|dc.description.abstract||Quantum computing offers the potential for efficiently solving otherwise classically difficult problems, with applications in material and drug design, cryptography, theoretical physics, number theory and more. However, quantum systems are notoriously fragile; interaction with the surrounding environment and lack of precise control constitute noise,
which makes construction of a reliable quantum computer extremely challenging. Threshold theorems show that by adding enough redundancy, reliable and arbitrarily long quantum computation is possible so long as the amount of noise is relatively low---below a ``threshold'' value. The amount of redundancy required is reasonable in the asymptotic sense, but in absolute terms the resource overhead of existing protocols is enormous when compared to current experimental capabilities.
In this thesis we examine a variety of techniques for reducing the resources required for fault-tolerant quantum computation. First, we show how to simplify universal encoded computation by using only transversal gates and standard error correction procedures, circumventing existing no-go theorems. The cost of certain error correction procedures is dominated by preparation of special ancillary states. We show how to simplify ancilla preparation, reducing the cost of error correction by more than a factor of four. Using this optimized ancilla preparation, we then develop improved techniques for proving rigorous lower bounds on the noise threshold. The techniques are specifically intended for analysis of relatively large codes such as the 23-qubit Golay code, for which we compute a lower bound on the threshold error rate of 0.132 percent per gate for depolarizing noise. This bound is the best known for any scheme.
Additional overhead can be incurred because quantum algorithms must be translated into sequences of gates that are actually available in the quantum computer. In particular, arbitrary single-qubit rotations must be decomposed into a discrete set of fault-tolerant gates. We find that by using a special class of non-deterministic circuits, the cost of decomposition can be reduced by as much as a factor of four over state-of-the-art techniques, which typically use deterministic circuits.
Finally, we examine global optimization of fault-tolerant quantum circuits. Physical connectivity constraints require that qubits are moved close together before they can interact, but such movement can cause data to lay idle, wasting time and space. We adapt techniques from VLSI in order to minimize time and space usage for computations in the surface code, and we develop a software prototype to demonstrate the potential savings.||en