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 . For every normalizable linear or nonlinear positive map ϕ on bounded linear operators X , there is an associated normalized PTP channel . 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.
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 . For every normalizable linear or nonlinear positive map ϕ on bounded linear operators X , there is an associated normalized PTP channel . 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.
Completely positive trace-preserving maps S, also known as quantum channels, arise in quantum physics as a description of how the density operator ρ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space H that ρ acts on is finite-dimensional, then every S must have a fixed point, i.e., a density operator ρ0 with S(ρ0)=ρ0. In infinite dimension, S need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that S leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal.
In quantum technologies, quantum channels are essential for transmitting quantum states. Quantum channels usually introduce noise, thereby reducing the information in a state. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. Precisely, given a pair of quantum channels Λ1,Λ2 and an arbitrary unknown input state, we show that the output state of the quantum channel Λ2 from the pair can be uniquely identified from the output statistics of the other channel Λ1 from the pair, using some POVM, iff Λ2 can be obtained from Λ1 by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, Λ2 may not always be obtained from Λ1 through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing Λ2 given that Λ1 has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.
There is great interest in generating and controlling entanglement in Bose-Einstein condensates and similar ensembles for use in quantum computation, simulation, and sensing. One class of entangled states useful for enhanced metrology are spin-squeezed states of N two-level atoms. After preparing a spin coherent state of width 1 / N centered at coordinates ( θ , ϕ ) on the Bloch sphere, atomic interactions generate a nonlinear evolution that shears the state's probability density, stretching it to an ellipse and causing squeezing in a direction perpendicular to the major axis. Here we consider the same setup but in the N → ∞ limit . This shrinks the initial coherent state to zero area. Large N also suppresses two-particle entanglement and squeezing, as required by a monogamy bound. The torsion (1-axis twist) is still present, however, and the center of the large N coherent state evolves as a qubit governed by a two-state Gross-Pitaevskii equation. The resulting nonlinearity is known to be a powerful resource in quantum computation. It can be used to implement single-input quantum state discrimination, an impossibility within linear one-particle quantum mechanics. We obtain a solution to the discrimination problem in terms of a Viviani curve on the Bloch sphere. We also consider an open-system variant containing both Bloch sphere torsion and dissipation. In this case it should be possible to generate two basins of attraction within the Bloch ball, having a shared boundary that can be used for a type of autonomous state discrimination. We explore these and other connections between spin squeezing in the large N limit and nonlinear quantum gates, and argue that a two-component condensate is a promising platform for realizing a nonlinear qubit.
There is growing interest in extensions of the standard model of gate-based quantum computation to include auxiliary degrees of freedom evolving according to a nonlinear Schrödinger equation. By reducing the
Boolean satisfiability problem SAT to quantum state discrimination, Abrams and Lloyd argued that the right type of nonlinearity can be used to solve NP and #P problems in polynomial time, at least in an
idealized noise-free limit. For practical implementation, however, we are restricted to simulated and emergent nonlinearities, such as that appearing in mean field models for ultracold atoms and similar
ensembles. A prominent example is the torsion model, which arises in two-component Bose-Einstein condensates and spin models with all-to-all Ising interaction. But torsion-based state discrimination appears to
fall short of solving SAT. Here we close this gap by constructing the filtered oracle of the Valiant-Vazirani theorem, providing a randomized polynomial-time reduction from SAT to UNIQUE SAT, a promise problem
where there is at most 1 satisfying assignment. In the noise-free limit, the UNIQUE SAT problem can be solved in polynomial time using torsion nonlinearity. Quantum Valiant-Vazirani reduction is no faster than
the efficient classical version, but a fault-tolerant implementation coupled to a nonlinear quantum coprocessor simulating torsion would enable polynomial time solution to NP (but not #P) problems.
Abstract The characterization of quantum noise is fundamental to quantum information processing, quantum control and reliable operation of near-term quantum devices. Conventional descriptions of quantum noise are based on completely positive trace-preserving (CPTP) maps acting on density operators. These models describe effective quantum dynamics after the underlying physical interactions have been represented as a quantum channel. However, in practical cases, imperfections such as calibration errors, control-field fluctuations, finite-resolution waveform generation, and analog hardware noise can act upon the probability amplitudes of a quantum state earlier. The effect of such perturbation in quantum state after normalization has largely been unexplored. In this work, we develop a rigorous mathematical analysis of the normalized additive amplitude noise in a finite-dimensional quantum system. Assuming a zero-mean complex perturbation, we derive a second-order closed form expression for the ensemble-averaged density operator. The analysis leads to state-dependent evolution of the quantum state which cannot be represented by a CPTP map. We further prove that in case of isotropic noise, the ensemble-averaged density operator becomes an effective depolarizing channel. For anisotropic perturbations, we derive the orientation-dependent Bloch-vector contraction which is determined by the covariance geometry of the additive noise. Closed-form expressions are further obtained for measurement probabilities, state fidelity, and ensemble purity, showing that fidelity degradation depends only on the component of the perturbation covariance orthogonal to the input state. Monte Carlo simulations closely match the theoretical analysis, especially for small noise regime. The proposed model establishes a connection between classical additive-noise models and effective quantum dynamics generated by normalized state-space perturbations.
From the point of view of communication theory, a quantum channel can be defined as a physical process mapping from initial to final state in the form of CPTP (Completely positive trace preserving) map. Quantum information theory provide a theoretical framework and some mathematical methods to accomplish this. In order to preserve information, we proposed a scheme for the protection of two-qubit state through various noise channel, where successive uses of noise channels are correlated. These successive uses of channels are called memory. Different types of noise channels with memory have been discussed in details. We compare the dynamics of entanglement of two-qubit state under different noise channel and it has been seen that one can preserve two-qubit steady state quantum correlation under noise channels with memory. The study of memory effect in quantum channel will emerge as an interesting phenomena to trace back information from the surrounding which is considered as a resource in quantum information task. However, we also define some mores measures of entanglement and their preservation. We hope that our new findings may help to study the two-qubit entanglement structure.