Characterizing lattices for cryptography

Following their divorce, Alice and Bob would like to split some of their possessions by flipping a coin. Unwilling to meet in person, and without a trusted third party, they must figure out a scheme to flip the coin over a telephone that guarantees that neither party can cheat. The preceding scenario is the traditional definition of two-party coin-flipping. In a classical setting, without limits on the available computational power, one player can always guarantee a coin-flipping victory by cheating. However, by employing quantum communication it is possible to guarantee, with only information-theoretic assumptions, that neither party can win by cheating, with a probability greater than two thirds. Along with the description of such a protocol, this thesis derives a tight lower bound on the bias for a large family of quantum weak coin-flipping protocols, proving such a protocol optimal within the family. The protocol described herein is an improvement and generalization of one examined by Spekkens and Rudolph. The key steps of the analysis involve Kitaev's description of quantum coin-flipping as a semidefinite program whose dual problem provides a certificate that upper bounds the amount of cheating for each party. In order for such quantum protocols to be viable, though, a number of practical obstacles involving the communication and processing of quantum information must be resolved. In the second half of this thesis, a scheme for processing quantum information is presented, which uses non-abelian anyons that are the magnetic and electric excitations of a discrete-group quantum gauge theory. In particular, the connections between group structure and computational power are examined, generalizing previous work by Kitaev, Ogburn and Preskill. Anyon based computation has the advantage of being topological, which exponentially suppresses the rate of decoherence and the errors associated with the elementary quantum gates. Though no physical systems with such excitations are currently known to exist, it remains an exciting open possibility that such particles could be either engineered or discovered in exotic two-dimensional systems.

Computer science will be radically transformed if ongoing efforts to build large-scale quantum computers eventually succeed and if the properties of these computers meet optimistic expectations. Nevertheless, computer scientists still lack a thorough understanding of the power of quantum computing, and it is not always clear how best to utilize the power that is understood. This dilemma exists because quantum algorithms are difficult to grasp and even more difficult to write. Despite large-scale international efforts, only a few important quantum algorithms are documented, leaving many essential questions about the potential of quantum algorithms unanswered. These unsolved problems are ideal challenges for the application of automatic programming technologies. Genetic programming techniques, in particular, have already produced several new quantum algorithms and it is reasonable to expect further discoveries in the future. These methods will help researchers to discover how additional practical problems can be solved using quantum computers, and they will also help to guide theoretical work on both the power and limits of quantum computing. This tutorial will provide an introduction to quantum computing and an introduction to the use of evolutionary computation for automatic quantum computer programming. No background in physics or in evolutionary computation will be assumed. While the primary focus of the tutorial will be on general concepts, specific results will also be presented, including human-competitive results produced by genetic programming. Follow-up material is available from the presenter's book, Automatic Quantum Computer Programming: A Genetic Programming Approach, published by Springer and Kluwer Academic Publishers.

The driving force in the pursuit for quantum computation is the exciting possibility that quantum algorithms can be more efficient than their classical analogues. Research on the subject has unraveled several aspects of how that can happen. Clever quantum algorithms have been discovered in recent years, although not systematically, and the field remains under active investigation. Richard Feynman was one of the pioneers who foresaw the power of quantum computers. In this issue dedicated to him, I give an introduction to how particle and wave aspects contribute to the power of quantum computers. Shor's and Grover's algorithms are analysed as examples.

Motivated by understanding the power of quantum computation with restricted number of qubits, we give two complete characterizations of unitary quantum space bounded computation. First we show that approximating an element of the inverse of a well-conditioned efficiently encoded 2^k(n) x 2^k(n) matrix is complete for the class of problems solvable by quantum circuits acting on O(k(n)) qubits with all measurements at the end of the computation. Similarly, estimating the minimum eigenvalue of an efficiently encoded Hermitian 2^k(n) x 2^k(n) matrix is also complete for this class. In the logspace case, our results improve on previous results of Ta-Shma by giving new space-efficient quantum algorithms that avoid intermediate measurements, as well as showing matching hardness results. Additionally, as a consequence we show that preciseQMA, the version of QMA with exponentially small completeness-soundess gap, is equal to PSPACE. Thus, the problem of estimating the minimum eigenvalue of a local Hamiltonian to inverse exponential precision is PSPACE-complete, which we show holds even in the frustration-free case. Finally, we can use this characterization to give a provable setting in which the ability to prepare the ground state of a local Hamiltonian is more powerful than the ability to prepare PEPS states. Interestingly, by suitably changing the parameterization of either of these problems we can completely characterize the power of quantum computation with simultaneously bounded time and space.

