Fault-tolerant quantum computation (FTQC) can be done in principle: the threshold theorems show that, for sufficiently low error rates, it is possible to do quantum computations of arbitrary size. However, current schemes that allow such scaling--using concatenated or surface codes--require very large overhead to achieve quantum computation at realistic error rates. One approach to reduce this overhead is to encode multiple logical qubits in a single code block. By combining two different codes--one for storage and Clifford gates, one for non-Clifford gates--it is possible to do a universal set of encoded quantum gates by measuring logical operators and performing logical teleportation between code blocks. We analyze the performance of such schemes, and for a few choices of codes show numerically that one can do quite large quantum computations at moderate error rates. One of the key requirements of this scheme is the ability to prepare a set of different ancilla states reliably. We present a scheme for distillation of these ancilla states, and give evidence that it is possible to get good distillation rates with low residual errors. We also look at some variations of this scheme using different codes.
