Optimal state discrimination and unstructured search in nonlinear quantum mechanics

Abstract

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. arXiv:1507.06334v2 [quant-ph]

Graph comparison via nonlinear quantum search

“…The nonlinearity of the system is introduced by the operator K. Nonlinear quantum search was first proposed by Abrams and Lloyd [7]. More recently, Childs and Young [8] developed a continuous-time search based upon the same properties of nonlinear quantum mechanics, using Eq. (23) as their specific nonlinear Schödinger equation.…”

“…• Foreword: This section discloses how to use the nonlinear evolution procedure from [8] in order to send the 0th and sth candidate states to the Bloch sphere poles |0 and |1 , respectively. The reader might note that the linear quantum circuit in Fig.…”

“…• In [8], candidate states are distributed along a small curve on the Bloch sphere. This curve is not an arc.…”

“…(23), let H = 0 for the time being. [8] show that the qubit's evolution across the sphere simplifies to d t (x, y, z) = gz(−y, x, 0) .…”

“…Our algorithm makes use of a two-part quantum dynamic process: in the first part, we encode information crucial for the comparison of the two graphs in a single qubit. In the second, we call upon techniques in nonlinear quantum computing [7,8] to achieve the speed-up necessary to extract this information efficiently. We use the number of edges in the maximum common edge subgraph as a measure of graph similarity, where the maximum common edge subgraph is defined as the subgraph common to both graphs having the maximal number of edges.…”