We present a computational framework based on geometric structures. No quantum mechanics is involved, and yet the algorithms perform tasks analogous to quantum computation. Tensor products and entangled states are not needed -- they are replaced by sets of basic shapes. To test the formalism we solve in geometric terms the Deutsch-Jozsa problem, historically the first example that demonstrated the potential power of quantum computation. Each step of the algorithm has a clear geometric interpetation and allows for a cartoon representation.
We present a brief survey of results where quantum information processing is useful to solve distributed computation tasks. We describe problems that are impossible to solve using classical resources but that become feasible with the help of quantum mechanics. We also give examples where the use of quantum information significantly reduces the need for communication. The main focus of the survey is on communication complexity but we also address other distributed tasks.
One of the fundamental theories of physics is that of quantum mechanics. Quantum mechanics tries to explain the inconsistencies in the behaviors of systems at the macro and micro scales. Quantum mechanics paved the way for quantum computing based on qubits. The existence of quantum computers up to 65 qubits is known. The advent of quantum computers weakens the security of many cryptographic algorithms. In this paper, we investigate quantum computing-based attacks and shed light on possible future developments.
Quantum Computing is a new and exciting field at the intersection of mathematics, computer science and physics. It concerns a utilization of quantum mechanics to improve the efficiency of computation. Here we present a gentle introduction to some of the ideas in quantum computing. The paper begins by motivating the central ideas of quantum mechanics and quantum computation with simple toy models. From there we move on to a formal presentation of the small fraction of (finite dimensional) quantum mechanics that we will need for basic quantum computation. Central notions of quantum architecture …
Since the introduction of quantum mechanics, it has been taught mostly as a theoretical subject. It is also viewed as a theory that provides a best understanding of the nature, but which does not have much practical applications in our day to day life. This notion has been considerably changed in the recent past with the advent of quantum information processing. Further, recently, IBM has introduced a set of quantum computers that are placed in the cloud and can be accessed freely from your class room though your mobile phones, PCs, Laptops having an internet connection. In this article, we wi…
Recent works have independently suggested that Quantum Mechanics might permit for procedures that transcend the power of Turing Machines as well as of `standard' Quantum Computers. These approaches rely on and indicate that Quantum Mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating 1) its principal computing capabilities from 2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to `existence' in mathematical log…
The quantum nature of bulk ensemble NMR quantum computing --the center of recent heated debate, is addressed. Concepts of the mixed state and entanglement are examined, and the data in a 2 qubit liquid NMR quantum computation are analyzed. It is pointed out that the key problem in the current debate is the understanding of entanglement in a mixed state system. The following points are concluded in this Letter: 1)Density matrix describes the "state" of an average particle in an ensemble. It can not describe the state of an individual particle in an ensemble in detail; 2) Entanglement is a prope…
We discuss the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler-Lagrange equations imply a fourth-order nonlinear partial differential equa…
A quantum computer is a machine that can perform certain calculations much faster than a classical computer by using the laws of quantum mechanics. Quantum computers do not exist yet, because it is extremely difficult to control quantum mechanical systems to the necessary degree. What is more, we do at this moment not know which physical system is the best suited for making a quantum computer (although we have some ideas). It is likely that a mature quantum information processing technology will use (among others) light, because photons are ideal carriers for quantum information. These notes a…
Much attention has been drawn to quantum computing and the exponential speed-up in computation the technology would be able to provide. Various claims have been made about what aspect of quantum mechanics causes this speed-up. Formulations of quantum computing have traditionally been made in orthodox (Copenhagen) and sometimes many-worlds quantum mechanics. We will aim to understand quantum computing in terms of de Broglie-Bohm Pilot-Wave Theory by considering different simple systems that may function as a basic quantum computer. We will provide a careful discussion of Pilot-Wave Theory and e…
Stochastic realization of the wave function in quantum mechanics, with the inclusion of soliton representation of extended particles, is discussed. The concept of Stochastic Qubits is used for quantum computing modeling.
Molecular science is governed by the dynamics of electrons, atomic nuclei, and their interaction with electromagnetic fields. A reliable physicochemical understanding of these processes is crucial for the design and synthesis of chemicals and materials of economic value. Although some problems in this field are adequately addressed by classical mechanics, many require an explicit quantum mechanical description. Such quantum problems represented by exponentially large wave function should naturally benefit from quantum computation on a number of logical qubits that scales only linearly with sys…
Quantum computing is an emerging paradigm that opens a new era for exponential computational speedup. Still, quantum computers have yet to be ready for commercial use. However, it is essential to train and qualify today the workforce that will develop quantum acceleration solutions to get the quantum advantage in the future. This tutorial gives a broad view of quantum computing, abstracting most of the mathematical formalism and proposing a hands-on with the quantum programming language Ket. The target audience is undergraduate and graduate students starting in quantum computing -- no prerequi…
Since Deutsch (1985), quantum computers have been modeled exclusively in the language of state vectors and the Schroedinger equation. We present a complementary view of quantum circuits inspired by the path integral formalism of quantum mechanics, and examine its application to some simple textbook problems.
