Wavelets as q-bits and q-bit states as wavelets
Steblinski, Pawel; Blachowicz, Tomasz
2009-01-01
In this short report it is argued that by the use of wavelets formalism it is possible to describe the q-bit state. The wavelet formalism address the real-valued physical signals, for example, obtained during typical physical measurements.
Bits and q-bits as versatility measures
Directory of Open Access Journals (Sweden)
José R.C. Piqueira
2004-06-01
Full Text Available Using Shannon information theory is a common strategy to measure any kind of variability in a signal or phenomenon. Some methods were developed to adapt information entropy measures to bird song data trying to emphasize its versatility aspect. This classical approach, using the concept of bit, produces interesting results. Now, the original idea developed in this paper is to use the quantum information theory and the quantum bit (q-bit concept in order to provide a more complete vision of the experimental results.Usar a teoria da informação de Shannon é uma estratégia comum para medir todo tipo de variabilidade em um sinal ou fenômeno. Alguns métodos foram desenvolvidos para adaptar a medida de entropia informacional a dados de cantos de pássaro, tentando enfatizar seus aspectos de versatilidade. Essa abordagem clássica, usando o conceito de bit, produz resultados interessantes. Agora, a idéia original desenvolvida neste artigo é usar a teoria quântica da informação e o conceito de q-bit, com a finalidade de proporcionar uma visão mais completa dos resultados experimentais.
The energy level splitting for Unharmonic dc SQUID to be used as phase Q-bit
DEFF Research Database (Denmark)
Klenov, Nicolai V.; Kornev, Victor K.; Pedersen, Niels Falsig
2006-01-01
Dc-SQUID with Josephson junctions characterized by nonsinusoidal current-phase relation is being considered as a basis for phase qubit. It has been shown that the second and third harmonic components each in the current-phase relation are able to provide double-well potential and the energy level...
Classical images as quantum entanglement: An image processing analogy of the GHZ experiment
Goldin, Matías A.; Francisco, Diego; Ledesma, Silvia
2011-04-01
In this paper we present an optical analogy of quantum entanglement by means of classical images. As in previous works, the quantum state of two or more qbits is encoded by using the spatial modulation in amplitude and phase of an electromagnetic field. We show here that bidimensional encoding of two qbit states allows us to interpret some non local features of the joint measurement by the assumption of “astigmatic” observers with different resolving power in two orthogonal directions. As an application, we discuss the optical simulation of measuring a system characterized by multiparticle entanglement. The simulation is based on a local representation of entanglement and a classical interferometric system. In particular we show how to simulate the Greenberger-Horne Zeilinger (GHZ) argument and the experimental results which interpretation illustrates the conflict between quantum mechanics and local realism.
Symbolic Quantum Computation Simulation in SymPy
Cugini, Addison; Curry, Matt; Granger, Brian
2010-10-01
Quantum computing is an emerging field which aims to use quantum mechanics to solve difficult computational problems with greater efficiency than on a classical computer. There is a need to create software that i) helps newcomers to learn the field, ii) enables practitioners to design and simulate quantum circuits and iii) provides an open foundation for further research in the field. Towards these ends we have created a package, in the open-source symbolic computation library SymPy, that simulates the quantum circuit model of quantum computation using Dirac notation. This framework builds on the extant powerful symbolic capabilities of SymPy to preform its simulations in a fully symbolic manner. We use object oriented design to abstract circuits as ordered collections of quantum gate and qbit objects. The gate objects can either be applied directly to the qbit objects or be represented as matrices in different bases. The package is also capable of performing the quantum Fourier transform and Shor's algorithm. A notion of measurement is made possible through the use of a non-commutative gate object. In this talk, we describe the software and show examples of quantum circuits on single and multi qbit states that involve common algorithms, gates and measurements.
A hybrid quantum-inspired genetic algorithm for multiobjective flow shop scheduling.
