Ph.D. Thesis - Physics

PC04b]. In both cases, the idea differs from those mentioned above, in that rather than
employing a sequence of discrete control pulses, they instead propose the creation of model
Hamiltonians using a limited set of continuously-varying controls. Similar ideas have been
set forth for the simulation of Bose-Hubbard models, also using trapped ions or neutral
atoms [GME + 02, PC04a]. In these examples, the opportunity arises to observe whether the
target system behaves qualitatively like the model system being studied for similar sets of
parameters.
1.3 Models of quantum simulation
The quantum simulations introduced above may be divided into two distinct types, or
models: in the first, of which the paper of Wu et al. [WBL02] is a fine example, a quantum
simulator is prepared in some initial state and then manipulated with a set of discrete
pulses, a situation akin to a digital computer. In fact, this type of simulation bears other
resemblances to classical digital computation, such as the fact that error correction codes
may be used [CS96, Ste96, Sho96, DS96, Got97]. We refer to this type of simulation as digital
quantum simulation. The second type, which the neutral atom community has recently
excelled at, involves creating an effective Hamiltonian that is controlled by continuously
adjusting the relevant parameters. This type is akin to classical analog computation, and
we refer to it as analog quantum simulation. This approach lends itself to more qualitative
questions, such as what phase (superfluid, insulator, etc.) the particles occupy within some
region of parameter space. Let us dig a bit more deeply into this distinction, because both
types are of interest in this work.
1.3.1 “Digital” quantum simulation
Digital quantum simulations are characterized by the use of discrete quantum gates to im-
plement the desired Hamiltonian. To clarify the terminology, a quantum gate is a (usually)
unitary operation that is applied to some set of qubits for a finite amount of time; any gate
involving more than two qubits may be decomposed into a sequence of one- and two-qubit
gates. Measurement is also considered a quantum gate, and is typically assumed to be of
the strong, projective variety. The weak measurements used in nuclear magnetic resonance
(NMR) are a notable exception.
This approach has some very appealing advantages: for one, any region of parameter
space may be probed, since the interactions are entirely engineered by control pulses from
the experimenter. The other very big advantage is that quantum error correction techniques
may be applied, and indeed will be necessary when the system grows to a large enough size.
However, quantum error correction schemes generally require a substantial increase in the
required number of control pulses, increasing the likelihood that systematic control errors
will reduce the accuracy of the simulation.
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