This paper investigates a variety of unconventional quantum computation devices, including fermionic quantum computers and computers that exploit nonlinear quantum mechanics. It is shown that unconventional quantum computing devices can in principle compute some quantities more rapidly than `conventional' quantum computers.
Faster algorithms, novel cryptographic mechanisms, and alternative methods of communication become possible when the model underlying information and computation changes from a classical mechanical model to a quantum mechanical one. Quantum algorithms perform a select set of tasks vastly more efficiently than any classical algorithm, but for many tasks it has been proved that quantum algorithms provide no advantage. The breadth of quantum computing applications is still being explored. Major application areas include security and the many fields that would benefit from efficient quantum simula…
A finite dimensional quantum mechanical system is modeled by a density rho, a trace one, positive semi-definite matrix on a suitable tensor product space H[N] . For the system to demonstrate experimentally certain non-classical behavior, rho cannot be in S, a closed convex set of densities whose extreme points have a specificed tensor product form. Two mathematical problems in the quantum computing literature arise from this context: (1) the determination whether a given rho is in S and (2) a measure of the ``entanglement'' of such a rho in terms of its distance from S. In this paper we descri…
Recent developments in quantum computing suggest that it could be possible to make conditional changes to the state of a quantum mechanical system without resorting to classical observation. It is accomplished through collective response of atoms comprising a lattice for the system and involves relative phase adjustments only. We exploit this possibility and describe experimental designs that could help elucidate quantum mechanical properties of these systems and distinguish between interpretations.
Quantum computing exploits the quantum-mechanical nature of matter to exist in multiple possible states simultaneously. This new approach promises to revolutionize the present form of computing. As an approach to quantum computing, we discuss ultrafast laser pulse shaping, in particular, the acousto-optic modulator based Fourier-Transform pulse-shaper, which has the ability to modulate tunable high power ultrafast laser pulses. We show that optical pulse shaping is an attractive route to quantum computing since shaped pulses can be transmitted over optical hardware and the same infrastructure …
Spatially Dense, Continuous-Variable Quantum Computing with Solid State Spin Nonlinearities
Massachusetts Institute of Technology（麻省理工学院）；Institute for Soldier Nanotechnologies, Massachusetts Institute of Technology（麻省理工学院士兵纳米技术研究所）；Research Laboratory of Electronics, Massachusetts Institute of Technology（麻省理工学院电子研究实验室）；DEVCOM, Army Research Laboratory（美国陆军研究实验室）
From the philosopher's perspective, the interest in quantum computation stems primarily from the way that it combines fundamental concepts from two distinct sciences: physics (especially quantum mechanics) and computer science, each long a subject of philosophical speculation and analysis in its own right. Quantum computing combines both of these more traditional areas of inquiry into one wholly new (if not quite independent) science. There are philosophical questions that arise from this merger, and philosophical lessons to be learned. Over the course of this chapter we discuss what I take to…
Information science is entering into a new era in which certain subtleties of quantum mechanics enables large enhancements in computational efficiency and communication security. Naturally, precise control of quantum systems required for the implementation of quantum information processing protocols implies potential breakthoughs in other sciences and technologies. We discuss recent developments in quantum control in optical systems and their applications in metrology and imaging.
The well known interpretational difficulties with nonlinear Schrödinger and von Neumann equations can be reduced to the problem of computing multiple-time correlation functions in the absence of Heisenberg picture. Having no Heisenberg picture one often resorts to Zeno-type reasoning which explicitly involves the projection postulate as a means of computing conditional and joint probabilities. Although the method works well in linear quantum mechanics, it completely fails for nonlinear evolutions. We propose an alternative way of performing the same task in linear quantum mechanics and show th…
Matrix quantum mechanics plays various important roles in theoretical physics, such as a holographic description of quantum black holes. Understanding quantum black holes and the role of entanglement in a holographic setup is of paramount importance for the development of better quantum algorithms (quantum error correction codes) and for the realization of a quantum theory of gravity. Quantum computing and deep learning offer us potentially useful approaches to study the dynamics of matrix quantum mechanics. In this paper we perform a systematic survey for quantum computing and deep learning a…
We show that differently constructed ensembles having the same density matrix may be physically distinguished by observing fluctuations of some observables. An explicit expression for fluctuations of an observable in an ensemble is given. This result challenges Peres's fundamental postulate and seems to be contrary to the widely-spread belief that ensembles with the same density matrix are physically identical. This leads us to suggest that the current liquid NMR quantum computing is truly quantum-mechanical in nature.
