Page 79 - Nanotechnology an introduction

180°. On the other hand, what might be called corresponds to rotation by 90°. This operation, the square root of inversion, has no classical equivalent. Two successive applications of correspond to classical
inversion, NOT.
The key features of quantum objects of interest for computational purposes are superposition—an object can be in several different states
simultaneously [42]—and entanglement [6]. Operations can be carried out internally, maintaining superposition, which is only destroyed at the very
end of the computation when a single output is required. The system must, however, be kept isolated from its environment during operations.
Entanglement with the environment implies decoherence and loss of useful information within the computer. It is easiest to avoid in an isolated
small system, hence the interest in realizing quantum computers using nanotechnology.
The architecture of computers needs to be wholly reconceived in order to exploit the peculiarities of quantum mechanics, which means in particular
that a particle can exist in two states simultaneously, whereas a cluster of electrons (physically instantiating a bit) in a conventional computer
represents either zero or one. The value of a qubit, on the other hand, which might be physically instantiated as a single electron localized on a
quantum dot (cf. Section 7.4.5), depends on its position relative to other electrons. For example, two electrons can exist in four different states—00,
01, 10, and 11—depending on their relative positions. If the electrons interact (are entangled) with each other, then any operation carried out on
one electron will simultaneously be carried out on the other—implying that one operation is carried out on four different states at the same time.
Hence a computer with just 32 bits could perform more than a thousand million operations simultaneously.
The physical embodiment of a bit of information—called a qubit in quantum computation—can be any absolutely small object capable of
possessing the two logic states 0 and 1 in superposition, e.g., an electron, a photon or an atom. Electron spin is an attractive attribute for quantum
computation. Qubits have also been installed in the energy states of an ion, or in the nuclear spins of atoms. A single photon polarized horizontally
(H) could encode the state |0〉 (using the Dirac notation) and polarized vertically (V) could encode the state |1〉 (Figure 7.8, upper left). The photon
2
2
can exist in an arbitrary superposition of these two states, represented as α|H〉 + β|〉, with |α| + |β| = 1. Any polarization can be represented on a
Poincaré sphere (Figure 7.8, lower left). The states can be manipulated using birefringent waveplates (Figure 7.8, upper right), and polarizing
beamsplitters are available for converting polarization to spacial location (Figure 7.8, lower right). With such common optical components, logic
gates can be constructed.

Figure 7.8 (Left) optically-encoded qubits. Various combinations of H(orizontal) and V(ertical) polarizations—D(iagonal), A(ntidiagonal), L(eft-circular diagonal), R(ight-circular diagonal)—are represented on the Poincaré sphere. (Right) a
birefringent waveplate is used to construct a logic gate, and birefringent waveplates and polarizing beamsplitters can spacially separate photons according to the H (≡ 0) and V (≡ 1) polarizations. Reproduced with permission from [134].
One of the biggest problems with current supercomputers is energy dissipation. They require tens of kilowatts of energy to run and generate vast
amounts of heat. Landauer showed in 1961 that almost all operations required in computation can be performed reversibly, thus dissipating no
heat. Reversible computation is possible on a quantum computer.
7.4. Electronic Devices

Electronic devices (such as the archetypical field effect transistor, Figure 7.6), depend on the movement of elementary packets of charge (i.e.,
electrons), carrying their electrostatic charge; that is, current flows.

7.4.1. Ballistic Transport

Even a simple electrical wire ranks as a device; a transistor (from several of which a logic gate can be constructed, and hence a digital computer)
can be thought of as based on a wire whose resistance depends on an external input. In an ideal binary device, the resistance is either infinite (i.e.,
the conductance is zero) or has some value R; for a piece of material of length l and cross-sectional area A it is given by

(7.1)
where ρ is the resistivity, which depends on the material properties:

(7.2)
where S is the Fermi surface area. Note in particular the inverse dependence on mean free path, ℓ. As is well known, an electron moving in a
F
perfect lattice experiences no resistance whatsoever; but lattice defects (e.g., impurities) and the thermal fluctuations inevitably present at any
temperature above absolute zero result in an effective mean free path of several tens of nanometers. But if a wire is shorter than a critical length l =
c
2
ℓ the resistance becomes ballistic, with a value of h/(2e ) = 25.75 kΩ per sub-band, independent of material parameters.Therefore, any device with
a characteristic length smaller than ℓ could justifiably be called a nanodevice.
7.4.2. Depletion Layers

The operation of an electronic device typically depends on junctions, including internal junctions between different kinds of materials and junctions