Ph.D. Thesis - Physics

Figure 3-6: Frequency-domain spectra of Hamiltonian H2 obtained using methods W1
(circles) and W2 (diamonds). The solid lines are fits to time-dependent data. The width
of the exact curve is taken to be the dephasing rate (1/T ∗ 2 ) of the 13 C nucleus, which has
the highest dephasing rate.
tion, the effects of control errors were very evident in the results for H2. Hamiltonian H2
was implemented using both W1 and W2 for εFT = 10·2π Hz and t0 = 0.5 ms. The shorter
time step was necessary because the larger ∆ made the simulation more sensitive to Trotter
errors. Comparing the W2 and W1 results shows that with no control error compensation,
a gap ∆exp is found that is ∆/5 away from the actual value. In contrast, with simple error
compensation ∆exp is ǫFT from the actual value. Future implementations should strive to
detect and bound control errors by verifying that ∆exp converges as t 3 0
t0, as theoretically expected.
3.6 Discussion
for small values of
To summarize this work, we have simulated the BCS Hamiltonian using the smallest prob-
lem instance that requires both adiabatic evolution and the Trotter approximation. We
find that our implementation on an NMR quantum computer saturates the bounds on the
precision that were predicted by us and by WBL. The most important aspect of this work
is understanding the limitations of the precision of the final result, both due to intrinsic as-
pects of the protocol such as the Fourier transform and the Trotter approximation, and due
to system-specific aspects such as control errors and natural decoherence. We summarize
our conclusions here.
1. Results obtained using digital quantum simulation are generally inefficient with re-
spect to the precision. Nevertheless, if the dynamics may be implemented efficiently,
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