Quantifying Uncertainty in Quantum Algorithms Using Possibility Theory

This case study is based on an academic research project conducted at the University of Stuttgart in 2025. The study introduces a computationally lightweight method to quantify uncertainty in quantum algorithms using Possibility Theory instead of traditional probabilistic sampling.

1. Executive Summary

Coherent noise is one of the main obstacles to reliable quantum computation. Standard probabilistic simulations require extensive sampling and scale poorly with circuit depth. By transforming empirical probability distributions into possibility distributions, this method evaluates the plausibility of deviations under noise and enables fast and interpretable robustness comparisons across quantum algorithm implementations.

The approach was applied to two real three qubit Quantum Fourier Transform circuits from IBM and Rigetti. The analysis revealed robustness differences that are not captured by gate count alone.

2. Objective

Quantum algorithms are highly sensitive to coherent gate errors, but existing robustness metrics are computationally expensive and often difficult to interpret.

The objective of this research was to develop a general purpose and low overhead uncertainty quantification method capable of:

• Modeling coherent gate errors using interval based perturbations
• Transforming empirical probability distributions into possibility distributions
• Computing a possibility based robustness index for algorithm outputs
• Comparing implementations of the same algorithm across different hardware platforms

The goal was not to replace full probabilistic simulation, but to provide a tactical, scalable and interpretable robustness metric for quantum algorithm evaluation.

3. Methodology

3.1 Noise Modeling

Gate errors were modeled within the interval ε ∈ {−15%, 0%, +15%} focusing on extreme deviations rather than full error sampling.

This reduces computational cost while capturing the worst case behavior relevant for robustness analysis.

3.2 Probability to Possibility Transformation

For each quantum operation:

• Empirical probability distributions were measured
• These distributions were transformed into possibility distributions using the standard Probability to Possibility transformation
• The resulting distributions express plausibility rather than frequency

This enables uncertainty quantification without Monte Carlo style sampling.

3.3 Robustness Index Computation

A possibility index ΠR was computed for each algorithm output.

This index provides:

• A scalar robustness score
• Direct comparability across implementations
• Independence from sampling density

The method was applied to two real QFT implementations:

• IBM three qubit QFT
• Rigetti three qubit QFT

with Rz gates treated as the noise source.

All simulations in this study were implemented in Python using the Qiskit framework, as part of an academic research project carried out at the University of Stuttgart in 2025.

4. Results

4.1 Robustness Comparison

The analysis showed that the IBM implementation of the three qubit QFT achieved a possibility index of 0.4633, while the Rigetti implementation reached 0.4624. Although the numerical difference is small, the interpretation is significant: the Rigetti circuit demonstrated slightly higher robustness, even though it contains a larger number of noisy Rz gates.

This result highlights an important insight. Robustness in quantum algorithms does not correlate directly with gate count. A circuit with more gates can still be more stable under coherent noise, depending on how those gates interact with the structure of the algorithm and the underlying hardware.

4.2 Independent Validation

The robustness ranking obtained through the possibility index was independently confirmed by results from Berberich in 2023. In that work, the author computed Lipschitz bounds for the same two QFT implementations. The IBM circuit presented a Lipschitz constant of 117.95, while the Rigetti circuit reached 106.79.

A lower Lipschitz constant indicates that the algorithm is less sensitive to perturbations in its inputs, which corresponds to higher robustness. This means that the Rigetti implementation, despite using more noisy Rz gates, is mathematically proven to be the more stable circuit.

The fact that two completely different methods — one based on possibility theory and one based on Lipschitz continuity — converge to the same conclusion reinforces the reliability of the analysis and validates the approach developed in this research.

5. Key Insights

• First practical application of Possibility Theory to quantum algorithm comparison
• Computationally lightweight and scalable with circuit structure
• Interpretable robustness ranking across hardware platforms
• Validated through independent mathematical analysis
• Reveals non intuitive behavior where more gates do not necessarily reduce robustness
• This demonstrates that possibility based metrics can uncover robustness characteristics that probabilistic methods may obscure.

6. Conclusion

This research, conducted at the University of Stuttgart in 2025, demonstrates that Possibility Theory provides an efficient and interpretable framework for quantifying uncertainty in quantum algorithms.

By combining interval based noise modeling, probability to possibility transformation and a scalar robustness index, the method enables meaningful comparison of algorithm implementations without the computational burden of full probabilistic simulation.

The value lies in redefining how robustness can be evaluated: faster, clearer and more scalable, especially for early stage quantum hardware.