The Quantum Internet is envisioned as the final stage of the quantum revolution, opening fundamentally new communications and computing capabilities, including the distributed quantum computing. But the Quantum Internet is governed by the laws of quantum mechanics. Phenomena with no counterpart in classical networks, such as no-cloning, quantum measurement, entanglement and teleporting, impose very challenging constraints for the network design. Specifically, classical network functionalities, ranging from error-control mechanisms to overhead-control strategies, are based on the assumption tha…
The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarise not just quantum computing, but the whole subject of quantum information theory. It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, the review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then …
The aim of the present paper is twofold. First, to give the main ideas behind quantum computingand quantum information, a field based on quantum-mechanical phenomena. Therefore, a shortreview is devoted to (i) quantum bits or qubits (and more generally qudits), the analogues of theusual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantummechanics, namely, linearity (which manifests itself through the superposition of qubits and theaction of unitary operators on qubits) and entanglement of certain multi-qubit states (a resourcethat is specific to quantum…
We clarify different definitions of the density matrix by proposing the use of different names, the full density matrix for a single-closed quantum system, the compressed density matrix for the averaged single molecule state from an ensemble of molecules, and the reduced density matrix for a part of an entangled quantum system, respectively. We show that ensembles with the same compressed density matrix can be physically distinguished by observing fluctuations of various observables. This is in contrast to a general belief that ensembles with the same compressed density matrix are identical. E…
An intense effort is being made today to build a quantum computer. Instead of presenting what has been achieved, I invoke here analogies from the history of science in an attempt to glimpse what the future might hold. Quantum computing is possible in principle - there are no known laws of Nature that prevent it - yet scaling up the few qubits demonstrated so far has proven to be exceedingly difficult. While this could be regarded merely as a technological or practical impediment, I argue that this difficulty might be a symptom of new laws of physics waiting to be discovered. I also introduce a…
Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the n…
Quantum image processing employs quantum computing to capture, manipulate, and recover images in various formats. This requires representations of encoded images using the quantum mechanical composition of any potential computing hardware. In this study, a quantum hue, saturation, and lightness (QHSL) color model is proposed to organize and conceptualize color-assignment attributes using the properties of quantum mechanics (i.e., entanglement and parallelism). The proposed color model is used to define representations of a two-dimensional QHSL image and investigate its data storage, chromatic …
This preprint contains a detailed Preface to Proceedinngs of the International Conference ``Foundations of Probability and Physics-3'' held in Växjö, Sweden, 7-12 June 2004; table of contents and round table. The main theme of the round table was {\it ``Fundamental problems in quantum mechanics, probabilistic description of reality, and quantum information.''} The topics that were specifically discussed were that of Quantum Cryptography, Quantum computing, and Quantum Macroscopic Structures. For each of these topics, the participants were asked to discuss which are the crucial Quantum features…
In this paper, we examine the state art of quantum computing and analyze its potential effects in scientific computing and cybersecurity. Additionally, a non-technical description of the mechanics of the listed form of computing is provided to educate the reader for better understanding of the arguments provided. The purpose of this study is not only to increase awareness in this nescient technology, but also serve as a general reference guide for any individual wishing to study other applications of quantum computing in areas that include finance, chemistry, and data science. Lastly, an educa…
Following the B. Hiley belief that unresolved problems of conventional quantum mechanics could be the result of a wrong mathematical structure, an alternative basic structure is suggested. Critical part of the structure is modification of the sense of commonly used terms state, observable, measurement giving them a clear unambiguous definition. This concrete definition, along with using of variable complex plane, is quite natural in geometric algebra terms. It helps to establish a feasible language for the area of quantum computing. The suggested approach is used then in the fiber optics quant…
Richard Feynman's observation that quantum mechanical effects could not be simulated efficiently on a computer led to speculation that computation in general could be done more efficiently if it used quantum effects. This speculation appeared justified when Peter Shor described a polynomial time quantum algorithm for factoring integers. 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 …
We present a number of quantum computing patterns that build on top of fundamental algorithms, that can be applied to solving concrete, NP-hard problems. In particular, we introduce the concept of a quantum dictionary as a summation of multiple patterns and algorithms, and show how it can be applied in the context of Quadratic Unconstrained Binary Optimization (QUBO) problems. We start by presenting a visual approach to quantum computing, which avoids a heavy-reliance on quantum mechanics, linear algebra, or complex mathematical notation, and favors geometrical intuition and computing paradigm…
