Ph.D. Thesis - Physics

Figure 3-5: The CHFBr2 molecule. Although drawn in a plane here, it is actually tetrahedral.
The method of implementing these Hamiltonians is as follows. The pulse sequences
presented here implement the individual parts of the longer pulse sequence discussed in
Sec. 3.2. We use composite rotations about the ˆx and ˆy axes to generate rotations about ˆz:
U0 =
m
R m ˆy (π/2)Rmˆx (πνm)R m −ˆy (π/2) , (3.6)
where m indexes the individual qubits. UXX and UY Y are generated by applying single
qubit pulses to rotate the scalar coupling from ˆz to ˆx or ˆy. Let us use the same convention
as above, that qubit a is 1 H, qubit b is 13 C, and qubit c is 19 F. Written in time order, the
pulse sequence to implement H1 is
UH1 = Râ y (π/2) − Rbˆy (π/2) − UZZ(t) − R â y (−π/2) − Rbˆy (−π/2) , (3.7)
where the superscripts of the rotation operators denote the qubit being acted upon, and the
UZZ evolution affects all qubits. The rotation operators for qubits 1 and 2 on either side of
UZZ(t) may be swapped, as these operations commute (and are assumed to take zero time).
To implement H2, one only needs to remove the coupling of the third nucleus since
Vac = Vbc = 0. This can be done using the refocusing technique presented in Sec. 2.1.5. H2
is simulated by replacing UZZ(t), in the above sequence, with
UZZ(t/2) − R ĉ x (π) − UZZ(t/2) . (3.8)
The initialization of the state |ΨI〉 = cG |G〉 + cE |E〉 requires, as mentioned in Sec. 3.1,
quasiadiabatic evolution. This discrete-step process was demonstrated in [SvDH + 03], but
here we speed it up somewhat to populate the first excited state |E〉. The Hamiltonian at
each discrete timestep s is Had(s) = (1 − s/S)H0 + (s/S)HBCS. We find S = 4 steps, with
time steps of tad = 1/700 s, to be sufficient. We do note that, for for HXX +HY Y ≫ H0
there can be a phase transition as s is changed [McK96]; as the gap goes to zero at the
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