Error channels and the threshold for fault-tolerant quantum computation

The threshold for fault-tolerant quantum computation depends on the available resources, including knowledge about the error model. I investigate the utility of such knowledge by designing a fault-tolerant procedure tailored to a restricted stochastic Pauli channel and studying the corresponding threshold for quantum computation. Surprisingly, I find that tailoring yields, at best, modest gains in the threshold, while substantial losses occur for error models only marginally different from the assumed channel. This result is shown to derive from the fact that the ancillae used in threshold estimation are of exceedingly high quality and, thus, difficult to improve upon. Motivated by this discovery, I propose a tractable algebraic algorithm for predicting the outcome of threshold estimates, one which approximates ancillae as having independent and identically distributed errors on their constituent qubits. In the limit of an infinitely large code, the algorithm simplifies tremendously, yielding a rigorous threshold bound given the availability of ancillae with i.i.d. errors. I use this bound as a metric to judge the relative performance of various fault-tolerant procedures in combination with different error models. Modest gains in the threshold are observed for certain restricted error models, and, for the assumed ancillae, Knill's fault-tolerant method is found to be superior to that of Steane. My algorithm generally yields high threshold bounds, reflecting the computational value of large, low-error ancillae. In an effort to render these bounds achievable, I develop a novel procedure for directly constructing large ancillae. Numerically, the scaling and average error properties of this procedure are found to be encouraging, and, though it is not fault-tolerant, I prove that each error can spread to only one additional location. Promising means of improving the ancillae are proposed, and I discuss briefly the challenges associated with preparing the cat states necessary for my procedure.