We show that it is impossible to obtain a quantum speedup for a faulty Hamiltonian oracle. The effect of dephasing noise to this continuous-time oracle model has first been investigated by Shenvi, Brown, and Whaley [Phys. Rev. A 68, 052313 (2003).]. The authors consider a faulty oracle described by a continuous-time master equation that acts as dephasing noise in the basis determined by the marked item. The analysis focuses on the implementation with a particular driving Hamiltonian. A universal lower bound for this oracle model, which rules out a better performance with a different driving Hamiltonian, has so far been lacking. Here, we derive an adversary-type lower bound which shows that the evolution time
T has to be at least in the order of
N, i.e., the size of the search space, when the error rate of the oracle is constant. This means that quadratic quantum speedup vanishes and the runtime assumes again the classical scaling. For the standard quantum oracle model this result was first proven by Regev and Schiff [in Automata, Languages and Programming, Lecture Notes in Computer Science Vol. 5125 (Springer, Berlin, 2008), pp. 773--781]. Here, we extend this result to the continuous-time setting.
This paper investigates the impact of noise in the quantum query model, a fundamental framework for quantum algorithms. We focus on the scenario where the oracle is subject to non-unitary (or irreversible) noise, specifically under the \textit{faulty oracle} model, where the oracle fails with a constant probability and acts as identity. Regev and Schiff (ICALP'08) showed that quantum advantage is lost for the search problem under this noise model. Our main result shows that every quantum query algorithm can be made robust in this noise model with a roughly quadratic blow-up in query complexity, thereby preserving quantum speedup for all problems where the quantum advantage is super-cubic. This is the first non-trivial robustification of quantum query algorithms against an oracle that is noisy.