Ph.D. Thesis - Physics
Ph.D. Thesis - Physics
Ph.D. Thesis - Physics
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state [BW92]. Quantum communication also enables unconditional exponential gains in<br />
communication complexity for certain problems, e.g. as discussed in Refs. [HBW98, Raz99,<br />
BCWdW01, GKK + 08].<br />
The application of quantum information processing that is of most interest to many<br />
physicists, and that motivates the work of this thesis, is quantum simulation. It is presented<br />
in its own section below.<br />
Quantum information primitives have been implemented in a number of different phys-<br />
ical systems, including trapped ions ([LBMW03], and many others cited in this thesis), nu-<br />
clear spins ([VSB + 01, VC05], and many others cited in this thesis), superconducting circuits<br />
[MNAU02, MOL + 99, NPT99, HSG + 07], neutral atoms [ALB + 07, BCJD99, MGW + 03], and<br />
quantum dots [LD98, KBT + 06, PJT + 05]. Each of these systems carries certain advantages<br />
and disadvantages. Solution-state NMR was the first system in which small algorithms,<br />
including the simplest nontrivial case of Shor’s algorithm, were implemented [VSB + 01].<br />
The long coherence times of nuclear spins combined with the superb control techniques<br />
already developed (and developed further still by quantum information researchers [VC05])<br />
enabled this system to get off the ground fairly quickly. However, NMR cannot scale to<br />
large numbers of qubits. Trapped ions have now emerged instead as the leading technology,<br />
because they combine coherence times even longer than nuclear spins in solution with a now<br />
highly-developed control apparatus, superior measurement accuracy, and several promising<br />
avenues to scalability.<br />
Of course, no technology has yet been scaled up to even more than ten qubits, meaning<br />
that a quantum computer or simulator that can outperform classical computation has not<br />
yet come close to being realized.<br />
1.2 What is quantum simulation?<br />
We now begin to explore the overarching theme of this thesis, quantum simulation. Quan-<br />
tum simulation is the use of one controllable quantum system, referred to in this thesis as<br />
the model system, to calculate properties of some other quantum system that is more diffi-<br />
cult to control, which we call the target system. For instance, one may map a Hamiltonian<br />
that describes a set of interacting fermions to a set of spin-1/2 systems which an experi-<br />
menter can control well, such as (in this thesis) nuclear spins in molecules or the electronic<br />
states of trapped ions. If controls are available to apply the same effective dynamics on the<br />
model system that is thought to govern the target system, then predictions regarding the<br />
target system can be inferred by performing experiments on the model system. Although<br />
deceptively simple as stated here, in practice it is difficult to discover good methods for<br />
performing quantum simulation. Of course, the person carrying out the quantum simula-<br />
tion also must make the assumption that the target system obeys a specific Hamiltonian.<br />
This could be stated in a positive light, in that simulation by model systems may inform<br />
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