QUANTUM COMPUTING - Bio-Inspired and Nanoscale Integrated Computing - page 92

Bernstein and Vazirani [50] and Brassard et al. [173, 202] observe that any QC can
be repeated to insure the output qubits are e -near classic in the final output
superposition after the repetitions. Note that if a QC with bounded amplitude
precision is reduced by an observation, the output qubits yield the correct value
with high likelihood. Hence we may consider simply not doing the observation
reduction to a basis state in the final step; in place of this (reduced) output
superposition we simply output the nonreduced quantum state superposition of
the QC that exists just prior to the final observation step. This alternative
approach can eliminate entirely the observation operation from many quantum
computations, and so provides small volume, but has the drawback of providing a
nonclassic output consisting of a nonreduced quantum state superposition. The
potential difficulty with this approach is as follows: If this (nonreduced quantum
state superposition) output is then processed by a classical computing machine, it
may propagate unwanted quantum effects to the classical computing machine.
3.12.6. What about Approximate Observation Operations?
An approach to this difficulty is to only do the observation operation approxi-
mately within accuracy e ; this may suffice for many QC applications. However,
even if the observation operation is done e -approximately by unitary operations, it
appears to require a number of additional qubits n 0 growing exponentially with the
n, the original number of qubits of the QC. In fact, we know of no upper bound on
n 0 better than 2 n log (1/ e ).
3.12.7. Why May the Volume Required by Observation Apparatus
Not Be Small?
We next consider whether it is reasonable to expect that a mathematical proof (or
such experimental demonstrations) of small volume quantum observation will
ever be done. We provide an informal argument that even an e -approximate
observation cannot be done in polynomial time using small volume, where e is the
inverse of a polynomial. (It should be emphasized that the following is not a
formal proof in any sense.) Since for n qubits, the size of the basis state space
grows as 2 n in the general case, it seems reasonable to assume (e.g., where the
physics of the strong measurement is modeled by a diffusion process [286-288]
that is rapidly mixing) that the likelihood of reversibility within polynomial time
bounds drops exponentially with the number n of qubits. Thus, in the context of
polynomial time computations, the e -approximate observation is assumed irre-
versible with high likelihood.
The argument will hinge on the assumption, made by conventional formula-
tions of quantum computation, that quantum computation (including both
unitary qubit operations as well as the nonunitary observation (or projection)
operation) can be exectuted for any given number n qubits, which makes an
implicit assumption that both unitary and nonunitary operations can be executed
at any scale.