Fast quantum state discrimination with nonlinear PTP channels

Abstract

We investigate models of nonlinear quantum computation based on deterministic positive trace-preserving (PTP) channels and evolution equations. The models are defined in any finite Hilbert space, but the main results are for dimension $N=2$. For every normalizable linear or nonlinear positive map $\varphi$ on bounded linear operators $X$, there is an associated normalized PTP channel $\varphi (X)/\mathrm{tr}[\varphi (X)]$. Normalized PTP channels include unitary mean field theories, such as the Gross-Pitaevskii equation for interacting bosons, as well as models of linear and nonlinear dissipation. They classify into 4 types, yielding 3 distinct forms of nonlinearity whose computational power we explore. In the qubit case these channels support Bloch ball torsion and other distortions studied previously, where it has been shown that such nonlinearity can be used to increase the separation between a pair of close qubit states, suggesting an exponential speedup for state discrimination. Building on this idea, we argue that this operation can be made robust to noise by using dissipation to induce a bifurcation to a novel phase where a pair of attracting fixed points create an intrinsically fault-tolerant nonlinear state discriminator.

References in corpus (45)

• Separability Criterion for Density Matrices
• Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard
• Quantum State Discrimination
• Quantum state discrimination and its applications
• Dividing Quantum Channels
• Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems
• Mixed-state evolution in the presence of gain and loss
• Physical accessibility of non-completely positive maps
• Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems?
• A general framework for complete positivity
• Quantum feedback with weak measurements
• Quantum Computational Complexity in the Presence of Closed Timelike Curves
• Comparison and unification of non-Hermitian and Lindblad approaches with applications to open quantum optical systems
• Nonlinearity without Superluminality
• Quantum nonlinear dynamics of continuously measured systems
• Localized closed timelike curves can perfectly distinguish quantum states
• Is the dynamics of open quantum systems always linear?
• Non-linear operations in quantum information theory
• Simulating nonlinear dynamics of collective spins via quantum measurement and feedback
• Assumptions that imply quantum dynamics is linear
• Nonlinear quantum state transformation of spin-1/2
• Complex chaos in the conditional dynamics of qubits
• On quantum error-correction by classical feedback in discrete time
• A Causal Framework for Non-Linear Quantum Mechanics
• Correlation experiments in nonlinear quantum mechanics
• Quantum Dynamics with Mean Field Interactions: a New Approach
• Optimal state discrimination and unstructured search in nonlinear quantum mechanics
• No-faster-than-light-signaling implies linear evolutions. A re-derivation
• Complete positivity of nonlinear evolution: A case study
• Quantum Search with General Nonlinearities
• Measurement induced chaos with entangled states
• Nonlinear Schroedinger equation and two-level atoms
• Nonlinear Quantum Search Using the Gross-Pitaevskii Equation
• Nonlinear extension of the quantum dynamical semigroup
• Exponential Sensitivity and its Cost in Quantum Physics
• Nonlinear evolution and signaling
• Quantum state matching of qubits via measurement-induced nonlinear transformations
• Quantum search with interacting Bose-Einstein condensates
• Stability of the Grabert master equation
• Instability in the Hartmann--Hahn double resonance
• Nonlinear Dynamics from Linear Quantum Evolutions
• What is tested when experiments test that quantum dynamics is linear
• Self-organized PT-symmetry of exciton-polariton condensate in a double-well potential
• Nonlinear quantum gates for a Bose-Einstein condensate
• Nonlinear speed-ups in ultracold quantum gases

Cited by in corpus (7)

• Quantum Dynamics with Stochastic Non-Hermitian Hamiltonians
• Quantum simulation of dissipation for Maxwell equations in dispersive media
• Nonlinear and non-CP gates for Bloch vector amplification
• Proposal for a Lorenz qubit
• Suppressing excitations in the nonlinear Landau-Zener model
• Quantum thermodynamics of Gross-Pitaevskii qubits
• The universe as a nonlinear quantum simulation: Large $n$ limit of the central spin model