Resilient Quantum Computation: Error Models and Thresholds

TL;DR

This work proves a threshold theorem for fault-tolerant quantum computation under realistic, quasi-independent error models: if the per-gate error is below a computable threshold, arbitrarily long quantum computations can be performed with arbitrary accuracy using concatenated quantum error-correcting codes, fault-tolerant recovery, and encoded operations. It introduces a complete fault-tolerant set of encoded gates built on the seven-qubit Steane code, including transversal operations, cat-state syndrome extraction, and a pi/8-state–based Toffoli construction, and shows the threshold persists when extending to a full universal gate set. Quantitative thresholds are derived for stochastic and monotone quasi-independent errors (roughly $p\lesssim 3\times10^{-6}$ with memory and $p\lesssim 1.3\times10^{-6}$ without memory, with extensions to $2.2\times10^{-6}$ and $0.9\times10^{-6}$ for the pi/8 gate), and leakage errors are mitigated with stop-leak gates, while overheads scale polylogarithmically in the problem size. The results establish that quantum computation, in principle, can tolerate noise below a finite bound, enabling scalable quantum devices and guiding practical implementations.

Abstract

Recent research has demonstrated that quantum computers can solve certain types of problems substantially faster than the known classical algorithms. These problems include factoring integers and certain physics simulations. Practical quantum computation requires overcoming the problems of environmental noise and operational errors, problems which appear to be much more severe than in classical computation due to the inherent fragility of quantum superpositions involving many degrees of freedom. Here we show that arbitrarily accurate quantum computations are possible provided that the error per operation is below a threshold value. The result is obtained by combining quantum error-correction, fault tolerant state recovery, fault tolerant encoding of operations and concatenation. It holds under physically realistic assumptions on the errors.

Figures (7)

• Figure 1: Construction of a Toffoli gate from the primitves given in Table 1. The first figure gives a controlled-controlled-NS. The second gate is a Tofoli gate for the first 3 qubit, whatever the 4th bit is.
• Figure 2: Fault tolerant calculation of the syndrome for the 7 bit code. The even state is obtained from the fault tolerant operation given by [shor]
• Figure 3: Verification of the syndrome of a qubyte. The syndrome for both bit flips $S_B$ and sign flips $S_S$ are calculated twice to learn if the error occured during the syndrome oepration.
• Figure 4: Fault tolerant implementation of a control not between two qubytes. Each qubit of a qubyte interacts at most with one qubit of another qubyte: this is the transversality property.
• Figure 5: Fault tolerant network giving purified $|\pi/8\rangle$ states.