To begin with, we need a quantum register made of many qubits to store information. Recall that a classical computer also requires memory to store information. The simplest way to realize a qubit physically is to use a two-level quantum system. For example, an electron, a spin 1/2 nucleus or two mutually orthogonal polarization states (horizontal and vertical, for example) of a single photon can be a qubit. We may also employ a two-dimensional subspace, such as the ground state and the first excited state, of a multi-dimensional Hilbert space, such as atomic energy levels. In the latter case, special care must be taken to avoid leakage of the state to the other part of the Hilbert space. In any case, the two states are identified as the basis vectors, |0〉 and |1〉, of the Hilbert space so that a general single qubit state takes the form |ψ〉 = α|0〉 + β|1〉, where |α|2 + |β|2 = 1. A multiqubit state is expanded in terms of the tensor products of these basis vectors. Each qubit must be separately addressable. Moreover it should be scalable up to a large number of qubits. The two-dimensional vector space of a qubit may be extended to be three-dimensional (qutrit) or, more generally, d-dimensional (qudit). A system may be made of several different kinds of qubits. Qubits in an ion trap quantum computer, for instance, may be defined as: (1) hyperfine/Zeeman sublevels in the electronic ground state of ions (2) a ground state and an excited state of a weakly allowed optical transition and (3) normal mode of ion oscillation. A similar scenario is also proposed for Josephson junction qubits, in which two flux qubits are coupled through a quantized LC circuit. Simultaneous usage of several types of qubits may be the most promising way to achieve a viable quantum computer.
To begin with, we need a quantum register made of many qubits to store information. Recall that a classical computer also requires memory to store information. The simplest way to realize a qubit physically is to use a two-level quantum system. For example, an electron, a spin 1/2 nucleus or two mutually orthogonal polarization states (horizontal and vertical, for example) of a single photon can be a qubit. We may also employ a two-dimensional subspace, such as the ground state and the first excited state, of a multi-dimensional Hilbert space, such as atomic energy levels. In the latter case, special care must be taken to avoid leakage of the state to the other part of the Hilbert space. In any case, the two states are identified as the basis vectors, |0〉 and |1〉, of the Hilbert space so that a general single qubit state takes the form |ψ〉 = α|0〉 + β|1〉, where |α|2 + |β|2 = 1. A multiqubit state is expanded in terms of the tensor products of these basis vectors. Each qubit must be separately addressable. Moreover it should be scalable up to a large number of qubits. The two-dimensional vector space of a qubit may be extended to be three-dimensional (qutrit) or, more generally, d-dimensional (qudit). A system may be made of several different kinds of qubits. Qubits in an ion trap quantum computer, for instance, may be defined as: (1) hyperfine/Zeeman sublevels in the electronic ground state of ions (2) a ground state and an excited state of a weakly allowed optical transition and (3) normal mode of ion oscillation. A similar scenario is also proposed for Josephson junction qubits, in which two flux qubits are coupled through a quantized LC circuit. Simultaneous usage of several types of qubits may be the most promising way to achieve a viable quantum computer.
Quantum computing operates in three stages: (i) preparation of the initial states of the n qubits of a register, (ii) step by step transformation of the state of this register by unitary operators which compose the quantum program, and (iii), measurement of all or some of the qubits of the register. The existence of quantum algorithms which are exponentially less complex than their classical counterparts for some classes of problems, stems from entangled states established by multi-qubit unitary operators within the quantum program. A register of n qubits is a quantum system composed of n quantum subsystems. The state |/spl psi/> of a quantum system composed of two quantum subsystems A and B is said to be entangled when |/spl psi/> is not reducible to a pair composed of a state |/spl psi/> of A and a state |/spl psi//sub B/> of B: such situations have no counterpart in the classical world. In quantum theory, such a pair of states is denoted by a tensor product: |/spl psi/> is entangled if it cannot be factorized into the tensor product |/spl psi//sub A/>/spl ominus/|/spl psi//sub B/>. This paper establishes conditions according to which it is possible to tell whether or not the state of a register of n qubits is entangled. The state of a single qubit is a vector /spl alpha/|0>+/spl beta/|1> of unit norm in a 2-dimensional vector space, where |0> and |1> are the two basis states and where a and are complex amplitudes. Then, if both A and B are qubits, the most general form of the state of a register composed of the 2 qubits A and B is also a vector of unit norm, but now in a 4-dimensional space: |/spl psi/>=/spl alpha/|00>+/spl beta/|01+/spl gamma/|10>+/spl sigma/|11>. It is straightforward to prove that |/spl psi/> can be factorized into |/spl psi//sub A/>/spl ominus/|/spl psi//sub B/> if and only if /spl alpha//spl delta/=/spl beta//spl gamma/. In such a case, |/spl psi/> is said to be separable, i.e. not entangled. This paper generalizes this form of condition to registers of n qubits. If |/spl psi/> is the state of a register of n qubits, two different questions about the separability of |/spl psi/> are answered: (i) is |/spl psi/> separable into a product |/spl psi//sub 1/>/spl ominus/|/spl psi//sub 2/>/spl ominus/L/spl ominus/|/spl psi//sub n/> of n single qubit states? and (ii), is |/spl psi/> separable into a product |/spl psi//sub A/>/spl ominus/|/spl psi//sub B/> of the states of two subregisters A and B, respectively containing p and q adjacent qubits with p+q=n? For both questions, necessary and sufficient conditions are given.
