# Sensitivity of quantum speedup by quantum annealing to a noisy oracle

## TL;DR

This paper investigates how different noise models affect the robustness of quantum speedup in the glued-trees problem, revealing that exponential speedup is fragile under noise, but polynomial speedup can still be achieved.

## Contribution

It introduces phenomenological noise models and analyzes their impact on quantum and classical algorithms for the glued-trees problem, highlighting the importance of noise modeling choices.

## Key findings

- Exponential quantum speedup is retained under long-range noise models.
- Long-range noise can make classical algorithms appear exponentially faster.
- Short-range noise destroys exponential speedup but preserves polynomial speedup.

## Abstract

The glued-trees problem is the only example known to date for which quantum annealing provides an exponential speedup, albeit by partly using excited state evolution, in an oracular setting. How robust is this speedup to noise on the oracle? To answer this, we construct phenomenological short-range and long-range noise models, and noise models that break or preserve the reflection symmetry of the spectrum. We show that under the long-range noise models an exponential quantum speedup is retained. However, we argue that a classical algorithm with an equivalent long-range noise model also exhibits an exponential speedup over the noiseless model. In the quantum setting the long-range noise is able to lift the spectral gap of the problem so that the evolution changes from diabatic to adiabatic. In the classical setting, long-range noise creates a significant probability of the walker landing directly on the EXIT vertex. Under short-range noise the exponential speedup is lost, but a polynomial quantum speedup is retained for sufficiently weak noise. In contrast to noise range, we find that breaking of spectral symmetry by the noise has no significant impact on the performance of the noisy algorithms. Our results about the long-range models highlight that care must be taken in selecting phenomenological noise models so as not to change the nature of the computational problem. We conclude from the short-range noise model results that the exponential speedup in the glued-trees problem is not robust to noise, but a polynomial quantum speedup is still possible.