QUANTUM COMPUTING - Bio-Inspired and Nanoscale Integrated Computing - page 77

3.6.2. Decoherence Errors in QC
Quantum decoherence is the gradual introduction of errors of amplitude in the
quantum superposition of basis states. All known experimental implementations of
QC suffer from the gradual decoherence of entangled states. The rate of decoher-
ence per step of QC depends on the specific technology implementing QC.
A significant property of Shor's algorithm is that the precision of the amplitudes
in the superpositions need be only a polynomial number of bits. Although the
addition of decoherence errors in the amplitudes may at first not have a major effect
on the QC, the effect of the errors may accumulate over time and completely destroy
the computation. Researchers have dealt with decoherence errors by extending
classical error correction techniques to quantum analogs. Generally, there is
assumed a decoherence error model where the errors introduced are assumed to
be uniform random with bounded magnitude, independently for each qubit.
3.6.3. Quantum Codes
Shor [127] and Steane [128] gave the first techniques for reducing quantum
decoherence by the addition of extra qubits which are then projected via observation
operations to eliminate errors in the superposition. Calderbank, Shor [129], and
Steane [130] then proved that QC can be done with bounded decoherence error,
assuming the error correction mechanism is without error itself. Bennett et al. [131],
and Laflamme [132] gave the first optimal 5-qubit codes, leading to asymptotically
optimal (for large code blocks) quantum error correction codes. Shor [133] and
Kitaev [134, 135] extended these techniques to do fault tolerant quantum computa-
tion on quantum networks, in the presence of bounded decoherence error, even if the
error correction mechanism also suffers from error decoherence errors. A final
innovation (Aharonov, Ben-O [136], and Knill et al. [137, 138]) was concatenated
versions of the above quantum codes that allow for arbitrarily long QC in the
presence of arbitrary (i.e., not necessarily random) decoherence error below a fixed
constant threshold. Current bounds on this threshold are very small, and it seems
likely (although it is not yet known) that they can be increased to above the
decoherance error bounds of experimental techniques for QC.
Also see the texts [38, 39] on quantum coding theory.
3.7. QUANTUM CRYPTOGRAPHY
Here we overview quantum cryptography; also see the following texts: [35-37].
3.7.1. Quantum Keys
Bennett et al. [139] and Bennett and Brassard [140] gave the first methods for
quantum cryptography using qubits as keys, which were proved to be secure