Improved Error-Scaling for Adiabatic Quantum State Transfer

We present a technique that dramatically improves the accuracy of adiabatic state transfer for a broad class of realistic Hamiltonians. For some systems, the total error scaling can be quadratically reduced at a fixed maximum transfer rate. These improvements rely only on the judicious choice of the total evolution time. Our technique is error-robust, and hence applicable to existing experiments utilizing adiabatic passage. We give two examples as proofs-of-principle, showing quadratic error reductions for an adiabatic search algorithm and a tunable two-qubit quantum logic gate.Comment: 10 Pages, 4 figures. Comments are welcome. Version substantially revised to generalize results to cases where several derivatives of the Hamiltonian are zero on the boundar

The Hilbertian Tensor Norm and Entangled Two-Prover Games

We study tensor norms over Banach spaces and their relations to quantum information theory, in particular their connection with two-prover games. We consider a version of the Hilbertian tensor norm $\gamma_2$ and its dual $\gamma_2^*$ that allow us to consider games with arbitrary output alphabet sizes. We establish direct-product theorems and prove a generalized Grothendieck inequality for these tensor norms. Furthermore, we investigate the connection between the Hilbertian tensor norm and the set of quantum probability distributions, and show two applications to quantum information theory: firstly, we give an alternative proof of the perfect parallel repetition theorem for entangled XOR games; and secondly, we prove a new upper bound on the ratio between the entangled and the classical value of two-prover games.Comment: 33 pages, some of the results have been obtained independently in arXiv:1007.3043v2, v2: an error in Theorem 4 has been corrected; Section 6 rewritten, v3: completely rewritten in order to improve readability; title changed; references added; published versio

Computational Indistinguishability between Quantum States and Its Cryptographic Application

We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is "secure" against any polynomial-time quantum adversary. Our problem, QSCDff, is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonly-used distinction problem between two probability distributions in computational cryptography. As our major contribution, we show that QSCDff has three properties of cryptographic interest: (i) QSCDff has a trapdoor; (ii) the average-case hardness of QSCDff coincides with its worst-case hardness; and (iii) QSCDff is computationally at least as hard as the graph automorphism problem in the worst case. These cryptographic properties enable us to construct a quantum public-key cryptosystem, which is likely to withstand any chosen plaintext attack of a polynomial-time quantum adversary. We further discuss a generalization of QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies on similar cryptographic properties of QSCDcyc.Comment: 24 pages, 2 figures. We improved presentation, and added more detail proofs and follow-up of recent wor

The relationship between minimum gap and success probability in adiabatic quantum computing

We explore the relationship between two figures of merit for an adiabatic quantum computation process: the success probability $P$ and the minimum gap $\Delta_{min}$ between the ground and first excited states, investigating to what extent the success probability for an ensemble of problem Hamiltonians can be fitted by a function of $\Delta_{min}$ and the computation time $T$. We study a generic adiabatic algorithm and show that a rich structure exists in the distribution of $P$ and $\Delta_{min}$. In the case of two qubits, $P$ is to a good approximation a function of $\Delta_{min}$, of the stage in the evolution at which the minimum occurs and of $T$. This structure persists in examples of larger systems.Comment: 13 pages, 6 figures. Substantially updated, with further discussion of the phase diagram and the relation between one- and two-qubit evolution, as well as a greatly extended list of reference

Quantum hypercomputation based on the dynamical algebra su(1,1)

An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is presented. The method that was used was to replace the Weyl-Heisenberg algebra by other dynamical algebra of low dimension that admits infinite-dimensional irreducible representations with naturally defined generalized coherent states. We have selected the Lie algebra $\mathfrak{su}(1,1)$, due to that this algebra posses the necessary characteristics for to realize the hypercomputation and also due to that such algebra has been identified as the dynamical algebra associated to many relatively simple quantum systems. In addition to an algebraic adaptation of KHQA over the algebra $\mathfrak{su}(1,1)$, we presented an adaptations of KHQA over some concrete physical referents: the infinite square well, the infinite cylindrical well, the perturbed infinite cylindrical well, the P{\"o}sch-Teller potentials, the Holstein-Primakoff system, and the Laguerre oscillator. We conclude that it is possible to have many physical systems within condensed matter and quantum optics on which it is possible to consider an implementation of KHQA.Comment: 25 pages, 1 figure, conclusions rewritten, typing and language errors corrected and latex format changed minor changes elsewhere and

