Fault-Tolerant Quantum Computation With Constant Error Rate
Dorit Aharonov, Michael Ben-Or
TL;DR
The paper proves a threshold theorem for quantum computation under a constant error rate $\eta$, applicable to very general noise models and to one-dimensional nearest-neighbor architectures. It achieves fault-tolerant universal quantum computation by combining two universal gate-sets, $G_1$ for CSS codes and $G_2$ for polynomial codes, with recursive simulations that drive the effective error rate down as long as $\eta<\eta_c$ (numerically estimated around $10^{-6}$ in a representative setup). It introduces polynomial codes and degree-reduction techniques to streamline fault-tolerant implementations, including a Toffoli gate construction, and provides universality proofs and a full generalization to non-probabilistic noise and decaying correlations. The results demonstrate robustness of quantum computation against local noise, quantify overheads, and extend applicability to higher dimensions and other universal gate sets, signaling a viable path toward practical fault-tolerant quantum computation under realistic noise. The threshold’s existence implies a concrete regime where quantum devices can perform reliable computations with polylogarithmic overhead per level of recursion.
Abstract
This paper proves the threshold result, which asserts that quantum computation can be made robust against errors and inaccuracies, when the error rate, $η$, is smaller than a constant threshold, $η_c$. The result holds for a very general, not necessarily probabilistic noise model, for quantum particles with any number of states, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation. The proceeding version was very succinct, and here we fill all the missing details, and elaborate on many parts of the proof. In particular, we devote a section for a discussion of universality issues and proofs that the sets of gates that we use are universal. Another section is devoted to a rigorous proof that fault tolerance can be achieved in the presence of general non probabilistic noise. The systematic structure of the fault tolerant procedures for polynomial codes is explained in length. The proof that the concatenation scheme works is written in a clearer way. The paper also contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.