Noise effects in the quantum search algorithm from the computational complexity point of view

We analyse the resilience of the quantum search algorithm in the presence of quantum noise modelled as trace preserving completely positive maps. We study the influence of noise on computational complexity of the quantum search algorithm. We show that only for small amounts of noise the quantum search algorithm is still more efficient than any classical algorithm.

💡 Research Summary

The paper investigates how various quantum noise channels affect Grover’s search algorithm from a computational‑complexity perspective. After a brief review of related work, the authors introduce the necessary formalism of quantum information theory, including density operators, partial traces, and completely positive trace‑preserving (CPTP) maps expressed in Kraus form. They then describe Grover’s algorithm in detail: the problem of locating a marked item in an unsorted database of size N = 2ⁿ, the initialization with a Hadamard transform, the oracle that flips the phase of the marked state, the diffusion operator that reflects about the uniform superposition, and the iteration count of approximately π/4 √N required to amplify the marked amplitude.

The core contribution is a systematic noise model. Six standard one‑qubit channels are considered—depolarizing, amplitude damping, phase damping, bit‑flip, phase‑flip, and bit‑phase‑flip—each parameterized by a noise strength α∈