Fault Tolerant Quantum Computation with Constant Error

TL;DR

This paper demonstrates fault tolerant quantum computation with a constant error threshold using concatenated quantum error correcting codes, achieving polylogarithmic resource costs and no measurements during computation.

Contribution

It improves previous bounds by establishing fault tolerance below a constant error threshold with general proper quantum codes and explicit constructions, including codes over larger fields.

Findings

Threshold error rate around 10^(-6)

Polylogarithmic time and space complexity

No measurements needed during quantum computation

Abstract

Recently Shor showed how to perform fault tolerant quantum computation when the error probability is logarithmically small. We improve this bound and describe fault tolerant quantum computation when the error probability is smaller than some constant threshold. The cost is polylogarithmic in time and space, and no measurements are used during the quantum computation. The result holds also for quantum circuits which operate on nearest neighbors only. To achieve this noise resistance, we use concatenated quantum error correcting codes. The scheme presented is general, and works with all quantum codes that satisfy some restrictions, namely that the code is ``proper''. We present two explicit classes of proper quantum codes. The first example of proper quantum codes generalizes classical secret sharing with polynomials. The second uses a known class of quantum codes and converts it to a proper code. This class is defined over a field with p elements, so the elementary quantum particle is not a qubit but a ``qupit''. With our codes, the threshold is about 10^(-6). Hopefully, this paper motivates a search for proper quantum codes with higher thresholds, at which point quantum computation becomes practical.