Page 187 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 175

While the maturity of NMR spectroscopy has enabled the successful
proof-of-concept implementation of various QC algorithms, the fact that the
technique must rely on measuring ensembles of spins to obtain a detectable
read-out signal is a limiting aspect of it, since this implies that one must
begin with the highly-mixed initial ensemble state; this is the result of there
being a very small energy difference between up and down spins at room
temperature, manifesting itself as a nearly random equilibrium distribution
[193].
A highly-mixed state possesses equally likely spin-up and spin-down
states, for example [193],
(1 ε− ) 2 ε+I/ 0 0 , (29)

ε ~ 10 , which is an almost random state with a small excess of the 0
−
5
state [193]. This expression for the equilibrium state follows from the
density matrix ȡ thermal which, being proportional to e − H/kT (where the
nuclear spins in a molecule posses the internal Hamiltonian H, T is
temperature and k is the Boltzmann constant), admits an expansion [190],

− H/kT − () 1 /kT − ( ) /kT
2
e ≈ e İ 1 ı z e İ 2 ı z ..., (30)
which with,
− İ 1 ı z () 1 /kT () 1
e ≈ I - ε σ /kT..., (31)
1 z
may be written as,
− H/kT () 1 ( ) 2
e ≈ I − İ 1 ı z /kT İ - 2 ı z /kT..., (32)

where I is the identity matrix and, for spin i, the parameter İ represents the
i
energy difference between up and down states. While the desired initial state
is a pure one, in which all spins are in the same state, e.g., 0 , the actual

randomness of the initial ensemble state may be overcome by a technique to
transform it into an almost pure state.
An almost pure state is one that produces a signal that is proportional to
that of a pure-state signal. It is generated by exploiting three facts [193],
namely: 1) That the magnetization is determined by the traceless part of the
density matrix; 2) That the completely mixed state 2I/ n is preserved under
both unitary and non-unitary transformations; and 3) That all scales are
relative, in particular, that only the ratio of two magnetizations determines
the final answer of a quantum computation, i.e., the deciding factor in a
measurement is, not the absolute magnetization, but its relative value
compared to the noise [193].