Quantum Amplitude Estimation for Investment Risk (2D)

Setup

True event probability a 2.0%

Quantum rounds / s 2

Classical samples / s 60

Canonical QAE (quantum)

Counting qubits m (M=2^m)

Total oracle queries M

Last measured â_Q

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Last round error

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Classical Monte Carlo

Samples N

Estimate â_C

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Std. error σ = √(a(1−a)/N)

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Speedup

Classical samples needed to match quantum's last error

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How it works

This is canonical quantum amplitude estimation: the event's true amplitude is encoded as θ = arcsin(√a). A register of m counting qubits runs quantum phase estimation on the Grover operator Q (reflect about "bad" ∘ reflect about the initial state), which has eigenvalues e^±2iθ. Measuring that register after M = 2^m total Grover applications yields outcome y with the exact probability derived by Brassard, Høyer, Mosca & Tapp (2002, Theorem 12):

Δ(y) = θ − πy/M
P(y) = sin²(M·Δ(y)) / (M²·sin²(Δ(y)))          [P(y)=1 if Δ(y)=0]
â(y) = sin²(πy/M)   ← the amplitude estimate read off y

Each round below draws a real outcome y from that exact distribution — genuine weighted random sampling, not a scripted number — then reports â(y) and its error against the true a. Because the distribution's width scales as 1/M, repeated rounds at exponentially growing M trace out the algorithm's real quadratic-speedup envelope:

Quantum (measured):  ε ~ O(1/M)          M = total Grover oracle calls
Classical (measured): ε ~ O(1/√N)         N = total Bernoulli samples

The classical curve on the right is not a formula either — it is the running error of an actual accumulating Monte Carlo estimator: every tick draws a real Bernoulli(a) sample and the plotted point is the true error of the sample mean so far. Both curves are logged on a log-log chart, so the quantum series should visibly fall with roughly twice the slope of the classical one — the same quadratic advantage proposed for pricing derivatives and estimating tail risk (e.g. the probability a portfolio's loss exceeds a Value-at-Risk threshold), where a is the rare "loss" event's true probability.

True event probability a — the rare tail event being estimated; smaller a needs more rounds/samples for the same relative precision.
Quantum rounds / s — how often a new canonical-QAE round is drawn; m cycles 0→13 then restarts, so M sweeps repeatedly while total spent queries keeps accumulating.
Classical samples / s — how fast independent Bernoulli(a) draws accumulate into the running Monte Carlo mean.

Quantum QAE round (measured) Classical Monte Carlo (measured) theory 1/M theory 1/√N