QUANTUM COMPUTING - Bio-Inspired and Nanoscale Integrated Computing - page 85

Disadvantages of Bulk QC. A possible disadvantage of bulk QC is that it
appears to allow only a weak measurement of the ensemble average that
does not provide a quantum state reduction; that is, the weak measure-
ment does not alter (at least by much) the superposition of states. Another
disadvantage of bulk QC is that it may require (for a variety of reasons)
macroscopic volumes and volumes which grow exponentially with the
number of qubits. Macroscopic volumes may be required for measure-
ment via conventional means. However, known quantum algorithms can
still be executed even in this case (e.g., see Gershenfeld, Chuang [22, 262]).
So the lack of strong measurement is not a major disadvantage.

Also, bulk QC requires the initialization close to a pure state. If bulk QC is
done at room temperature, the initialization methods of Cory, Fahmy, Havel [261]
(using logical labeling) and Gershenfeld, Chuang [22, 262] (using spatial aver-
aging) yield a pseudo-pure state, where the number of molecules actually in the
pure state drops exponentially as 1/c n with the number n of qubits, for some
constant c (as noted by Warren [271]). If we approximate the resulting measure-
ment error by a normal distribution, the measurement error is (with high
likelihood) at least a multiplicative factor of 1 c 0

p , for some constant c 0 .

=

c n 4 c 0 = N

p

To overcome this measurement error, we need 1 = ; and so we require

2

that the volume be at least N 4ð c n c 0 Þ Hence, for the output of the bulk QC to be

(weakly) measured, the volume (the number N molecules) of bulk QC needs to
grow exponentially with the number n of qubits. Recently, there have been various
other proposed methods for initialization to a pure state:

= :

Barnes [272] proposes the use of very low temperatures.
Gershenfeld, Chuang [22, 262] suggest the use of gradient fields.
Knill et al. [273] suggest a randomization technique they call temporal
averaging.
Recent work of Schulman, Vazirani [274] provides polynomial volume for
initialization, with the assumption of an exponential decrease in spin-spin
correlations with the distance between the nuclei located within a molecule.
(In particular, they assume that the statistical correlation between and two
bits on a molecule falls of exponentially with the distance between these
bits). Although their methods may provide a solution in practice, known
inter-atomic interactions such as the spin-spin correlations are generally
considered to be governed by potential force laws which decrease by inverse
polynomial powers rather than by an exponential decrease.

It has not yet been experimentally established which of these pure state
initialization methods scale to a large number of qubits without large volume.
(Note: Some physicists feel that it has not been clearly established whether
(a) NMR is actually a quantum phenomenon with quantum superposition of basis
states, or (b) if NMR just mimics a quantum phenomenon and is actually just