Li, Bin-Bin; Wang, Ling
2007-06-01
This paper proposes a hybrid quantum-inspired genetic algorithm (HQGA) for the multiobjective flow shop scheduling problem (FSSP), which is a typical NP-hard combinatorial optimization problem with strong engineering backgrounds. On the one hand, a quantum-inspired GA (QGA) based on Q-bit representation is applied for exploration in the discrete 0-1 hyperspace by using the updating operator of quantum gate and genetic operators of Q-bit. Moreover, random-key representation is used to convert the Q-bit representation to job permutation for evaluating the objective values of the schedule solution. On the other hand, permutation-based GA (PGA) is applied for both performing exploration in permutation-based scheduling space and stressing exploitation for good schedule solutions. To evaluate solutions in multiobjective sense, a randomly weighted linear-sum function is used in QGA, and a nondominated sorting technique including classification of Pareto fronts and fitness assignment is applied in PGA with regard to both proximity and diversity of solutions. To maintain the diversity of the population, two trimming techniques for population are proposed. The proposed HQGA is tested based on some multiobjective FSSPs. Simulation results and comparisons based on several performance metrics demonstrate the effectiveness of the proposed HQGA.
Phase Slips in Topological Superconductor Wire Devices
Goldberg, Samuel; Bergman, Doron; Pekker, David; Refael, Gil
2012-02-01
We make a detailed study of phase slips in topological superconducting wires and devices based on topological wires. We begin by investigating a device composed of a topological superconducting wire connected to a non-topological wire (T-S). In the T-segment only slips of the phase by multiples of 4π are allowed, while in the S-segment slips by 2π are also allowed. We show that near the interface, 2π phase slips are also allowed and we comment on the consequences of such phase slips for the Aharonov-Casher effect. We also consider an implementation of a q-bit consisting of a T-S-T device, where the quantum information is stored in the parity of the two topological segments via the four Majorana modes. We show that the central S-segment of this type of device can support 2π phase-slips which result in the decoherence of the q-bit.
Dual Field Theories of Quantum Computation
Vanchurin, Vitaly
2016-01-01
Given two quantum states of $N$ q-bits we are interested to find the shortest quantum circuit consisting of only one- and two- q-bit gates that would transfer one state into another. We call it the quantum maze problem for the reasons described in the paper. We argue that in a large $N$ limit the quantum maze problem is equivalent to the problem of finding a semiclassical trajectory of some lattice field theory (the dual theory) on an $N+1$ dimensional space-time with geometrically flat, but topologically compact spatial slices. The spatial fundamental domain is an $N$ dimensional hyper-rhombohedron, and the temporal direction describes transitions from an arbitrary initial state to an arbitrary target state. We first consider a complex Klein-Gordon field theory and argue that it can only be used to study the shortest quantum circuits which do not involve generators composed of tensor products of multiple Pauli $Z$ matrices. Since such situation is not generic we call it the $Z$-problem. On the dual field the...
System Framework for a Multi-Band, Multi-Mode Software Defined Radio
2014-06-01
the differential encoder, Q-bit delay, and precoder are required per IRIG 106 specifications. In addition, the shaping filter is different for the...Parallel sT 0 dt BIT DELAY SYMBOL PRECODER I Q IRIG 106 RANDOMIZER nT sin L nT cos L nT I L...International Telemetry Conference, Oct 2014. [4] IRIG Standard 106-11, Chapter 2: Transmitter and Receiver Systems, www.irig106.org, June 2011. [5
Remote Stimulated Triggering of Quantum Entangled Nuclear Metastable States of 115mIn
Van Gent, D L
2004-01-01
We report experiments in which two indium foils were quantum entangled via photoexcitation of stable 115In to radioactive 115mIn by utilizing Bremsstrahlung gamma photons produced by a Varian Compact Linear Accelerator (CLINAC). After photo-excitation, remote triggering of the "master" foil with low energy gamma photons, yielded stimulated emissions of 336 keV gamma photons from quantum entangled 115mIn in the "slave" foil located up to 1600 m away from the "master" foil. These experiments strongly demonstrate that useful quantum information can be transferred through quantum channels via modulation of quantum noise (accelerated radioactive decay of 115mIn metastable nuclei). Thus, this modality of QE transmission is fundamentally different from optical QE information transfer via quantum entangled space "q-bits" as developed by information theorists for quantum channel information transfer. Additionally, there is no obvious potential for signal degradation with increasing distance nor the problems associated...
