Toward fault-tolerant quantum computation without concatenation

It has been known that quantum 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. Other, unconcatenated codes with their own attractive features—improved accuracy threshold, local operations—have also been studied. By iteratively distilling a certain two-qubit entangled state it is shown how to perform an encoded Toffoli gate, important for universal computation, on CSS codes that are either unconcatenated or, for a range of very large block sizes, singly concatenated. 1 Codes and computation At the end of the long tunnel of experimental quantum computing there is the light of accuracy thresholds provided by quantum error correcting codes. The lengthy computations necessary for efficient factorization and simulation of quantum systems are all of a sudden possible if error rates for qubits are reduced below certain critical values in a sufficiently parallel quantum computer. The quantum codes which give rise to this intriguing phase transition work on the same basic principle as their classical precursors—they keep information secure by using more physical (qu)bits per logical (qu)bit of the code. What is desired of quantum codes is that, as this redundancy is increased, there should be exponential improvement in the storage/computation fidelity of the code. Once the basic problem of correcting fully quantum errors was solved by the advent of quantum codes, a general recipe was obtained for expanding a given fewqubit code to achieve these exponential fidelity gains. This recipe generates a new higher level code by mimicking the old code, but with the role of physical qubits played by logical qubits of the old code. Detailed methods of error correction and computation have been obtained for versions of this recipe with an arbitrary number of iterations or “concatenations”—even when faults might occur in error correction processes themselves [1]–[7]. However, different frameworks for fault-tolerant error correction, e.g. topological quantum codes [8]–[11], might prove superior, and more general methods