A real quantum computer theoretically can perform a large number of parallel processing at the same time by superposition states and unitary transformations in a Hilbert space. We show that arbitrary unitary transformation can be operated in a framework of classic physics such as optics. We also show that, using the optics, the same processing as what quantum computing logic does can be implemented and that image recognition can be done by that logic. Although our system currently does not utilize entanglement physics like a quantum computer using electronic spin system, we can construct a high dimensional Hilbert space (e.g. one thousand dimension). We consider that our system may be another approach to realize quantum computing paradigm.
A real quantum computer theoretically can perform a large number of parallel processing at the same time by superposition states and unitary transformations in a Hilbert space. We show that arbitrary unitary transformation can be operated in a framework of classic physics such as optics. We also show that, using the optics, the same processing as what quantum computing logic does can be implemented and that image recognition can be done by that logic. Although our system currently does not utilize entanglement physics like a quantum computer using electronic spin system, we can construct a high dimensional Hilbert space (e.g. one thousand dimension). We consider that our system may be another approach to realize quantum computing paradigm.
Summary form only given. In quantum cryptography, non-cloning properties of quantum states are exploited to achieve quantum security. At the same time, typically one optical mode is used, for instance that of a single-mode optical fiber, effectively reducing the problem to a one-dimensional system. Only recently have researchers begun to exploit higher dimensions for quantum cryptography [1]. High dimensional spaces appear naturally in imaging applications.We here experimentally investigate the question if it is possible to recognize an arbitrary pattern in a physical object using much fewer photons than the complexity of the pattern. Theoretically it has already been proposed to use a quantum computer for pattern recognition and that this can be achieved with very few photons [2]. To avoid using a quantum computer, we exploit a spatial light modulator, which can map a single incoming optical mode on thousands outgoing optical modes in a programmable way. In this way, a weak light pulse containing only a few or even a single photon can be distributed over the high number of pixels of a complex 2-dimensional image. The inverse is just as feasible: a known complex pattern can be mapped onto a single mode. We use this principle to analyze the light from a physical object when illuminated with, e.g., a plane wave. After reflection of or transmission through the object under investigation, the response can be projected onto a single mode. The signal in that mode can be analyzed with the standard repertoire of quantum optical measurements, determining, e.g., the field amplitude, the intensity or the number of photons.We use coherent pulses of light which have mean photon numbers much smaller than the thousands of modes of the complex image. We introduce the parameter S=K/n as the ratio of the number of modes, K, of the response field pattern and the number of photons, n, that is used for the illumination. This measurement is in the quantum domain if S>>1. We read out objects with S=4, well in the quantum regime. The method can be extended by preshaping the illumination light, with applications in authentication of physical objects [3,4].
Summary form only given, as follows. Gives a system theoretic approach to the concept of quantum computing. A quantum computer is envisaged to be a system of quantum circuits, acting on a state space, which is a finite-say 2/sup n/-dimensional-complex Hilbert space. The circuitry is a sequence of unitary transforms U/sub t/ /spl epsiv/ SU(2/sup n/) followed by a measurement. These transforms, so-called quantum gates, are controlled by a classical computer, usually especially electromagnetic fields. The state space of a quantum computer has the structure of a Hermitian vector space. Thus it allows "simultaneous" superposition of orthogonal basis states (corresponding to classical states) with the possibility of constructive and destructive interference between different paths of computation. This principle allows the usage of so-called entangled states by preparing the superposition of special "bent" configurations of basis vectors in the product Hilbert space, e.g., as they are known from error control and cryptography. This latter "entanglement" not only promises to make a quantum computer much more powerful than a probabilistic one, but it also allows a method of parallelism. The reason for this lies not only in the fact that the 2/sup n/ dimensional Hilbert space is the n-fold tensor product of 2-dimensional spaces as it is given by spin-1/2-particles, like photons, representing a quantum bit. First of all, we shall address the fact that the power of quantum computing lies in the properties of entangled states as opposed to those of separable states which essentially would resemble classical circuits without giving the feature of exponential speed-up. We shall briefly describe possible physical realisations of quantum (optic, electronic, magnetic,...) devices allowing the engineering of Hamiltonians needed for quantum systems. Basic quantum gates, especially the quantum Horner-Toffoli gate will be discussed to display the principles and methodology for designing quantum circuits. An outlook towards the possible availability of nano- and mesoscopic technologies supporting this new architecture in future generations concludes based on a speculative worst/best case forecast of possible applications, e.g., in public key and conventional cryptography.
Hilbert spaces provide the fundamental mathematical framework for describing quantum mechanical systems. Their structure, characterized by an inner product and completeness, allows for the representation of quantum states as vectors and physical observables as self-adjoint operators. Key quantum phenomena such as superposition and entanglement find natural expression within this formalism. Superposition, where a quantum system can exist in multiple states simultaneously, is represented by linear combinations of basis vectors in the Hilbert space. Entanglement, a non-classical correlation between quantum systems, is described by non-separable state vectors in a tensor product of Hilbert spaces. These concepts are pivotal in quantum computing, where the unit of information, the qubit, is a two-level quantum system whose state is a vector in a two-dimensional complex Hilbert space (ℂ²). Quantum gates, which perform operations on qubits, are represented by unitary operators acting on these state vectors. The power of quantum computation, particularly in algorithms like Shor's or Grover's, stems from the ability to exploit superposition and entanglement, processes intrinsically described within the Hilbert space framework. Thus, a thorough understanding of Hilbert spaces is indispensable for grasping the principles of quantum mechanics and for advancing the field of quantum information and computation [1, 7]. The transition from classical bits to quantum qubits, and from classical logic gates to quantum unitary operations, is entirely predicated on the mathematical properties endowed by Hilbert spaces.
A quantum associative memory, much more natural than those of “quantum computers”, is presented. Neural-net-like processing with real-valued variables is transformed into processing with quantum waves. Successful computer simulations of image storage and retrieval are reported. Our Hopfield-like algorithm allows quantum implementation with holographic procedure using present-day quantum-optics techniques. This brings many advantages over classical Hopfield neural nets and quantum computers with logic gates.
Quantum Image Processing has exploded in recent years with dozens of papers trying to take advantage of quantum parallelism in order to offer a better alternative to how current computers are dealing with digital images. The vast majority of these papers define or make use of quantum representations based on very large superposition states spanning as many terms as there are pixels in the image they try to represent. While such a representation may apparently offer an advantage in terms of space (number of qubits used) and speed of processing (due to quantum parallelism), it also harbors a fundamental flaw: only one pixel can be recovered from the quantum representation of the entire image, and even that one is obtained non-deterministically through a measurement operation applied on the superposition state. We investigate in detail this measurement bottleneck problem by looking at the number of copies of the quantum representation that are necessary in order to recover various fractions of the original image. The results clearly show that any potential advantage a quantum representation might bring with respect to a classical one is paid for dearly with the huge amount of resources (space and time) required by a quantum approach to image processing.