Directory of Open Access Journals (Sweden)
Jinwei Gu
2015-01-01
Full Text Available A mutualism quantum genetic algorithm (MQGA is proposed for an integrated supply chain scheduling with the materials pickup, flow shop scheduling, and the finished products delivery. The objective is to minimize the makespan, that is, the arrival time of the last finished product to the customer. In MQGA, a new symbiosis strategy named mutualism is proposed to adjust the size of each population dynamically by regarding the mutual influence relation of the two subpopulations. A hybrid Q-bit coding method and a local speeding-up method are designed to increase the diversity of genes, and a checking routine is carried out to ensure the feasibility of each solution; that is, the total physical space of each delivery batch could not exceed the capacity of the vehicle. Compared with the modified genetic algorithm (MGA and the quantum-inspired genetic algorithm (QGA, the effectiveness and efficiency of the MQGA are validated by numerical experiments.
Concepts of a quantum information theory of many letters
Boström, K J
2000-01-01
A theoretical framework is presented allowing the treatment of quantum messages with components of variable length. To this aim a many-letter space is constructed which turns out to be a natural space for quantum messages of this type, generalizing the usual fixed-length quantum information theory which is fully contained within this framework. An observable is defined measuring the amount of quantum information carried by a particular message, with the "qbit" obtaining a second meaning as its unit. A general characterization of quantum codes is given where compression codes are defined by their property of decreasing the average information content of a given a priori message ensemble. A lossless quantum coding scheme - analog to the classical Huffman scheme but different from the Braunstein scheme - is implemented, which not only ensures perfect fidelity in retrieving the original messages but also provides optimal compression.
Using the J1-J2 Quantum Spin Chain as an Adiabatic Quantum Data Bus
Chancellor, Nicholas
2012-01-01
This paper investigates numerically a phenomenon which can be used to transport a single q-bit down a J1-J2 Heisenberg spin chain using a quantum adiabatic process. The motivation for investigating such processes comes from the idea that this method of transport could potentially be used as a means of sending data to various parts of a quantum computer made of artificial spins, and that this method could take advantage of the easily prepared ground state at the so called Majumdar-Ghosh point. We examine several annealing protocols for this process and find similar result for all of them. The annealing process works well up to a critical frustration threshold.
On the ergodic secret message capacity of the wiretap channel with finite-rate feedback
Rezki, Zouheir
2012-07-01
We study the secret message capacity of an ergodic block fading wiretap channel with partial channel state information at the transmitter and perfect channel state information at the receivers. We consider that in addition to the statistics of the main and the eavesdropper channel state information (CSI), the sender is provided by the legitimate receiver with a q-bit feedback, at the beginning of each coherence block, through an error-free feedback channel, with capacity q bits. We establish upper and lower bounds on the secrecy capacity. We show that a positive secrecy rate is achievable even when the feedback is at the end of each coherence block and q = 1. We also show that the lower and the upper bounds coincide asymptotically as q → ∞. Finally, asymptotic analysis at high Signal-to-Noise Ratio (SNR) are presented where it is found that the capacity is bounded at high-SNR and present a simple suboptimal scalar quantizer that is capacity achieving, without the need of any numerical optimization, as q → ∞. When applied to Rayleigh fading channels, we show that, at high-SNR, a 4-bit feedback achieves 90% of the secrecy capacity when perfect main CSI is available at the transmitter. © 2012 IEEE.
Directory of Open Access Journals (Sweden)
M Soltani
2015-12-01
Full Text Available In this work, we generalize the entanglement of three-qbit Bosonic systems beyond the single-mode approximation when one of the observers is accelerated. For this purpose, we review the effects of acceleration on field modes and quantum states. The single-mode approximation and beyond the single-mode approximation methods are introduced. After this brief introduction, the main problem of this paper, tripartite entanglement of bosonic systems in a noninertial frame beyond the single- mode approximation is investigated. The tripartite entangled states have different classes with GHZ and W states being most important. Here, we choose &pi-tangle as a measure of tripartite entanglement. If the three parties share GHZ state, the corresponding &pi-tangle will increase by increasing acceleration for some Unruh modes. This phenomenon, increasing entanglement, has never been observed in the single-mode approximation for bosonic case. Moreover, the &pi-tangle dose not exhibit a monotonic behavior with increasing acceleration. In the infinite acceleration limit, the &pi-tangle goes to different nonzero values for distinct Unruh modes. Unlike GHZ state, the entanglement of the W state shows only monotonically increasing and decreasing behaviors with increasing acceleration. Also, the entanglement for all possible choices of Unruh modes approaches only 0.176 in the high acceleration limit. Therefore, according to the quantum entanglement, there is no distinction between the single-mode approximation and beyond the single-mode approximation methods in this limit.
