This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when $\eta$, the probability for an error in a qubit or a gate, is smaller than a constant threshold $\eta_c$. This result improves on Shor's result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56-65], which shows how to perform fault-tolerant quantum computation when the error rate $\eta$ decays polylogarithmically with the size of the computation, an assumption which is physically unreasonable. The cost of making the quantum circuit fault-tolerant in our construction is polylogarithmic in time and space. Our result holds for a very general local noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault-tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with $p>2$ states, namely, $p$-qudits, 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. We estimate the threshold of our construction to be $\eta_c\simeq 10^{-6}$, in the best case. By this we show that local noise is in principle not an obstacle for scalable quantum computation. The main ingredient of our proof is the computation on states encoded by a quantum error correcting code (QECC). To this end we introduce a special class of Calderbank-Shor-Steane (CSS) codes, called polynomial codes (the quantum analogue of Reed-Solomon codes). Their nice algebraic structure allows all of the encoded gates to be transversal. We also provide another version of the proof which uses more general CSS codes, but its encoded gates are slightly less elegant. To achieve fault tolerance, we encode the quantum circuit by another circuit by using one of these QECCs. This step is repeated polyloglog many times, each step slightly improving the effective error rate, to achieve the desired reliability. The resulting circuit exhibits a hierarchical structure, and for the analysis of its robustness we borrow terminology from Khalfin and Tsirelson [Found. Phys., 22 (1992), pp. 879-948] and Gacs [Advances in Computing Research: A Research Annual: Randomness and Computation, JAI Press, Greenwich, CT, 1989]. The paper is to a large extent self-contained. In particular, we provide simpler proofs for many of the known results we use, such as the fact that it suffices to correct for bit-flips and phase-flips, the correctness of CSS codes, and the fact that two-qubit gates are universal, together with their extensions to higher-dimensional particles. We also provide full proofs of the universality of the sets of gates we use (the proof of universality was missing in Shor's paper). This paper thus provides a self-contained and complete proof of universal fault-tolerant quantum computation in the presence of local noise.
['Dorit Aharonov', 'Michael Ben-Or']
Fault-Tolerant Quantum Computation with Constant Error Rate