Ph.D. Thesis - Physics

Figure 2-3: The four traces here are NMR peak signals as a function of simulated time
T. The temporal variation of the signal depends on the initial state. The top trace, (a),
corresponds to the state |0〉, which is an eigenstate and does not oscillate. In (b), the
state |0〉 + i |2〉 is simulated, and accordingly, it oscillates at 2Ω. In (c) and (d), the state
|0〉 + |1〉 + |2〉 + |3〉 is used, and oscillations at both Ω and 3Ω are observed. Image taken
from Ref. [STH + 98].
late values that were directly programmed into the system. However, this relatively simple
experiment demonstrates the type of reasoning that will be used with a more complicated
system in the next chapter. We have seen, generally, how one can bound the error on a
quantum simulation due to a classical Fourier transform, if such a method is used. But is
this the only error?
We expect that errors due to faulty controls will also affect the precision of the final
answer. Since the number of gates needed grows with the desired precision in the result,
we would expect control errors to become more and more important as greater precision is
required. Furthermore, the simulated Hamiltonian in Sec. 2.2 consists entirely of mutually-
commuting terms. This allows for a fairly simple implementation, since approximating
non-commuting terms using composite pulses is not necessary.
In Ch. 3, we explore the additional errors that arise when such approximations are
needed. We shall see that control errors and natural decoherence (due to the larger number
of required pulses) take on increased importance. Most noteworthy is that such errors may
introduce a systematic shift in the answer, rather than only a broadening of the line.
2.4 Conclusions and further questions
In this chapter, we have seen how an NMR system can be used to implement small quantum
simulations. The necessary controls are present to implement a universal set of quantum
gates, and we have also seen an example of mapping a target system Hamiltonian to a
model system Hamiltonian, in this case an NMR Hamiltonian. We have shown in some
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