Ahn's ground-state algorithm computes the far end of a model's yield curve directly.

Start with the bond-pricing equation. In a one-factor short-rate model, the zero-coupon price satisfies a Feynman-Kac equation, with the short rate acting as a potential. A Lamperti transform makes the diffusion coefficient constant; a similarity transform removes the first-derivative term. The result is a Schrödinger operator H. Expanding the bond price in its eigenstates gives terms that decay like exp(-E_n τ). Only the lowest term survives at long maturities. The model's asymptotic yield therefore equals the ground-state energy E0, while the gap E1 - E0 determines how quickly the forward curve flattens. This spectral account of the term structure is established mathematical finance. The paper cites Hansen-Scheinkman and Linetsky. Ahn's contribution is to pass H to a variational quantum eigensolver.

The hardware work depends on a second idea. H is stoquastic, the paper's term for having no positive off-diagonal entries. Its ground-state amplitudes are strictly positive and can be read as the square roots of a probability distribution. This lets Ahn estimate the energy from bitstring counts alone, using E[p] = ⟨√p|H|√p⟩. It takes one measurement setting; the Pauli decomposition takes nine commuting groups. Any histogram supplies a valid state through √p, so E[p] must equal or exceed E0. Noise can only raise it. Extrapolation is the only step that can take the estimate below.

The models are Vasicek, Cox-Ingersoll-Ross and Black-Karasinski, the last of which has no closed form. In noiseless simulation, the six-qubit Vasicek ansatz uses 30 CNOTs and comes within 0.058 bp of the grid energy. The same six-qubit comparison yields +0.92 bp for CIR and +0.12 bp for Black-Karasinski. Hardware errors on ibm_cleveland range from 33 to 125 bp relative to the grid ground state, averaging 90 bp. These are binary-encoded four-qubit circuits with 16 grid points and 9 CNOTs. The results follow deconvolution and zero-noise extrapolation across ten runs in two sessions. Separately, Ahn calibrates Vasicek and CIR monthly to the Gürkaynak-Sack-Wright zero curve for 492 months, November 1985 to October 2026. E0 and the 30-year zero yield have a correlation of 0.98 over that period.

What does the 0.98 Treasury correlation measure?

Nothing tradable.

Both E0 and the 30-year yield come from the same 1-30 year curve. Median fitting errors are 7.5 bp for Vasicek and 8.3 bp for CIR. With a median error below 9 bp on the 1-30 year yields, a fitted model will place its asymptote near the 30-year yield. The paper supplies the reason for the residual offset: the 30-year yield "still contains the 1/τ term" in the spectral expansion and has yet to reach the asymptote. Ahn explicitly describes the correlation as something that "reflects the in-sample consistency of the model-implied long-term yield with the calibrated curves; it is not an out-of-sample forecast of future yields."

The level depends on calibration choices too. An unconstrained fit pushes κ toward zero in about 16% of months. The Vasicek formula θ - σ²/(2κ²) loses its meaning there, prompting a lower bound of 0.1 on κ. Changing the bound to 0.05 or 0.2 puts the correlation between 0.94 and 0.99, with a mean offset between 38 and 50 bp. The monthly offset from the 30-year yield is 46±38 bp for Vasicek. Its standard deviation is nearly as large as its mean.

Ahn uses two other measures to assess model uncertainty on a given date. At most of eight selected dates, the three models' E0 estimates agree within 50 bp. And the κ bound moves the mean offset across 38 to 50 bp. In Ahn's words, the hardware accuracy "is of the same order as the model and calibration uncertainties found here."

Most of the correction comes from the WKB tail set

Session 1 shows how the correction works. After readout mitigation alone, errors remain at 566 to 1,139 bp. The physical diagnosis is convincing: gate noise behaves like bit flips in the most significant qubits. For Vasicek, the fitted rates are 1.65% and 1.25%. Flipping the top qubit moves weight from the central peak at grid points 7 and 8 to the edges at 15 and 0. The potential is highest at those edges.

Ahn estimates the flip rates by requiring the corrected distribution to vanish in a tail set. A WKB criterion identifies those tails, using a 10^-2 threshold fixed before the hardware data were analyzed. Deconvolution followed by linear zero-noise extrapolation (ZNE) reduces the errors to 33 to 115 bp. CNOT folding supplies the noise levels at λ = 1 and 3.

