Optimal state discrimination and unstructured search in nonlinear quantum mechanics

Nonlinear variants of quantum mechanics can solve tasks that are impossible in standard quantum theory, such as perfectly distinguishing nonorthogonal states. Here we derive the optimal protocol for distinguishing two states of a qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum mechanics that arises as an effective description of Bose-Einstein condensates. Using this protocol, we present an algorithm for unstructured search in the Gross-Pitaevskii model, obtaining an exponential improvement over a previous algorithm of Meyer and Wong. This result establishes a limitation on the effectiveness of the Gross-Pitaevskii approximation. More generally, we demonstrate similar behavior under a family of related nonlinearities, giving evidence that the ability to quickly discriminate nonorthogonal states and thereby solve unstructured search is a generic feature of nonlinear quantum mechanics.

Figure 1

(a) Orientation of states on the Bloch sphere with overlap $cos\frac{\alpha }{2}$. (b) The flow on the Bloch sphere induced by the Gross-Pitaevskii nonlinearity.

Figure 2

Optimally distinguishing two states of a qubit using the Gross-Pitaevskii nonlinearity $\kappa \left(x\right)=g{x}^2$. (a) Inner product between states as a function of time, as given by Eq. (22), for initial states with ${\alpha }_0=0.1$. (b) Time to distinguish states separated by an angle ${\alpha }_0$ on the Bloch sphere, as given by Eq. (23).

Figure 3

Rate of change of the overlap for the Gross-Pitaevskii and logarithmic nonlinearities. The solid curve is Eq. (33) with $g=1$; the dashed curve is Eq. (21) with $g=2$.

Figure 4

Hadamard test. The first register stores a qubit. The Hadamard gate $H$ acts as $H|0\rangle =\frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$ and $H|1\rangle =\frac{1}{\sqrt{2}}\left(\right|0\rangle -|1\rangle )$.