Comparing the Overhead of Topological and Concatenated Quantum Error Correction

TL;DR

This paper compares the resource overhead of topological and concatenated quantum error correction codes using numerical estimates for different quantum technologies, highlighting their suitability based on error rates and gate efficiencies.

Contribution

It provides a detailed numerical comparison of topological and concatenated quantum error correction overheads across various quantum hardware platforms.

Findings

Topological codes require fewer resources at high physical error rates.

Concatenated codes are more efficient below approximately 10E-7 error rate.

Different quantum technologies benefit from different error correction strategies.

Abstract

This work compares the overhead of quantum error correction with concatenated and topological quantum error-correcting codes. To perform a numerical analysis, we use the Quantum Resource Estimator Toolbox (QuRE) that we recently developed. We use QuRE to estimate the number of qubits, quantum gates, and amount of time needed to factor a 1024-bit number on several candidate quantum technologies that differ in their clock speed and reliability. We make several interesting observations. First, topological quantum error correction requires fewer resources when physical gate error rates are high, white concatenated codes have smaller overhead for physical gate error rates below approximately 10E-7. Consequently, we show that different error-correcting codes should be chosen for two of the studied physical quantum technologies - ion traps and superconducting qubits. Second, we observe that the composition of the elementary gate types occurring in a typical logical circuit, a fault-tolerant circuit protected by the surface code, and a fault-tolerant circuit protected by a concatenated code all differ. This also suggests that choosing the most appropriate error correction technique depends on the ability of the future technology to perform specific gates efficiently.