Implementing “Option Pricing using Quantum Computers” in Qiskit — Part 6: Section 5
This is the final part of the series. Every option type from the paper (vanilla, portfolios, basket, Asian, barrier) has been built, verified against classical ground truth, and run on real hardware in earlier parts. This one is Section 5’s real-hardware experiment on a vanilla European call — and how closely it can actually be reproduced.
The paper states, for S0=2, exact discretized prices and probabilities: [1.21, 1.74, 2.28, 2.81], [0.001, 0.554, 0.425, 0.019]. Fig. 15 gives the actual circuit, gate by gate, angles included.
The next formula to use, converting a measured probability back into a dollar payoff, is Eq. 12. Deriving its algebraic inverse and applying it gave a result nowhere near the classically-computed ground truth — off by roughly 25%, even using the exact statevector value with zero estimation noise involved.
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The fix came from a different equation in Section 3.3 — the rotation angle’s actual construction, θ(i) = c·f̃(i) + π/4 — which turns out not to invert the same way Eq. 12's literal form suggested. Reverse-engineering f̃(i) directly from the paper's own known-correct angles (it's a clean, symmetric ramp from -π/4 to +π/4, centered exactly at the strike) gave the actual correct inversion. Once that was in place, the corrected post-processing matched ground truth almost exactly, at S0=2.
Extending to the full spot-price sweep
The paper sweeps S0 from 1.8 to 2.5, stating plainly that only the distribution-loading part of the circuit changes across this sweep — the payoff-rotation angles stay fixed. That's a real, useful fact, and it's confirmed by the shape of what happens if you take it slightly too literally: keeping the exact same payoff angles fixed across every spot price, recomputing only the loaded distribution, produces a curve that tracks the true expected payoff closely near S0=2 and visibly flattens out at the sweep's edges — exactly the signature of an approximation calibrated at one point, degrading the further you move from it.
Getting the full sweep working meant rebuilding the entire circuit fresh for each spot price rather than reusing fixed angles — and getting that right required reconstructing the paper’s discretization scheme itself, which is not written down anywhere in the text as an explicit algorithm. It has to be reverse-engineered from the one worked example. The truncation range, it turns out, is built from the linear-space mean and standard deviation of the price distribution, not the more obvious log-space ones — a distinction that only became visible by testing both and comparing against the paper’s stated [1.21, ..., 2.81].
The practical consequence: away from S0=2, this reproduction's absolute dollar values don't match the paper's Fig. 16 numbers exactly. The shape does — a rising curve, real hardware measurements tracking the exact simulated curve reasonably closely, mitigated results at least as close to exact as unmitigated ones.
