Ph.D. Thesis - Physics

⎡ ⎤
a 0 0 0
⎢ ⎥
⎢
ρ2 = ⎢
0 c 0 0 ⎥
⎢
⎣ 0 0 d 0
⎥ ; (2.9)
⎦
0 0 0 b
⎡ ⎤
a 0 0 0
⎢ ⎥
⎢
ρ3 = ⎢
0 d 0 0 ⎥
⎢
⎣ 0 0 b 0
⎥ . (2.10)
⎦
0 0 0 c
A quantum computation (or simulation), not including measurement, is a unitary op-
eration, which we will write as U. The sum of the density matrices P =
i=1,2,3 ρ′ i , where
ρ ′ i = U † ρiU, is
⎡ ⎤ ⎡ ⎤
1 0 0 0
1 0 0 0
⎢ ⎥ ⎢ ⎥
⎢
P = (4a − 1) U ⎢
0 0 0 0 ⎥ ⎢
⎥
⎢
⎣ 0 0 0 0
⎥ + (1 − a) ⎢
0 1 0 0 ⎥
⎢
⎦ ⎣ 0 0 1 0
⎥ .
⎦
(2.11)
0 0 0 0
0 0 0 1
The measured NMR signal is given by Tr(P) =
i=1,2,3 ρ′ iM, where M is the observable
in question. The portion of the signal that is proportional to the identity matrix above does
not produce any signal when measured, and does not evolve under unitary transforms.
Therefore, by performing this experiment three times with the initial states ρ1, ρ2, and ρ3,
we obtain a signal that is proportional to what the result would have been on a pure state
〈↑↑ | ↑↑ 〉:
i=1,2,3
Tr ρ ′ iM
= (4a − 1)Tr U〈↑↑ | ↑↑ 〉U †
. (2.12)
As can clearly be seen, two qubits produce a four-dimensional Hilbert space, which
requires three averaging steps. From the above example, for n qubits one will require,
in general, 2 n − 1 steps. Therefore, this method is inefficient, and nulls the speedup of
quantum computation, but is sufficient for the implementation of algorithms involving a
small number of qubits.
2.1.3 Single-qubit operations
The rf coils along the ˆx and ˆy directions generate magnetic fields that oscillate at a frequency
ω, whose strength is characterized by a frequency ω1, which is related to the magnetic field
produced by ω1 = γB1. Here γ = gNµN is the magnetic moment of the nucleus being
addressed, with gN the nuclear g-factor and µN the nuclear magneton. The Hamiltonian
for the interaction of this field with a given nucleus is
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