Are all reversible computations tidy?
Maroney, O J E
2004-01-01
It has long been known that to minimise the heat emitted by a deterministic computer during it's operation it is necessary to make the computation act in a logically reversible manner\\cite{Lan61}. Such logically reversible operations require a number of auxiliary bits to be stored, maintaining a history of the computation, and which allows the initial state to be reconstructed by running the computation in reverse. These auxiliary bits are wasteful of resources and may require a dissipation of energy for them to be reused. A simple procedure due to Bennett\\cite{Ben73} allows these auxiliary bits to be "tidied", without dissipating energy, on a classical computer. All reversible classical computations can be made tidy in this way. However, this procedure depends upon a classical operation ("cloning") that cannot be generalised to quantum computers\\cite{WZ82}. Quantum computations must be logically reversible, and therefore produce auxiliary qbits during their operation. We show that there are classes of quantu...
Abdallah, Mohamed M.
2013-11-01
In this work, we develop joint interference-aware random beam and spectrum selection scheme that provide enhanced performance for the secondary network under the condition that the interference observed at the primary receiver is below a predetermined acceptable value. We consider a secondary link composed of a transmitter equipped with multiple antennas and a single-antenna receiver sharing the same spectrum with a set of primary links composed of a single-antenna transmitter and a single-antenna receiver. The proposed schemes jointly select a beam, among a set of power-optimized random beams, as well as the primary spectrum that maximizes the signal-to-interference-plus-noise ratio (SINR) of the secondary link while satisfying the primary interference constraint. In particular, we consider the case where the interference level is described by a q-bit description of its magnitude, whereby we propose a technique to find the optimal quantizer thresholds in a mean square error (MSE) sense. © 2013 IEEE.
Black hole evaporation without an event horizon
Bardeen, James M
2014-01-01
A reformulation of the calculation of the semi-classical energy-momentum tensor on a Schwarzschild background, the Bousso covariant entropy bound, and the ER=EPR conjecture of Maldacena and Susskind taken together suggest a scenario for the evaporation of a large spherically symmetric black hole formed in gravitational collapse in which 1) the classical r = 0 singularity is replaced by an initially small non-singular core inside an inner apparent horizon, 2) the radius of the core grows with time due to the increasing entanglement between Hawking radiation quanta outside the black hole and the Hawking partner quanta in the core contributing to the quantum back-reaction, and 3) by the Page time the trapped surfaces disappear and all quantum information stored in the interior is free to escape. The scenario preserves unitarity without any need for a "firewall" in the vicinity of the outer apparent horizon. Qbits in the Hawking radiation are never mutually entangled, and their number never exceeds the Bekenstein...
Ergodic secret message capacity of the wiretap channel with finite-rate feedback
Rezki, Zouheir
2014-06-01
We study the secret message capacity of an ergodic block fading wiretap channel with partial channel state information at the transmitter and perfect channel state information at the receivers, under both a short term power constraint (STPC) and a long term power constraint (LTPC). We consider that in addition to the statistics of the main and the eavesdropper channel state information (CSI), the sender is provided by the legitimate receiver with a q-bit feedback, at the beginning of each coherence block, through an error-free public channel, with capacity q bits. We establish upper and lower bounds on the secrecy capacity. We show that the lower and the upper bounds coincide asymptotically as q → ∞. When applied to Rayleigh fading channels, we show that, a 4-bit feedback achieves about 90% of the secrecy capacity when perfect main CSI is available at the transmitter. Finally, asymptotic analysis at high and low Signal-to-Noise Ratio (SNR) is presented. It is found that the capacity is bounded at high-SNR, whereas at asymptotically low-SNR, the lower bounds and the upper bound scale linearly with SNR under STPC. Furthermore, subject to LTPC, the capacity at low-SNR is equal to the capacity of the main channel without secrecy constraint and with perfect CSI at both the transmitter and the receiver, under a mild condition on the fading statistics. We also show that a positive secrecy rate is achievable even when the feedback is at the end of each coherence block and q=1. © 2002-2012 IEEE.
