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3.11. SUMMARY AND ACKNOWLEDGMENTS
We have have provided an overview of the field of quantum computing and
surveyed its major algorithmic results and applications as well as physical
implementations and their limitations. Some issues such as energy costs as well
as volume of observation apparatus for quantum computing are still unresolved,
and the later are further addressed in the Appendix. We would like to thank
G. Brassard for his clear explanation of numerous results in the field of QC. Also,
I would like to thank P. Shor and U. Vazirani for references and for illuminating
discussions on quantum computation, particularly on the issue of volume bounds
for quantum observation.
3.12. APPENDIX: VOLUME OF OBSERVATION APPARATUS
FOR QUANTUM COMPUTING
Here we discuss the challenge of providing volume bounds for observation
apparatus when doing QC.
3.12.1. A Potentially Fallacious Proof of Small Volume
First we note that one might be tempted to give a constructive proof that
observation can be done on n qubits in small volume, along the following lines:
(i) Basis Step. We begin with a simple, well established experimental method
for observation of a single qubit in small quantum system with say n 0 qubits for a
constant n 0 . There are many other examples of experimentally verified methods
for observation using macroscopic measurement apparatus. (For example, a
number of proposed QC architectures (e.g., the Cirac and Zoller [233] 3, and
Pellizzari et al. [234] proposed ion trap QC and Kane's [264] silicon-based NMR
QC) give specific descriptions of measuring apparatus that have been experimen-
tally verified for observation of a single qubit within a quantum computing
systems with a constant number of qubits. While their measuring apparatus is
macroscopic, it still must have just some finite volume.
(ii) Inductive Step. However, then we just scale up by using the same
experimental apparatus to do observation on each of n qubits (that is, repeating
the observation for each of the other qubits). This seems to result in a small
volume (perhaps even linear size) apparatus for observation.
The potential fallacy of this line of argument is:
(a) In the basis step, the experiments of [230, 260] did not provide bounds on
the errors (or fidelity) of the measurement as a function of the volume of
the measuring apparatus.
(b) The inductive step fails to take into account quantum effects involving
both the measuring apparatus and the n qubits, as might be predicted by
 
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