p2

He showed that his algorithm rises only polynomially so that a 500-digit number takes only eight times as
many computational steps to factor as a 250-digit number. And by using the quantum factoring
algorithm, a 250-digit number requires only ≈ 5×10 10 steps or < 1 second to factor at terahertz speed, so
that a 500-digit number will take ≤ 1 second to factor. No classical polynomial-time algorithm for this
problem exists at present. This breakthrough generated a cottage industry of research into quantum
computing and quantum information theory.
IBM (2001) constructed a prototype quantum computer that uses the nuclear spins of seven atoms that
are part of a large molecule with the iron-based chemical composition H 5 C 11 O 2 F 5 Fe. The computer uses
entangled nuclear spins for storage and has a capacity of seven qubits (qubits are defined in the bulleted
list in the next two paragraphs below). All of the Fluorine atoms in the large molecule are Fluorine
isotope 19 and two of the Carbon atoms are Carbon isotope 13. All the other non-hydrogen atoms have
even isotope numbers and no nuclear spins. The objective of the prototype quantum computer was to
factor the number 15 into its two prime factors 3 and 5 by using Shor’s quantum factoring algorithm. The
quantum computation required that a sample of ≈ 10 18 of the large molecules be placed in a magnetic field
and manipulated by nuclear magnetic resonance (NMR) techniques. This mechanism allows the spins to
function as qubits, whereby Schor’s algorithm can be performed via manipulation of the NMR fields.
NMR was used to implement quantum computing in this prototype, because the nuclear spins are well
isolated from decoherence as a result of the very long decoherence time (the time after which quantum
coherence is lost due to environmental noise) in the system.
To factor larger numbers will require a system that uses more than seven qubits. It is estimated that a
quantum computer using ≈ 36 qubits could very quickly perform computations that would require a
conventional computer ≈ 13 billion years to perform. And such a computer could solve one of the
technical problems of human teleportation discussed in Section 3.1. However, a scale-up in the number
of qubits is difficult because the IBM prototype has reached the technology limit of NMR quantum
computing. The prototype’s operation requires that all of the qubits must be in the same molecule. And
molecules with more than seven spins that can be used as qubits are not feasible at present. However,
there are alternative technologies for quantum computing that show promise for scaling up the number of
qubits. The technologies of nuclear spin orientation of single atom impurities in semiconductors, electron
spin orientation in quantum dots, and the manipulation of magnetic flux quanta in superconductors all
show promise of providing a basis for scalable quantum computers. Finally, the primary technical
problem in quantum computing at the present time is decoherence, and this must be eliminated or
otherwise mitigated before new quantum technology can become competitive with conventional computer
technology.
A byproduct of the recent quantum computing and information research is that a modern theory of
entanglement has emerged. Researchers now treat entanglement as a quantifiable physical resource that
enables quantum information processing and computation. Entanglement is no longer treated as a
paradox of quantum theory. It has been recently discovered that (Nielsen and Chuang, 2000; Nielsen,
2003; Terhal et al., 2003):
• various kinds of pure and mixed entangled states may be prepared in addition to the simple purestate
superpositions that was described in the previous section
• the members of an entangled group of objects do not have their own individual quantum states,
only the group as a whole has a well-defined state (i.e., “the whole is greater than the sum of its
parts”)
• entangled objects behave as if they were physically connected together no matter how far apart
they actually are, distance does not attenuate entanglement in the slightest – it has been
demonstrated that information can be teleported over 40 km using existing technology (H.
Everitt, Army Research Office, 2000)
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