Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size~$N$ and margin~$gamma$ (how well the given data are separated), the query complexity of the best-known quantum training
algorithm scales as either $( icefrac{sqrt{N}}{gamma^2})log( icefrac1{gamma^2)}$ or $ icefrac{N}{sqrt{gamma}}$, which is achieved by a hybrid of classical and quantum search. In this paper, we improve the version space quantum training method for perceptrons such that the query complexity of our algorithm scales as $sqrt{ icefrac{N}{gamma}}$. This is achieved by constructing an oracle for the perceptrons using quantum counting of the number of data elements that are correctly classified. We show that query complexity to construct such an oracle has a quadratic improvement over classical methods. Once such an oracle is constructed, bounded-error quantum search can be used to search over the hyperplane instances. The optimality of our algorithm is proven by reducing the evaluation of a two-level AND-OR tree (for which the query complexity lower bound is known) to a multi-criterion search. Our quantum training algorithm can be generalized to train more complex machine learning models such as neural networks, which are built on a large number of perceptrons.
Unsupervised machine learning is one of the main techniques employed in artificial intelligence. Quantum computers offer opportunities to speed up such machine learning techniques. Here, we introduce an algorithm for quantum assisted unsupervised dat
a clustering using the self-organizing feature map, a type of artificial neural network. We make a proof-of-concept realization of one of the central components on the IBM Q Experience and show that it allows us to reduce the number of calculations in a number of clusters. We compare the results with the classical algorithm on a toy example of unsupervised text clustering.
This is a pre-publication version of a forthcoming book on quantum atom optics. It is written as a senior undergraduate to junior graduate level textbook, assuming knowledge of basic quantum mechanics, and covers the basic principles of neutral atom
matter wave systems with an emphasis on quantum technology applications. The topics covered include: introduction to second quantization of many-body systems, Bose-Einstein condensation, the order parameter and Gross-Pitaevskii equation, spin dynamics of atoms, spinor Bose-Einstein condensates, atom diffraction, atomic interferometry beyond the standard limit, quantum simulation, squeezing and entanglement with atomic ensembles, quantum information with atomic ensembles. This book would suit students who wish to obtain the necessary skills for working with neutral atom many-body atomic systems, or could be used as a text for an undergraduate or graduate level course (exercises are included throughout). This is a near-final draft of the book, but inevitably errors may be present. If any errors are found, we welcome you to contact us and it will be corrected before publication. (TB: tim.byrnes[at]nyu.edu, EI: ebube[at]nyu.edu)
We study the dynamics of the massive Schwinger model on a lattice using exact diagonalization. When periodic boundary conditions are imposed, analytic arguments indicate that a non-zero electric flux in the initial state can unwind and decrease to a
minimum value equal to minus its initial value, due to the effects of a pair of charges that repeatedly traverse the spatial circle. Our numerical results support the existence of this flux unwinding phenomenon, both for initial states containing a charged pair inserted by hand, and when the charges are produced by Schwinger pair production. We also study boundary conditions where charges are confined to an interval and flux unwinding cannot occur, and the massless limit, where our results agree with the predictions of the bosonized description of the Schwinger model.
We develop a general framework for applying the Kelly criterion to stock markets. By supplying an arbitrary probability distribution modeling the future price movement of a set of stocks, the Kelly fraction for investing each stock can be calculated
by inverting a matrix involving only first and second moments. The framework works for one or a portfolio of stocks and the Kelly fractions can be efficiently calculated. For a simple model of geometric Brownian motion of a single stock we show that our calculated Kelly fraction agrees with existing results. We demonstrate that the Kelly fractions can be calculated easily for other types of probabilities such as the Gaussian distribution and correlated multivariate assets.