A finite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane. The key to the algorithm lies in a formulation that generates a plane in n-dimensional space that minimizes the sum of the squares of the 2-norm distances to each of m1 given points in the space. The plane is generated by an eigenvector corresponding to a smallest eigenvalue of an n × n simple matrix derived from the m1 points. The algorithm was tested on the publicly available Wisconsin Breast Prognosis Cancer database to generate well separated patient survival curves. In contrast, the k-mean algorithm did not generate such well-separated survival curves.

Photonic crystals are periodic dielectric structures, where the periodicity varies on the wavelength scale. Analogous to electrons in semiconductors, the photon propagation can be described using a band structure in which transmission bands are separated by bandgaps, energy ranges at which light cannot exist inside the photonic crystals. This analogy suggests that photonic crystals may be suitable to fabricate the components needed for integrated optics. As yet most of the research done on photonic crystals has been focussed on using these bandgaps. However, many novel properties of light propagating in the transmission bands may also serve this purpose. This thesis studies theoretically and experimentally the light propagation in two dimensional planar photonic crystals with the aim of creating new components for integrated optics. A theoretical model based on Fourier analysis of electromagnetic Bloch waves was developed to describe light propagation in photonic crystals. This model gives an intuitive understanding of the novel phenomena observed in the transmission bands of the photonic crystal: negative refraction, auto-collimation and super-dispersion. This new approach clarifies the fundamental physics of these phenomena enabling their advantages and disadvantages to be easily evaluated for new optical functions; particular attention is paid to auto-collimation. The influence of the structural properties of the photonic crystal (lattice type, filling factor, etc.) is also studied and has been used to create two integrated optic components: a light condenser and a wavelength demultiplexer. The theoretical predictions have been tested experimentally on two dimensional planar photonic crystals fabricated in heterostructure GaAs/AlxGa1-xAs. The fabrication of these photonic crystals was the first experimental work undertaken in this thesis. Advanced microfabrication techniques enabled the fabrication of planar photonic close to those of the state-of-the-art. The optical properties of the fabricated photonic crystals fabricated were studied using the internal light source technique. This method uses photoluminescence generated by quantum wells or quantum dots as an internal light source to probe the optical properties of the photonic crystals. Initially the experimental setup was designed to study the bandgaps. However after some modifications the transmission bands could also be investigated. Another experimental improvement made was the introduction of a technique using coupled waveguides which reduces the losses by a factor of 10 and allowed the light in the photonic crystals to be studied over long distances. Adding a spatially resolved vertical detection, enabled the analysis of the light diffracted out of the plane in order to measure the propagation losses and to map the electromagnetic field. These two modifications were used to study components for integrated optics based on photonic crystals: waveguides and auto-collimators.

This study conceptually investigates the impact of quantum computers on blockchains within the supply chain context. Powerful quantum computers enable attackers to break into blockchains by rapid inverse calculations of mathematical problems that are the core of one of the main blockchain security foundations, known as asymmetric cryptography. They are also able to violate the integrity of public blockchains like bitcoin through mining acceleration. Hence, quantum computers can engender threats to the supply chain users of blockchain. On the other hand, there are ongoing efforts to create a quantum-resistant solution. One approach for such a solution is to utilize quantum tools themselves. Moreover, sufficiently powerful quantum computers are still being developed, and it is still unclear whether a quantum solution will arrive first or vice versa. The contrasting duality of quantum computers and lack of a clear picture over the timing of the arrival of a solution and threats give rise to the uncertainty that might hinder the attractiveness of blockchains for supply chains.

