Ph.D. Thesis - Physics

electrodes [TKK + 99]. This heating has been observed to scale roughly as 1/d 4 , where d is
the distance from the ion to the nearest electrode [DOS + 06].
Several steps have been taken to reduce these sources of decoherence in ion trap sys-
tems. Decoherence due to fluctuating magnetic fields has been suppressed by the encod-
ing of qubits in a decoherence-free subspace [LOJ + 05, HSKH + 05], and also, for hyperfine
qubits, by using a magnetic field-insensitive transition [LOJ + 05]. Errors due to sponta-
neous scattering can theoretically be eliminated by performing gates using magnetic fields
and radiofrequency pulses rather than laser light [CW08, OLA + 08, JBT + 08]. Although
decoherence of internal (electronic) states of the ions is of considerable importance, our
efforts to develop ion trap architectures are mainly driven by the need to minimize mo-
tional state decoherence, or heating. The problem of motional decoherence has been par-
tially and practically addressed by the demonstration of suppression of heating rates by
several orders of magnitude through cryogenic cooling [LGA + 08, DOS + 06]. Recently, the
temperature-dependence of this heating rate has been more systematically characterized
[LGL + 08]. However, the sources of this noise are still not completely understood.
Considerations of the effects of decoherence have a major impact on the next two limi-
tations of quantum simulation presented.
1.4.2 Limitations to Precision
In quantum simulation, one seeks to calculate some specific property of a quantum system,
for instance an eigenvalue of a Hamiltonian. The precision to which a given quantity can be
calculated depends on any inherent limitations in the algorithm used, as well as on factors
such as imperfect controls and decoherence. Therefore, we may frame our investigations by
two questions:
1. For a given quantum simulation algorithm for calculating some quantity ∆, what is,
in principle, the expected error ǫ in the measurement of ∆, as a function of the space
and time resources required?
2. How does ǫ change as a certain systematic or random error in the control pulses
applied to the system is introduced?
We begin the discussion of the first question with an example. Suppose that one wished
to calculate the difference ∆ between the energy eigenvalues of two states |1〉 and |2〉. One
way to do this is to prepare a superposition of the two states, then evolve under some
simulated Hamiltonian for a set of discrete times tn, for N total timesteps. Measurements
of a suitable operator on the system at each time will oscillate at a rate ∆. Determination
of ∆ can then be done by classically Fourier-transforming the measurement results. With
how much precision can ∆, in principle, be measured? Certainly, in the above scheme, the
sampling rate plays a role: more points will lead to more precision. Therefore, we expect
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