Adaptive and non-adaptive data hiding methods for grayscale images based on modulus function
Directory of Open Access Journals (Sweden)
Najme Maleki
2014-07-01
Full Text Available This paper presents two adaptive and non-adaptive data hiding methods for grayscale images based on modulus function. Our adaptive scheme is based on the concept of human vision sensitivity, so the pixels in edge areas than to smooth areas can tolerate much more changes without making visible distortion for human eyes. In our adaptive scheme, the average differencing value of four neighborhood pixels into a block via a threshold secret key determines whether current block is located in edge or smooth area. Pixels in the edge areas are embedded by Q-bit of secret data with a larger value of Q than that of pixels placed in smooth areas. Also in this scholar, we represent one non-adaptive data hiding algorithm. Our non-adaptive scheme, via an error reduction procedure, produces a high visual quality for stego-image. The proposed schemes present several advantages. 1-of aspects the embedding capacity and visual quality of stego-image are scalable. In other words, the embedding rate as well as the image quality can be scaled for practical applications 2-the high embedding capacity with minimal visual distortion can be achieved, 3-our methods require little memory space for secret data embedding and extracting phases, 4-secret keys have used to protect of the embedded secret data. Thus, level of security is high, 5-the problem of overflow or underflow does not occur. Experimental results indicated that the proposed adaptive scheme significantly is superior to the currently existing scheme, in terms of stego-image visual quality, embedding capacity and level of security and also our non-adaptive method is better than other non-adaptive methods, in view of stego-image quality. Results show which our adaptive algorithm can resist against the RS steganalysis attack.
Sayed, Mostafa M.
2014-11-01
Spectrum sharing systems have been introduced to alleviate the problem of spectrum scarcity by allowing an unlicensed secondary user (SU) to share the spectrum with a licensed primary user (PU) under acceptable interference levels to the primary receiver (PU-Rx). In this paper, we consider a secondary link composed of a secondary transmitter (SU-Tx) equipped with multiple antennas and a single-antenna secondary receiver (SU-Rx). The secondary link is allowed to share the spectrum with a primary network composed of multiple PUs communicating over distinct frequency spectra with a primary base station. We develop a transmission scheme where the SU-Tx initially broadcasts a set of random beams over all the available primary spectra for which the PU-Rx sends back the index of the spectrum with the minimum interference level, as well as information that describes the interference value, for each beam. Based on the feedback information on the PU-Rx, the SU-Tx adapts the transmitted beams and then resends the new beams over the best primary spectrum for each beam to the SU-Rx. The SU-Rx selects the beam that maximizes the received signal-to-interference-plus-noise ratio (SINR) to be used in transmission over the next frame. We consider three cases for the level of feedback information describing the interference level. In the first case, the interference level is described by both its magnitude and phase; in the second case, only the magnitude is considered; and in the third case, we focus on a q-bit description of its magnitude. In the latter case, we propose a technique to find the optimal quantizer thresholds in a mean-square-error sense. We also develop a statistical analysis for the SINR statistics and the capacity and bit error rate of the secondary link and present numerical results that study the impact of the different system parameters.
Sirtori, Carlo
2017-02-01
Superradiance is one of the many fascinating phenomena predicted by quantum electrodynamics that have first been experimentally demonstrated in atomic systems and more recently in condensed matter systems like quantum dots, superconducting q-bits, cyclotron transitions and plasma oscillations in quantum wells (QWs). It occurs when a dense collection of N identical two-level emitters are phased via the exchange of photons, giving rise to enhanced light-matter interaction, hence to a faster emission rate. Of great interest is the regime where the ensemble interacts with one photon only and therefore all of the atoms, but one, are in the ground state. In this case the quantum superposition of all possible configurations produces a symmetric state that decays radiatively with a rate N times larger than that of the individual oscillators. This phenomenon, called single photon superradiance, results from the exchange of real photons among the N emitters. Yet, to single photon superradiance is also associated another collective effect that renormalizes the emission frequency, known as cooperative Lamb shift. In this work, we show that single photon superradiance and cooperative Lamb shift can be engineered in a semiconductor device by coupling spatially separated plasma resonances arising from the collective motion of confined electrons in QWs. These resonances hold a giant dipole along the growth direction z and have no mutual Coulomb coupling. They thus behave as a collection of macro-atoms on different positions along the z axis. Our device is therefore a test bench to simulate the low excitation regime of quantum electrodynamics.
Optically simulated universal quantum computation
Francisco, D.; Ledesma, S.