Relativity and quantum mechanics are the pillars of modern physics. Relativity deals with the observable phenomenon in macroscopic level, while quantum mechanics engages in the sub-atomic particles. The quantum properties of sub-atomic particles such as electron behave both like wave and particle at the same time. The same properties can be applied in computing by using a concept called qubits. In digital computers, we have 0s and 1s as the input, whereas, in quantum computing, we have qubits, the state of the bit in superposition as both 0s and 1s at the same time. Using this superposition bits, the extremely difficult calculations can be cracked by the power of quantum computers. One such extreme case is breaking the cryptography. This is a piece of bad news for government agencies, militaries, and financial institutions. To overcome this problem, researchers are coming with the idea of quantum cryptography, which theoretically acts as a secure solution to the existing cryptography problems.

Advanced driver assistance systems (ADAS) have been developed to automate and modify vehicles for safety and better driving experience. Among all computer vision modules in ADAS, 360-degree surround view generation of immediate surroundings of the vehicle is very important, due to application in on-road traffic assistance, parking assistance etc. This paper presents a novel algorithm for fast and computationally efficient transformation of input fisheye images into required top down view. This paper also presents a generalized framework for generating top down view of images captured by cameras with fish-eye lenses mounted on vehicles, irrespective of pitch or tilt angle. The proposed approach comprises of two major steps, viz. correcting the fish-eye lens images to rectilinear images, and generating top-view perspective of the corrected images. The images captured by the fish-eye lens possess barrel distortion, for which a nonlinear and non-iterative method is used. Thereafter, homography is used to obtain top-down view of corrected images. This paper also targets to develop surroundings of the vehicle for wider distortion less field of view and camera perspective independent top down view, with minimum computation cost which is essential due to limited computation power on vehicles.

We execute the quantum eraser, the Elitzur–Vaidman bomb, and the Hardy’s paradox experiment using high-level programming language on a generic, gate-based superconducting quantum processor made publicly available by IBM. The quantum circuits for these experiments use a mixture of one-qubit and multi-qubit gates and require high entanglement gate accuracy. The results aligned with theoretical predictions of quantum mechanics to high confidence on circuits using up to 3 qubits. The power of quantum computers and high-level language as a platform for experimenting and studying quantum phenomena is henceforth demonstrated.

In order to delineate which minimalistic physical primitives can enable the full power of universal quantum computing, it has been fruitful to consider various measurement based architectures which reduce or eliminate the use of coherent unitary evolution, and also involve operations that are physically natural. In this context previous works had shown that the triplet-singlet measurement of two qubit angular momentum (or equivalently two qubit exchange symmetry) yields the power of quantum computation given access to a few additional different single qubit states or gates. However, Freedman, Hastings and Shokrian-Zini1 recently proposed a remarkable conjecture, called the 'STP=BQP' conjecture, which states that the two-qubit singlet/triplet measurement is quantum computationally universal given only an initial ensemble of maximally mixed single qubits. In this work we prove this conjecture. This provides a method for quantum computing that is fully rotationally symmetric (i.e. reference frame independent), using primitives that are physically very-accessible, naturally resilient to certain forms of error, and provably the simplest possible.

Richard Feynman's observation that certain quantum mechanical effects cannot be simulated efficiently on a computer led to speculation that computation in general could be done more efficiently if it used these quantum effects. This speculation proved justified when Peter Shor described a polynomial time quantum algorithm for factoring intergers. In quantum systems, the computational space increases exponentially with the size of the system, which enables exponential parallelism. This parallelism could lead to exponentially faster quantum algorithms than possible classically. The catch is that accessing the results, which requires measurement, proves tricky and requires new nontraditional programming techniques. The aim of this paper is to guide computer scientists through the barriers that separate quantum computing from conventional computing. We introduce basic principles of quantum mechanics to explain where the power of quantum computers comes from and why it is difficult to harness. We describe quantum cryptography, teleportation, and dense coding. Various approaches to exploiting the power of quantum parallelism are explained. We conclude with a discussion of quantum error correction.

