-vector simulation. The estimate is exact quantum mechanics with no hardware noise. The point is the scaling of accuracy with evaluation qubits, not a hardware benchmark.
- Inner estimate only. This demo runs the single tail-probability estimate (the QAE inner loop). The sensitivity analysis — the outer Grover search over parameters that yields the quartic speedup — is described above but not executed here.
- Best-case readout. The QAE estimate shown is the most likely evaluation-register outcome (the closest grid point), not a sampled measurement. A real run draws from a distribution around it, so on hardware the per-shot estimate would scatter; here we report the noiseless centre.