2008-04-01
Recently, classical optics based systems to emulate quantum information processing have been proposed. The analogy is based on the possibility of encoding a quantum state of a system with a 2N-dimensional Hilbert space as an image in the input of an optical system. The probability amplitude of each state of a certain basis is associated with the complex amplitude of the electromagnetic field in a given slice of the laser wavefront. Temporal evolution is represented as the change of the complex amplitude of the field when the wavefront pass through a certain optical arrangement. Different modules that represent universal gates for quantum computation have been implemented. For instance, unitary operations acting on the qbits space (or U(2) gates) are represented by means of two phase plates, two spherical lenses and a phase grating in a typical image processing set up. In this work, we present CNOT gates which are emulated by means of a cube prism that splits a pair of adjacent rays incoming from the input image. As an example of application, we present an optical module that can be used to simulate the quantum teleportation process. We also show experimental results that illustrate the validity of the analogy. Although the experimental results obtained are promising and show the capability of the system for simulate the real quantum process, we must take into account that any classical simulation of quantum phenomena, has as fundamental limitation the impossibility of representing non local entanglement. In this classical context, quantum teleportation has only an illustrative interpretation.
The high-order quantum coherence of thermal light
Chen, Hui
Thermal light, such as sunlight, is usually considered classical light. In a macroscopic picture, classical theory successfully explained the first-order coherence phenomena of thermal light. The macroscopic theory, based on the statistical behavior of light intensity fluctuations, however, can only phenomenologically explain the second- or higher-order coherence phenomena of thermal light. This thesis introduces a microscopic quantum picture, based on the interferences of a large number of randomly distributed and randomly radiated subfields, wavepackets or photons, to the study of high-order coherence of thermal light. This thesis concludes that the second-order intensity fluctuation correlation is caused by nonlocal interference: a pair of wavepackets, which are randomly paired together, interferes with the pair itself at two distant space-time coordinates. This study has the following practical motivations: (1) to simulate N-qbits. Practical quantum computing requires quantum bits(qubits) of N-digit to represent all possible integers from 0 to 2N-1 simultaneously. A large number of independent particles can be prepared to represent a large set of N orthogonal |0> and |1> bits. In fact, based on our recent experiments of simulating the high-order correlation of entangled photons, thermal radiation is suggested as a promising source for quantum information processing. (2) to achieve sunlight ghost imaging. Ghost imaging has three attractive non-classical features: (a) the ghost camera can "see" targets that can never be seen by a classic camera; (2) it is turbulence-free; and (3) its spatial resolution is mainly determined by the angular diameter of the light source. For example, a sunlight ghost image of an object on earth may achieve a spatial resolution of 200 micrometer because the angular diameter of sun is 0.53 degree with respect to Earth. Although ghost imaging has been experimental demonstrated by using entangled photon pairs and "pseudo-thermal light
Bimberg, Dieter
2011-01-01
satellite telephones, in some ways continuing his work for the candidate degree when he had developed power rectifiers based on Ge and Si. Then, in 1970, he presented the first solar cells with efficiency >30% based on heterojunctions. Soon the Soviet Space Administration became aware of these results, and in 1986 the Soviet space station MIR was partially powered by solar cells developed by Alferov and Andre'ev. Finally in 1992, a joint research program between the author of this Editorial and Zhores Alferov, being both guest scientists at the same time at the University of California, on semiconductor quantum dots for the active zone of (nowadays many different) optoelectronic devices was proposed and inaugurated. Quantum dot lasers today have the lowest threshold current density of any semiconductor lasers. They are far superior to quantum wells as amplifiers, and their nonlinear optical applications such as cross-gain modulation in local area networks, present the basis for novel types of solar cells, nanoflash memories, single q-bit emitters for quantum cryptography etc. The story of inventing a concept and inventing applications seems to repeat in some way. This Semiconductor Science and Technology special edition presents contributions from about 100 researchers around the globe, who use in their work concepts invented by Zhores Alferov during his long active scientific life spanning six decades. They would like to pay a tribute to him and honour him on the occasion of his 80th birthday. This very personal way of saying thank you thus adds to the many prizes he has received during the past 40 years, starting with the Ballantine Gold Medal of the Franklin Institute, via the Nobel Prize for Physics 2000 to many honorary doctorates from institutes around the world.