Abstract
The standard approach to quantum fault tolerance is to calculate error thresholds on basic gates in the limit of arbitrarily many concatenation levels. In contrast this paper takes the number of qubits and the target implementation accuracy as given, and provides a framework for engineering the constrained quantum system to the required tolerance. The approach requires solving the full dynamics of the quantum system for an arbitrary admixture (biased or unbiased) of Pauli errors. The inaccuracy between ideal and implemented quantum systems is captured by the supremum of the Schatten k-norm of the difference between the ideal and implemented density matrices taken over all density matrices. This is a more complete analysis than the standard approach, where an intricate combination of worst case assumptions and combinatorial analysis is used to analyze the special case of equiprobable errors. Conditions for fault tolerance are now expressed in terms of error regions rather than a single number (the standard error threshold). In the important special case of a stochastic noise model and a single logical qubit, an optimization over all 2x2 density matrices is required to obtain the full dynamics. The complexity of this calculation is greatly simplified through reduction to an optimization over only three projectors. Error regions are calculated for the standard 5- and 7-qubit codes. Knowledge of the full dynamics makes it possible to design sophisticated concatenation strategies that go beyond repeatedly using the same code, and these strategies can achieve target fault tolerance thresholds with fewer qubits.