Degradation of Grover’s search under collective phase flips in queries to the oracle

Abstract

We address the case, when querying to the oracle in Grover's algorithm is exposed to noise including phase distortions. The oracle-box wires can be altered by an opposite party that tries to prevent correct receiving data from the oracle. This situation reflects an experienced truth that any access to prophetic knowledge cannot be common and direct. To study the problem, we introduce a simple model of collective phase distortions on the base of phase damping channel. In the model used, the success probability is not altered via the oracle-box wires per se. Phase distortions of the considered type can hardly be detected via any one-time query to the oracle. However, the success probability is significantly changed, when such errors are introduced as an intermediate step into the Grover iteration. We investigate the success probability with respect to variations the parameter that characterizes the amount of phase errors. It turns out that the success probability is decreased significantly even if the error amount is not very high. Moreover, this probability quickly reduces to the value of one half, which corresponds to the completely mixed state. We also study trade-off relations between quantum coherence and the success probability in the presence of noise of the considered type.

Grover search under localized dephasing

“…This case is studied also in Ref. [16], where exact results are presented. The reference contains some slight differences in the action of the noise, but it is possible to make a direct comparison with the results here.…”

Quantum search degeneration under amplitude noise in queries to the oracle

“…This approach allows us to formulate results in a closed analytic form. Also, our scenario does not mean that distortions act in all qubits simultaneously [16].…”

“…Using simplified model of collective amplitude errors, we address one of such scenarios. In a certain sense, the treatment of [16] is reformulated with another kind of noise. Another approach to quantum search under localized dephasing was considered [17].…”

“…Let us consider a model with amplitude damping. Following [16], we restrict a consideration to density matrices effectively two-dimensional with respect to the basis |w , |m . Such matrices can be represented via the Bloch vector r = (r x , r y , r z ), so that…”

“…Let us proceed to the case of noise in the oracle-box wires [16]. Each iteration acts on density matrices of the register according to the formula…”