Space, time, parallelism and noise requirements for reliable quantum computing

Abstract

Quantum error correction methods use processing power to combat noise. The noise level which can be tolerated in a fault-tolerant method is therefore a function of the computational resources available, especially the size of computer and degree of parallelism. I present an analysis of error correction with block codes, made fault-tolerant through the use of prepared ancilla blocks. The preparation and verification of the ancillas is described in detail. It is shown that the ancillas need only be verified against a small set of errors. This, combined with previously known advantages, makes this 'ancilla factory' the best method to apply error correction, whether in concatenated or block coding. I then consider the resources required to achieve 2 · 1010 computational steps reliably in a computer of 2150 logical qubits, finding that the simplest [[n, 1, d]] block codes can tolerate more noise with smaller overheads than the 7L-bit concatenated code. The scaling is such that block codes remain the better choice for all computations one is likely to contemplate.