Exciton-polaritons are a coherent electron-hole-photon (e-h-p) system where condensation has been observed in semiconductor microcavities. In contrast to equilibrium Bose-Einstein condensation (BEC) for long lifetime systems, polariton condensates ha
ve a dynamical nonequilibrium feature owing to the similar physical structure that they have to semiconductor lasers. One of the distinguishing features of a condensate to a laser is the presence of strong coupling between the matter and photon fields. Irrespective of its equilibrium or nonequilibrium nature, exciton-polariton have been observed to maintain strong coupling. We show that by investigating high density regime of exciton-polariton condensates, the negative branch directly observed in photoluminescence. This is evidence that the present e-h-p system is still in the strong coupling regime, contrary to past results where the system reduced to standard lasing at high density.
We analyze a similar scheme for producing light-mediated entanglement between atomic ensembles, as first realized by Julsgaard, Kozhekin and Polzik [Nature {bf 413}, 400 (2001)]. In the standard approach to modeling the scheme, a Holstein-Primakoff a
pproximation is made, where the atomic ensembles are treated as bosonic modes, and is only valid for short interaction times. In this paper, we solve the time evolution without this approximation, which extends the region of validity of the interaction time. For short entangling times, we find this produces a state with similar characteristics as a two-mode squeezed state, in agreement with standard predictions. For long entangling times, the state evolves into a non-Gaussian form, and the two-mode squeezed state characteristics start to diminish. This is attributed to more exotic types of entangled states being generated. We characterize the states by examining the Fock state probability distributions, Husimi $Q$ distributions, and non-local entanglement between the ensembles. We compare and connect several quantities obtained using the Holstein-Primakoff approach and our exact time evolution methods.
We propose the use of stimulated Raman adiabatic passage (STIRAP) to offer a fast high fidelity method of performing SU(2) rotations on spinor Bose Einstein condensates (BEC). Past demonstrations of BEC optical control suffer from difficulties arisin
g from collective enhancement of spontaneous emission and inefficient two-photon transitions originating from selection rules. We present here a novel scheme which allows for arbitrary coherent rotations of two-component BECs while overcoming these issues. Numerical tests of the method show that for BECs of ce{^{87}Rb} with up to $ 10^4 $ atoms and gate times of $ SI{1}{microsecond} $, decoherence due to spontaneous emission can be suppressed to negligible values.
When vortices are displaced in Bose-Einstein condensates (BEC), the Magnus force gives the system a momentum transverse in the direction to the displacement. We show that Bose-Einstein condensates (BEC) in long channels with vortices exhibit a quanti
zation of the current response with respect to the spatial vortex distribution. The quantization originates from the well-known topological property of the phase around a vortex --- it is an integer multiple of $ 2 pi $. In a similar way to the integer quantum Hall effect, the current along the channel is related to this topological phase, and can be extracted from two experimentally measurable quantities: the total momentum of the BEC and the spatial distribution. The quantization is in units of $ m/2h $, where $ m $ is the mass of the atoms and $ h $ is Plancks constant. We derive an exact vortex momentum-displacement relation for BECs in long channels under general circumstances. Our results presents the possibility that the configuration described here can be used as a novel way of measuring the mass of the atoms in the BEC using a topological invariant of the system. If an accurate determination of the plateaus are experimentally possible, this gives the possibility of a topological quantum mass standard and precise determination of the fine structure constant.
Quantum teleportation is the transfer of quantum information between two locations by the use of shared entanglement. Current teleportation schemes broadly fall under one of two categories, of either qubit or continuous variables teleportation. Spin
coherent states on spin ensembles can be teleported under the continuous variables approximation for states with small deviations from a given polarization. However, for spin coherent states with large deviations, no convenient teleportation protocol exists. Recently, we introduced a teleportation scheme where a spin coherent state lying on the equator of the Bloch sphere is teleported between distant parties (A.N. Pyrkov and T. Byrnes, New. J. Phys. 16, 073038 (2014)). Here we generalize the protocol to a spin coherent state with an arbitrary position on Bloch sphere. Our proposed scheme breaks classical bounds based on communication in the unconditional case and quantum state estimation with postselection.
