Fast Quantum State Discrimination with Nonlinear Positive Trace‐Preserving Channels

Abstract

Models of nonlinear quantum computation based on deterministic positive trace-preserving (PTP) channels and evolution equations are investigated. 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 𝝓 on bounded linear operators X, there is an associated normalized PTP channel 𝝓(X)∕tr[𝝓(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 four types, yielding three distinct forms of nonlinearity whose computational power are explored. 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, the authors 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.

Nonlinear and non-CP gates for Bloch vector amplification

“…Class (ii) leads to a restricted form of nonlinearity, where a diagonal nonlinear term is added to the master equation to conserve trace. This type of evolution equation extends a pure-state nonlinear Schrödinger equation first introduced by Gisin [18] in 1981, to mixed states [23,24,26,30,35]. Rembieliński and Caban [36] recently argued that this type of nonlinearity is causal (does not support superluminal signaling) and should not be excluded from a fundamental theory.…”

“…In this paper we examine Bloch vector amplification gates based on the following NINO model [23,24,26,30,35]:…”

“…Let us examine the equations of motion for the model (54), which is similar to a model investigated in [35]…”

“…, where f(X) is a continuous 1-parameter positive linear or nonlinear map satisfying [ ( )] X tr 0 f ¹ for all positive semidefinite (PSD) operators X. Normalized PTP channels fall into 4 classes, yielding 3 distinct forms of nonlinearity [35]:…”

Spontaneous Disentanglement and Thermalization

“…[ 3–5 ] This long‐standing problem has motivated some proposals for nonlinear extensions to quantum theory. [ 6–11 ] Moreover, processes giving rise to spontaneous collapse have been explored. [ 12–21 ] In some cases, however, nonlinear quantum dynamics may give rise to conflicts with well‐established physical principles, such as causality [ 22–27 ] and separability.…”