⚛️ quantum physics

Long-term yields of short-rate models as Feynman-Kac ground states: a variational quantum eigensolver with single-setting energy estimation on two NISQ platforms

This paper demonstrates that the long-term yields of short-rate models can be computed as ground-state energies using a variational quantum eigensolver with a single-setting energy estimator, achieving significantly higher accuracy and lower error on NISQ hardware compared to traditional Pauli-measurement approaches.

Original author: Doyeol "David" Ahn (Department of Electrical,Computer Engineering, University of Seoul, Seoul 02504, Republic of Korea,,Singularity Quantum Inc., Newport Beach, CA 92660, USA)

Published 2026-11-03

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CC BY 4.0

Original author: Doyeol "David" Ahn (Department of Electrical,Computer Engineering, University of Seoul, Seoul 02504, Republic of Korea,,Singularity Quantum Inc., Newport Beach, CA 92660, USA)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of finance, predicting the future value of money is a constant challenge. Banks and investors rely on mathematical models to estimate interest rates, which determine the cost of borrowing and the return on savings. These models often treat the interest rate as a single number that wanders up and down over time, influenced by random market forces. To make sense of this wandering, mathematicians use equations that describe how the probability of different interest rates changes as time passes. For decades, these equations have been solved using powerful classical computers, but as financial models become more complex, they demand more computing power than even the fastest supercomputers can easily provide. This has led researchers to look toward quantum computers, machines that use the strange laws of quantum physics to process information in ways classical machines cannot. However, current quantum computers are still in their early stages, prone to errors and limited in size, making it difficult to trust their results for real-world financial decisions.

A team of researchers has now demonstrated a new way to use these early quantum machines to solve a specific, long-standing financial question: what will interest rates settle at in the very long run? Instead of trying to simulate the entire history of the market, they focused on the "ground state" of the system, a concept borrowed from physics that represents the most stable, lowest-energy configuration of a system. By translating the financial problem into a language that quantum computers understand, they showed that the long-term interest rate is essentially the energy of this stable state. The researchers tested this idea on three different financial models, including one that had never been solved with a closed-form formula before. They ran their calculations on two different types of quantum hardware—one using superconducting circuits and another using trapped ions—and found that their method could estimate the long-term rate with remarkable accuracy, far outperforming standard quantum techniques.

The core of this work lies in a clever transformation of a standard financial equation. Usually, calculating the price of a bond involves tracking how interest rates change over time, a process that requires simulating a path through many possible futures. The researchers realized that for the specific question of the long-term yield, this complex time-evolution could be replaced by a simpler static problem. They transformed the financial equation into a form that looks like a Schrödinger equation, a fundamental equation in quantum mechanics that describes how particles behave. In this new form, the interest rate acts like a landscape or a potential field, and the long-term yield corresponds to the lowest point in that landscape, known as the ground-state energy. This shift is crucial because finding the lowest energy state is a task that quantum computers are naturally designed to handle, whereas simulating time evolution is much harder for them.

To make this work on real, imperfect hardware, the team had to overcome the noise that plagues current quantum devices. Quantum bits, or qubits, are fragile and easily disturbed by their environment, leading to errors in the calculation. The researchers discovered that on the superconducting processor they used, the most common errors were "bit flips," where a qubit accidentally switches from a zero to a one or vice versa. These flips tended to push the calculation toward the edges of the possible range, creating a bias that skewed the results. To fix this, they developed a statistical method to identify and remove these errors. They used a principle from physics to predict where the true answer should be zero and then mathematically "deconvolved" the noisy data to pull the result back toward the center. This technique allowed them to correct for the hardware's imperfections without needing to know the exact answer beforehand.

The team tested their approach on three models: the Vasicek model, the Cox–Ingersoll–Ross model, and the Black–Karasinski model. The first two have known solutions, which served as a check to ensure their method was working correctly. The third model, Black–Karasinski, is more complex and does not have a known simple formula, making it a true test of the method's power. They calibrated these models to real historical data from the U.S. Treasury market, spanning from 1985 to 2026. In simulations without noise, their six-qubit quantum setup reproduced the known answers with an error of less than one hundredth of a basis point, a unit used to measure tiny differences in interest rates. When they moved to actual hardware, the results remained impressively close. On the superconducting processor, their four-qubit circuits estimated the long-term yield with an error ranging from 33 to 125 basis points. On the trapped-ion processor, the error was reduced from over 1,000 basis points to about 250 basis points using their correction method.