A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations

We are interested in the problem of characterizing the correlations that arise when performing local measurements on separate quantum systems. In a previous work [Phys. Rev. Lett. 98, 010401 (2007)], we introduced an infinite hierarchy of conditions necessarily satisfied by any set of quantum correlations. Each of these conditions could be tested using semidefinite programming. We present here new results concerning this hierarchy. We prove in particular that it is complete, in the sense that any set of correlations satisfying every condition in the hierarchy has a quantum representation in terms of commuting measurements. Although our tests are conceived to rule out non-quantum correlations, and can in principle certify that a set of correlations is quantum only in the asymptotic limit where all tests are satisfied, we show that in some cases it is possible to conclude that a given set of correlations is quantum after performing only a finite number of tests. We provide a criterion to detect when such a situation arises, and we explain how to reconstruct the quantum states and measurement operators reproducing the given correlations. Finally, we present several applications of our approach. We use it in particular to bound the quantum violation of various Bell inequalities.Comment: 33 pages, 2 figure

Richard Stallmanin tekijänoikeuskritiikin yhteiskuntafilosofinen analyysi

Käsittelen työssäni Richard Stallmanin vapaiden ohjelmistojen filosofiaa ja siihen sisältyvää tekijänoikeuskritiikkiä. Analyysin taustana ovat tekijänoikeuden perustasta esitetyt teoriat ja liberalistinen yhteiskuntafilosofia sekä kommunitaristien siihen kohdistama kritiikki. Tekijänoikeuden oikeutusta koskevat kolme keskeistä teoriaa ovat työteoria, persoonallisuusteoria ja utilitaristinen teoria. John Locken luonnonoikeusfilosofiaan nojaavassa työteoriassa katsotaan, että jostakin resurssista tulee tekijänsä omaisuutta, kun siihen sekoittuu tekijän työpanos. Persoonallisuusteoria perustuu Immanuel Kantin ja G. W. F. Hegelin näkemykseen omaisuudesta osana henkilön persoonallisuutta, jonka vapautta on suojattava. Utilitaristinen teoria katsoo tekijänoikeuden olevan perusteltua siihen liittyvän kannustimen vuoksi: enemmän teoksia syntyy, mikäli niiden tekijöille annetaan teokseen erityinen yksinoikeus. Stallman kyseenalaistaa tietokoneohjelmiin saatavan tekijänoikeuden oikeutuksen viittaamalla käyttäjien oikeuksiin. Stallmanin mukaan tietokoneohjelman käyttäjälle kuuluvat neljä vapautta: 1) vapaus käyttää ohjelmaa rajoituksetta, 2) vapaus muuttaa ohjelmaa tarpeiden mukaan, 3) vapaus levittää ohjelmaa muille, 4) vapaus parannella ohjelmaa ja jakaa se muiden kanssa. Näitä vapauksia Stallman perustelee viittaamalla yhteisöön. Hän ottaa moraalifilosofiseksi lähtökohdaksi vasta­vuoroisuuden periaatteen, jonka mukaan yhteisön jäsenten tulee auttaa toinen toisiaan esimerkiksi jakamalla tietokoneohjelmia. Tekijänoikeuden asettamat rajoitukset ovat esteitä tämäntyyppiselle solidaarisuudelle ja yhteisössä sekä yhteiskunnassa vallitsevalle avunannon ilmapiirille. Yhteiskuntafilosofiselta perusluonteeltaan Stallmanin filosofia edustaa liberalismia. Väite käyttäjien yksilöllisistä oikeuksista perustuu liberalistiseen ihmiskuvaan. Se on myös universalistinen: Stallman pitää käyttäjien oikeuksia joko luonnollisina oikeuksina tai ihmisoikeuksina. Toisaalta teoriassa on keskeisellä sijalla ajatus yhteisön eheydestä ja yhteisöllisen hyvän edistämisestä, mikä tuo siihen kommunitaristisia piirteitä. Toteuttamalla ohjelmistojen vapauden elämäntapaa yhteisö toteuttaa tietynlaista hyvän elämän politiikkaa. Teoriaa ei kuitenkaan voi kokonaisuudessaan pitää kommunitaristisena siihen sisältyvän vahvan universalismin vuoksi. Vapautta korostavan luon­teensa ja negatiivisen omaisuuskäsityksensä vuoksi osuvampi luonnehdinta Stallmanin filosofialle on digitaalinen anarkismi. Asiasanat: tekijänoikeus, liberalismi, kommunitarismi, Richard Stallman, vapaat ohjelmistot, avoin lähdekoodi, tietokoneohjelmisto