Signatures of quantum noise in the operation of Deutsch's algorithm

We use Deutsch's algorithm as a stand in for more complex quantum algorithms in order to determine how quantum properties of an environment manifest themselves in results that can be obtained on quantum computers. We model pure dephasing in two different ways; one keeps the full density matrix of the qubits and environments (quantum) while the other uses Kraus operators (classical). We find that a single run of the algorithm yields the same effect in both cases, but running the algorithm twice leads to stark differences. Taking correlations and interplay between different decoherence processes into account leads to a slowing of decoherence effects for balanced functions. For constant functions, the effect is much more pronounced, and there is a qualitative change in the dependence of measurement outcomes on decoherence. We present results obtained on one of the IBM Quantum processors, which fully reproduce the predicted effect regardless of the assumptions made in the derivation. We further illustrate the findings on NV center spin qubits, which show more complex behavior due to a small size of the environment.