Perhaps the most significant finding was the efficiency of their approach compared to standard methods. Typically, to measure the energy of a quantum system, one must measure many different combinations of properties, requiring nine separate measurement settings for these models. The researchers' method, however, relied on a single measurement setting because the mathematical structure of their problem guaranteed that the answer could be found by simply counting how often the system landed in each state. This single-setting approach not only cut the measurement time by a factor of nine but also reduced the final error by a factor of ten compared to the standard method. This suggests that for certain types of financial problems, quantum computers can be useful even in their current, noisy state, provided the problem is framed correctly and the errors are handled with the right statistical tools.

The researchers also demonstrated that their method works across different types of quantum hardware. The fact that the same error patterns and correction techniques worked on both the superconducting and trapped-ion machines suggests that the bias they identified is a fundamental feature of the algorithm and the encoding they used, rather than a quirk of a specific machine. Furthermore, they showed that the method could be trained directly on the quantum device itself. By adjusting the parameters of the quantum circuit while it was running on the hardware, they improved the accuracy of the result from an initial error of nearly 5,000 basis points down to just 130 basis points. This "on-device" training proves that the quantum computer can learn to find the best answer even when starting from a poor guess, a vital capability for practical applications.

Ultimately, this work connects the abstract world of quantum algorithms with the concrete reality of financial markets. By calibrating their models to the U.S. Treasury curve from October 2026, the researchers showed that their quantum hardware estimates agreed with the theoretical grid values within 23 to 67 basis points. This completes a full chain from market data to a hardware-based estimate, demonstrating that the long-term yield of a short-rate model can indeed be computed as a ground-state energy. While the current results are limited to one-factor models, which are simpler than the multi-factor models used by major banks, the success of this approach opens the door for more complex applications. The study suggests that with better error correction and more qubits, quantum computers could eventually tackle the most difficult problems in interest-rate modeling, offering a new tool for understanding the long-term behavior of the global economy.

Technical Summary: Long-term Yields of Short-Rate Models as Feynman–Kac Ground States

1. Problem Statement

The paper addresses the challenge of computing the long-term yield of short-rate interest rate models using quantum algorithms. While most quantum finance proposals focus on option pricing via amplitude estimation or variational imaginary-time evolution, the Variational Quantum Eigensolver (VQE) has seen limited application in derivative pricing because the ground state of standard pricing operators (like Black–Scholes) does not directly encode option prices. Furthermore, existing variational imaginary-time evolution methods require complex measurements (ancilla-controlled Hadamard tests) for metric tensors and energy gradients at every time step.

The authors identify a specific financial quantity—the long-term yield of a short-rate model—that corresponds exactly to the ground-state energy of a Hermitian operator. This allows the problem to be framed as a ground-state search, solvable directly by VQE without intermediate time-evolution steps.

2. Methodology

Theoretical Formulation:
The authors derive a Schrödinger operator $H$ from the Feynman–Kac bond-pricing equation. Starting with a one-factor short-rate model $dr_t = b(r_t)dt + s(r_t)dW_t$ , they apply:

Lamperti Transform: To convert the state-dependent diffusion coefficient into a constant.
Similarity Transform: To eliminate the first-derivative term, resulting in a Hermitian Hamiltonian $H = -\frac{s_\infty^2}{2}\frac{\partial^2}{\partial y^2} + V(y)$ .
The long-term yield $y_\infty$ is shown to be the ground-state energy $E_0$ of this Hamiltonian. The models analyzed are Vasicek, Cox–Ingersoll–Ross (CIR), and Black–Karasinski.

Stoquastic Property and Single-Setting Estimator:
A critical feature of the derived Hamiltonian is that it is stoquastic (all off-diagonal elements are non-positive). By the Perron–Frobenius theorem, the ground state $\psi_0$ can be chosen to have strictly positive components ( $\psi_0 = \sqrt{p_0}$ ).

Implication: The energy can be estimated using only computational-basis samples (bitstrings) from a single measurement setting, rather than decomposing the Hamiltonian into Pauli strings.
Estimator: The energy is calculated as $E[p] = \langle \sqrt{p} | H | \sqrt{p} \rangle$ , where $p$ is the measured probability distribution. This estimator is a variational upper bound to the true ground-state energy.

Error Mitigation Strategy:
The authors diagnose that gate noise on superconducting hardware manifests primarily as bit flips of high-order qubits, which shift probability mass to the "tails" of the grid (high-potential edges), significantly inflating the energy estimate.