The paper's own ablation gives the tail set most of the credit. Across four-qubit runs in both sessions, simply zeroing the WKB tail bins and renormalizing produces errors of 33 to 149 bp at λ = 1. Deconvolution gives 56 to 144 bp. For session 1 alone, truncation gives 39 to 149 bp. The six-qubit result separates the methods sharply: after extrapolation, truncation has an error of 77 bp and deconvolution 518 bp. The abstract acknowledges this: "An ablation shows that this WKB tail set carries most of the correction." Its next sentence nevertheless presents deconvolution plus extrapolation as the result, at 33 to 125 bp.

Ahn gives several reasons for keeping deconvolution in the headline. The noise model is checked on synthetic data, where it recovers the top two flip rates within 0.12 percentage points. Its fitted rates are comparable across runs and platforms. It also returns removed weight to the bins from which it leaked. And its extrapolated values remain above E0, whereas extrapolated truncation drops below E0 for Black-Karasinski.

Most of that case concerns diagnosis. Remaining above E0 is an accuracy property, and it matters given the Black-Karasinski undershoot from extrapolated truncation. Even so, truncation wins by 441 bp at six qubits. For accuracy, the useful transferable ingredient is the WKB prior identifying where the true distribution is nearly zero, with true weight below 3.3×10^-4. The threshold matters. At four qubits, setting it to 10^-1 increases deconvolution errors to 87 to 381 bp; at six qubits the error reaches 1,137 bp. At 10^-3, the CIR tail set is empty.

The cross-platform experiment lends support to the noise explanation. On IonQ's trapped-ion processor Forte Enterprise, the Vasicek error drops from 1,021±117 bp to 254±53 bp in the 5,000-shot run. This uses λ = 1 without ZNE. The comparable IBM errors, likewise at λ = 1, fall from 704 to 144 bp. IonQ's fitted flip rates are 2.2% and 1.4%, against 1.7% and 1.3% on IBM. Ahn declines to rank the machines: "A difference of two standard deviations at these sample sizes is not sufficient to rank the two processors."

The grid consumes as much accuracy as the hardware

All the hardware errors above use the same grid's ground state as their benchmark: 16 points at four qubits. Against the continuum, the 16-point grid is off by -42.9 bp for Vasicek and -22.5 bp for CIR. Black-Karasinski has no closed form; its grid misses the reference value by -64.9 bp. Ahn identifies this as a limitation.

In the Treasury-calibrated session, readout mitigation leaves errors against the grid of 86 to 657 bp. Deconvolution and ZNE bring them down to 23 to 67 bp. The hardware estimates are 6.37%, 6.03% and 6.18%, compared with continuum values of 5.90 to 5.91% and a 30-year zero yield of 5.46%. Grid and hardware errors partly offset each other here. That leaves the estimates 12 to 45 bp from the continuum values, a cancellation Ahn declines to rely on.

The headline tenfold improvement over Pauli-measurement VQE compares 791 to 1,129 bp with 69 to 111 bp. Those Pauli results use TREX, IBM's built-in readout mitigation, plus ZNE. The √p approach takes 1.5 minutes of QPU time per model, including folding and readout calibration. The two Estimator runs take 3.6. As the paper discloses, the methods use different mitigation and shot budgets: 2×10^4 shots per group for the Estimator, against 10^5 shots per folding factor for √p. Its answer is that the Estimator's statistical errors, 55 to 475 bp, are far smaller than its bias. On that account, sampling does not explain the difference.

Ahn also makes the cleaner comparison. With readout mitigation alone, √p has errors of 663 to 1,124 bp and is already closer to E0 than Pauli with TREX, at 1,290 to 1,525 bp. The readout-only √p error is 25 to 57% lower per model. It uses one measurement setting against nine.

We ran no hardware tests. We have no access to IBM or IonQ machines. A classical eigensolver or simulated VQE could check the noiseless benchmarks, while leaving the mitigation and platform comparison untested. The paper acknowledges the economics of these one-factor models: "a classical eigensolver on a fine grid is inexpensive, and we use it as the reference." Ahn is equally explicit about the hardware: "The hardware results demonstrate the method; they do not claim a computational advantage."

Valuation at the long end

The application is valuation. Ahn proposes E0 as a model-consistent anchor for the ultimate forward rate toward which insurance regulators extrapolate beyond the last liquid maturity. The paper makes no comparison between E0 and a regulatory ultimate forward rate, leaving that use untested. A median Vasicek gap of 0.23 per year, or 0.25 for CIR, implies roughly four years for the forward curve to flatten.

We found no test of E0 against subsequent yield changes. A Treasury futures position on the spread between E0 and the 30-year yield is a trade we raise ourselves; the paper never proposes it. For Vasicek, that spread is 46±38 bp in-sample. Without a test, the trade has no evidence behind it. An out-of-sample result showing that the spread predicts long-end moves would change this view.