The elementary particle of information used by modern digital computers is the bit—a register or memory element that can be in one of two distinct states, 0 or 1. But we live in a quantum world, and one can design computers in which each elementary unit of information is a quantum bit, or qubit, which can be in any superposition of two quantum states, |0〉 and |1〉. A quantum computer built with n such components could itself be in a superposition of 2n distinct states, each splinter of the superposition performing its own computation in parallel with all the rest.
A mathematical quantisation of a Random Access Memory (RAM) is proposed starting from its matrix representation. This quantum RAM (q-RAM) is employed as the neural unit of q-RAM-based Neural Networks, q-RbNN, which can be seen as the quantisation of the corresponding RAM-based ones. The models proposed here are direct realisable in quantum circuits, have a natural adaptation of the classical learning algorithms and physical feasibility of quantum learning in contrast to what has been proposed in the literature. 1 Quantum computation and Mathematical Quantisation Quantum computing [14] was originally proposed by Richard Feynman [7] in the 1980s and had its formalisation with David Deutsch which proposed the quantum Turing machine [4]. Quantum Computing has been popularised through the quantum circuit model [5] which is a quantisation [15] of the classical boolean circuit model of computation. Quantum computer also became a potential parallel device for improving the computational efficiency of neural networks [6]. The quantum information unit is the quantum bit or ”qubit”. A very intuitive view of the quantisation procedure is put forward by Nik Weaver in the Preface of his book Mathematical Quantization [15] with just a phrase which says it all: ”The fundamental idea of mathematical quantisation is sets are replaced with Hilbert spaces”. The quantisation of the boolean circuit logic starts by simply embedding the the classical bits {0, 1} in a convenient Hilbert space. The natural way of doing this is to represent them as (orthonormal) basis of a Complex Hilbert space. In this context these basis elements are called the computational-basis states.Linear combinations (from Linear Algebra [9]) of the basis spans the whole space whose elements, called states, are said to be in superposition. Any basis can be used (recall from Linear Algebra [9] that there usually are many!). But in Quantum Computing it is customary to use the most conventional and well known one: |0〉, |1〉 are a pair of orthonormal basis vectors representing each classical bit, or ”cbit”, as column vector, |0〉 = [1 0]T and |1〉 = [0 1]T. A general state of the system (a vector) can be written as: |ψ〉 = α |0〉+β |1〉, where α, β are complex coefficients (called probability amplitudes) constrained by the normalization condition: |α| + |β| = 1. This is the model of one qubit. Multiple qubits are obtained via tensor products. By linearity we just need to say how tensor behaves on the basis: |i〉 ⊗ |j〉 = |i〉 |j〉 = |ij〉 , where i, j ∈ {0, 1} If that were all the story, quantum computing would be just a trivial extension of classical computing. But Quantum Mechanics Principles [14] restrain the kind of permissible operations. Operations on qubits are carried out only by unitary operators (i.e. ∗Supported by MCT-CNPq and PRONEX/FACEPE. On sabbatical leave at the School of Computer Science, University of Birmingham ESANN'2009 proceedings, European Symposium on Artificial Neural Networks Advances in Computational Intelligence and Learning. Bruges (Belgium), 22-24 April 2009, d-side publi., ISBN 2-930307-09-9.
The quantum world is fascinating. It presents a description of nature that defies our most rooted concepts about what reality is. For example, quantum objects possess \lq\lq spooky\rq\rq\ properties that allow them to be in multiple places at the same time, to move in different directions simultaneously, or to exist and not to exist. In other words, to live several parallel stories. Making use of these parallel stories to compute is the realm of quantum computation, a world in which information is processed using the laws of quantum physics. Nowadays, quantum computers are an object of an extensive experimental and theoretical research, since it is known that, on paper, these can solve many of the current computational challenges. These are the problems that even the most sophisticated supercomputers of today, or tomorrow, will not be able to solve.
Holonomic quantum computation (HQC) realizes quantum gates through non-Abelian geometric phases, providing an experimentally accessible approach to quantum control. While the nonadiabatic HQC framework has been extensively developed for three-level Λ systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a d-pod configuration, where a single excited state is coupled to d ground states, the latter forming the computational subspace. This scheme enables universal holonomic single- and two-qudit gates using only optical or microwave pulses on trapped atoms or ions, offering an efficient route to implement a discrete universal gate set with minimal pulse coordination. As an explicit example, we analyze in detail the qutrit (d=3) case, demonstrating compact realizations of single- and two-qutrit holonomic gates, each gate requiring at most two loops in the Grassmannian generated by at most three pulses.
As we have seen, even a single qubit is a surprisingly interesting object. However, the real power of quantum information processing begins with systems of two or more qubits. Before studying these in detail we need to expand our notation a little. Consider a system of two qubits, labeled a and b , each of which has two basis states, |0⧽ and |1⧽. The whole system then has four basis states, which can be written as |0 a 0 b ⧽, |0 a 1 b ⧽, |1 a 0 b ⧽ and |1 a 1 b ⧽, and can be found in any general superposition of these states, so that it occupies a four-dimensional Hilbert space. In the same way, a system of three qubits inhabits an eight-dimensional Hilbert space, and so on. This exponential increase in the size of the Hilbert space with a linear increase in the number of qubits underlies the power of quantum computers. Direct products The size of the Hilbert spaces involved can also be a huge problem, however, making it difficult to describe states of systems with many qubits. A partial solution is to note that some states can be described in a simpler way, using the concept of direct products . These states, in which the individual qubits can in principle be discussed separately, make up a tiny minority of the states accessible to a multi-qubit system, but include many important states, most notably the basis states. States of this kind are said to be separable , and states which are not separable are said to be entangled . Entangled states are much more interesting than separable ones, but it is wise to begin with the simpler case.