Optimal Universal Quantum Error Correction via Bounded Reference Frames

TL;DR

This paper constructs approximate quantum error correcting codes that are covariant under local unitary groups using quantum reference frames, achieving near-optimal error correction scaling and addressing fundamental limitations in universal quantum error correction.

Contribution

It introduces a method to create covariant quantum codes with optimal error scaling using quantum reference frames, overcoming the Eastin-Knill theorem constraints.

Findings

Error scales as 1/n^2 for small erasures, akin to Heisenberg limit

Error scales as 1/n when all qudits are at risk of erasure

Codes are capable of efficiently correcting various erasure errors

Abstract

Error correcting codes with a universal set of transversal gates are a desideratum for quantum computing. Such codes, however, are ruled out by the Eastin-Knill theorem. Moreover, the theorem also rules out codes which are covariant with respect to the action of transversal unitary operations forming continuous symmetries. In this work, starting from an arbitrary code, we construct approximate codes which are covariant with respect to the entire group of local unitary gates in dimension $d$, using quantum reference frames. We show that our codes are capable of efficiently correcting different types of erasure errors. When only a small fraction of the $n$ qudits upon which the code is built are erased, our covariant code has an error that scales as $1/n^2$, which is reminiscent of the Heisenberg limit of quantum metrology. When every qudit has a chance of being erased, our covariant code has an error that scales as $1/n$. We show that the error scaling is optimal in both cases. Our approach has implications for fault-tolerant quantum computing, reference frame error correction, and the AdS-CFT duality.