Fault Tolerant Quantum Computation with Constant Error
Dorit Aharonov, Michael Ben-Or
TL;DR
This work advances fault-tolerant quantum computation by proving that constant noise levels can be tolerated with polylogarithmic overhead using proper quantum computation codes and concatenated simulations. It introduces a mixed-state circuit model and formalizes encoded computation through block codes, with explicit constructions in linear $F_p$-codes and polynomial codes. The key contributions are the concatenation framework, the notion of code spread, and the demonstration that a nonzero fault threshold exists (approximately $10^{-6}$) for these codes, including nearest-neighbor variants. The results motivate the search for higher-threshold, practically realizable proper codes to push quantum computing closer to feasibility.
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.