Ph.D. Thesis - Physics

Figure 2-2: The 2,3-dibromothiophene molecule. The two unlabeled vertices represent
hydrogen atoms, whose nuclei were used in this experiment.
success; the frequencies at which the signal evolves for each initial state depend on the
energy difference between the two states.
In the same paper, they also implement a simulation of a truncated, driven, anharmonic
oscillator, but the above is sufficient for our purposes of illustrating the basic methods
of quantum simulation with NMR. In what follows, we begin to address the question of
precision using this experiment as a model.
2.3 The question of precision
Having seen qualitative evidence for the success of quantum simulation, we move to dis-
cussing the quantitative issue of how much precision may be obtained in a measured result.
For instance, we might ask in the case of the truncated oscillator experiment, how could one
extract the harmonic oscillator frequency from the simulation results? These peaks could
be Fourier-transformed to yield an energy (or frequency) spectrum; depending on the initial
states used, the spectra would show peaks at Ω, 2Ω, or 3Ω. How well can Ω be measured,
and how does this precision depend on the number of pulses used, or the total simulation
time?
We wish to give a general idea for how the limitation to the precision may be estimated.
To begin with, the precision of the final result is limited by the width of the peak in
frequency space; thus, it depends on the Fourier sampling rate. Also, the minimum time
step determines the maximum energy that can be measured. Therefore, we may suppose
that the error in the final result ǫ scales as Ω/Q, where Q is the maximum number of time
steps. Since the total number of gates NG is proportional to Q, NG ∝ Ω/ǫ. A bound on the
precision may then be obtained by calculating the total simulation time NGtg, where tg is
the time to perform a single iteration of the simulated Hamiltonian, and setting this equal
to the decoherence time.
Admittedly, the above example is somewhat artificial, in that one is attempting to calcu-
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