Fault-tolerant Computation without Concatenation

It has been known that error-correction via concatenated codes can be done with exponentially small failure rate if the error rate for physical qubits is below a certain accuracy threshold (probably ∼ 10–10). Other, un-concatenated codes with their own attractive features—e.g., an accuracy threshold ∼ 10—have also been studied. A method to obtain universal computation is presented here which does not rely on any concatenated structure within the code itself, but instead emulates this structure with logical qubits in order to construct an encoded Toffoli gate. This realizes ∼ 10 as a threshold for fault-tolerant quantum computation. 1 QEC codes and universal computation In the “space” of all possible quantum error-correcting codes, much recent work has focused on a relatively small class, namely concatenated codes [1]–[7]. The basic idea behind these is to improve the results of a given few-qubit code by replacing physical qubits in its blocks with logical qubits of the code. We look closely at the qubits used in a block of, say, the 7-qubit code and find that each of them itself comprises 7 qubits, and that each of these comprises 7 more, etc. It turns out, if the error/decoherence rate for physical qubits is below a certain accuracy threshold pc, the chances of quantum information stored in this way being corrupted go down exponentially with the total block size of the code. Estimates for pc range from 10 −3 to 10 errors per qubit per recovery round [2]–[5]. However, a different framework for error-correcting codes has also been proposed [8][9][10] whose properties have a natural geometric interpretation in terms of qubits arranged in a lattice. Recently, fully fault-tolerant methods of recovery within this framework were presented [11]; they exhibit the same kind of failure rate scaling and accuracy threshold as concatenated codes. In fact, pc is significantly improved to 10 . Here, I show how to achieve universal fault-tolerant computation for these codes, and in fact a much wider class, by