Tail-Constrained Bit-Flip Deconvolution: They propose a method to correct this noise. A "tail set" $T$ is defined using a WKB criterion (based on the potential $V(y)$ and an energy estimate) to identify bins where the true ground-state weight is negligible.
Procedure: The measured distribution is deconvolved under the assumption of independent bit flips, constrained such that the resulting distribution vanishes on the tail set $T$ . This is combined with readout mitigation and Zero-Noise Extrapolation (ZNE) via CNOT folding.

Experimental Setup:

Ansatz: A shallow real-amplitude ansatz with $L$ layers of single-qubit rotations and CNOT ladders.
Hardware: Experiments were conducted on the IBM superconducting processor ibm_cleveland and the IonQ trapped-ion processor Forte Enterprise.
Calibration: Models were calibrated to the U.S. Treasury zero curve (1985–2026).

3. Key Contributions

Formulation: The paper formulates the long-term yield of one-factor short-rate models as the ground-state energy of a qubit Hamiltonian, providing explicit Lamperti and similarity transforms for Vasicek, CIR, and Black–Karasinski models.
Single-Setting Estimator: It introduces a single-measurement-setting energy estimator for stoquastic Hamiltonians, leveraging the positivity of the ground state to avoid the overhead of Pauli-string decomposition.
Noise Characterization and Correction: The authors identify that hardware noise in binary-encoded circuits acts predominantly as bit flips of high-order qubits. They propose a tail-constrained deconvolution method that corrects this bias without requiring the exact solution, using a WKB criterion to define the tail set.
Hardware Demonstration: The method is demonstrated on two distinct quantum platforms (IBM and IonQ), showing consistent bias structures and error reduction across platforms.
End-to-End Pipeline: The paper demonstrates a complete chain from market data (U.S. Treasury curves) to a hardware-based estimate of the long-term yield.

4. Results

Classical Simulation:

In noiseless simulations, a six-qubit real-amplitude ansatz reproduces the grid ground-state energy for the Vasicek model within 0.06 basis points (bp).
The grid error converges as $h^2$ (where $h$ is the grid spacing).
For the Black–Karasinski model (which has no closed-form solution), the method provides a numerical ground-state energy consistent with fine-grid benchmarks.

Hardware Experiments (IBM ibm_cleveland):

Four-Qubit Circuits: Across ten runs in two sessions for all three models, the error of the hardware estimates (after readout mitigation, tail-constrained deconvolution, and ZNE) ranged from 33 to 125 bp relative to the grid ground-state energies.
Comparison with Standard VQE: Under identical conditions, a standard Pauli-measurement VQE with readout mitigation and ZNE yielded errors of 791–1,129 bp. The single-setting estimator achieved errors roughly ten times smaller with only one measurement setting (vs. nine for Pauli grouping).
On-Device Training: An on-device VQE trained with SPSA reduced the noiseless energy error from 4,939 bp to 130 bp in 150 iterations.

Cross-Platform Validation (IonQ):

On the IonQ Forte Enterprise processor, the same analysis pipeline reduced the Vasicek circuit error from ~1,021 bp to 254 bp (at $\lambda=1$ ), demonstrating that the bias structure (high-order qubit bit flips) is not specific to superconducting hardware.

Market Calibration:

When parameters were calibrated to the U.S. Treasury curve of October 2026, the hardware estimates agreed with the grid ground-state energies within 23–67 bp.
The long-term yield derived from the models followed the 30-year market yield with an in-sample correlation of 0.98.

5. Significance and Claims

The paper claims that the long-term yield is a natural and effective target for ground-state quantum algorithms because:

It is exactly a ground-state energy, avoiding the complexity of time-evolution.
The associated Hamiltonian is stoquastic, enabling a computationally cheaper single-setting measurement strategy.
The variational nature of the estimator ( $E[p] \ge E_0$ ) is preserved even after readout mitigation and deconvolution, as these operations return valid probability distributions.

The authors are modest regarding computational advantage, noting that for one-factor models, classical solvers are already inexpensive. The significance lies in demonstrating the feasibility of the method on current NISQ hardware, establishing a robust error-mitigation pipeline for stoquastic problems, and validating the approach across different hardware architectures. The work serves as a proof-of-concept for extending these techniques to multi-factor models and non-affine systems where classical solutions are intractable.