Quantum Search with General Nonlinearities

Abstract

Evolution by the Gross-Pitaevskii equation, which describes Bose-Einstein condensates under certain conditions, solves the unstructured search problem more efficiently than does the Schrödinger equation, because it includes a cubic nonlinearity, proportional to $|\psi {|}^2\psi$. This is not the only nonlinearity of the form $f(|\psi {|}^2)\psi$ that arises in effective equations for the evolution of real quantum physical systems, however: The cubic-quintic nonlinear Schrödinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrödinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a "database" when there is a single correct answer may be easier than searching when there are multiple correct answers. In each of these cases the nonlinear equation is an effective approximation to a multi-particle Schrödinger equation, for search by which Grover's algorithm is optimal. Thus our results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically.

30 pages, 7 figures

References in corpus (2)

• Spatial search by quantum walk
• Nonlinear Quantum Search Using the Gross-Pitaevskii Equation

Cited by in corpus (20)

• Connectivity is a Poor Indicator of Fast Quantum Search
• Fast quantum state discrimination with nonlinear PTP channels
• Optimal state discrimination and unstructured search in nonlinear quantum mechanics
• On the dynamical nature of nonlinear coupling of logarithmic quantum wave equation, Everett-Hirschman entropy and temperature
• Attractor-repeller pair of topological zero-modes in a nonlinear quantum walk
• Quantum computation speedup limits from quantum metrological precision bounds
• Grover search and the no-signaling principle
• Irreconcilable Difference Between Quantum Walks and Adiabatic Quantum Computing
• Enhancing precision thermometry with nonlinear qubits
• Nonlinear flip-flop quantum walks through potential barriers
• Nonlinear and non-CP gates for Bloch vector amplification
• Unitary Evolutions Sourced By Interacting Quantum Memories: Closed Quantum Systems Directing Themselves Using Their State Histories
• Quantum Cellular Automaton Model for the Generalized Dirac Equation
• Constant-Time Quantum Search with a Many-Body Quantum System
• Quantum thermodynamics of Gross-Pitaevskii qubits
• Searching via nonlinear quantum walk on the 2D-grid
• Example Exact Solutions of the Time-independent Gross-Pitaevskii and Schrödinger Equations
• The universe as a nonlinear quantum simulation: Large $n$ limit of the central spin model
• Tomogram-based quantifiers of nonclassicality dynamics in Kerr and cubic media
• Graph comparison via nonlinear quantum search