Shor's algorithms for the integer factorization and the discrete logarithm problems can be regarded as a negative effect of the quantum mechanism on publickey cryptography. From the computational point of view, his algorithms illustrate that quantum computation could be more powerful. It is natural to consider that the power of quantum computation could be exploited to withstand even quantum adversaries. Over the last decade, quantum cryptography has been discussed and developed even from the computational complexity-theoretic point of view. In this paper, we will survey what has been studied in quantum computational cryptography.

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i)transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii)transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii)projecting this ancilla with near unity success probability.

The following article has been included in a multiple retraction: Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract In 2015 SAGE were made aware of concerns regarding the Special Issue of Mathematics & Mechanics of Solids on Advances in Canonical Duality Theory, guest-edited by Professor David Gao. At the request of the Guest Editor, the Special Issue has been retracted, due to conflict of interest regarding Professor Gao’s role as Guest Editor and co-author on a number of submitted papers. In addition the peer review process was less rigorous than the journal requires. The Guest Editor takes full responsibility for the retraction. The following articles that were due to appear in the Special Issue have therefore been retracted: Canonical Duality-Triality: Bridge between Nonconvex Analysis/Mechanics and Global Optimization in complex systems; David Y Gao, Ning Ruan, and Vittorio Latorre. http://mms.sagepub.com/content/early/2015/02/24/1081286514566533.abstract Canonical Dual Approach for Contact Mechanics Problems with Friction; Vittorio Latorre, Simone Sagratella, David Y Gao. http://mms.sagepub.com/content/early/2015/01/20/1081286514566534.abstract Canonical Duality Theory for Solving Non-Monotone Variational Inequality Problems; Guoshan Liu, David Y Gao, Shouyang Wang. http://mms.sagepub.com/content/early/2015/02/04/1081286514566535.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part I; Shu-Cherng Fang, David Y Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wen-Xun Xing. http://mms.sagepub.com/content/early/2015/02/24/1081286514566704.abstract Double Well Potential Function and Its Optimization in The n-dimensional Real Space. Part II; Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. http://mms.sagepub.com/content/early/2015/02/09/1081286514566723.abstract Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant-Kirchhoff Material; David Y Gao and E. Hajilarov. http://mms.sagepub.com/content/early/2015/07/06/1081286515591084.abstract Triality Theory and Complete Post-buckling Solutions of Large Deformed Beam by Canonical Dual Finite Element Method; Kun Cai, David Y Gao, Qinghua Qin. http://mms.sagepub.com/content/early/2015/06/28/1081286515591085.abstract Global Solutions to Spherically Constrained Quadratic Minimization via Canonical Duality Theory; Yi Chen, David Y Gao. http://mms.sagepub.com/content/early/2015/04/08/1081286515577122.abstract Unified Canonical Duality Methodology for Global Optimization; Vittorio Latorre, David Y Gao and N. Ruan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591305.abstract A Framework of Canonical Dual Algorithms for Global Optimization; Xiaojun Zhou, David Y Gao, Chunhua Yang. http://mms.sagepub.com/content/early/2015/07/22/1081286515592190.abstract Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems; Ning Ruan, David Y Gao. http://mms.sagepub.com/content/early/2015/07/08/1081286515591087.abstract Global Optimization Solutions to a Class of Non-convex Quadratic Minimization Problems with Quadratic Constraints; Yubo Yuan. http://mms.sagepub.com/content/early/2015/07/06/1081286515591086.abstract On Minimal Distance between Two Non-Convex Surfaces; Daniel Morales-Silva, David Y Gao. http://mms.sagepub.com/content/early/2015/07/27/1081286515592949.abstract The Editor-in-Chief and SAGE strive to uphold the very highest standards of publication ethics and are committed to supporting the high standards of integrity of Mathematics & Mechanics of Solids. Authors, reviewers, editors and interested readers are encouraged to consult SAGE’s ethics statements and the Committee on Publication Ethics (COPE) website for guidelines on publication ethics.