Approximate quantum error correcting codes from conformal field theory

TL;DR

This paper investigates how conformal field theories can be used as quantum error correcting codes, identifying conditions for error correction thresholds and the number of protected qubits, with implications for holography and quantum gravity.

Contribution

It provides a general analysis of CFT-based quantum codes under dephasing noise, establishing error correction thresholds and bounds on protected logical qubits.

Findings

Finite decoding threshold depends on minimal scaling dimension > 1/2

Number of protected qubits grows at least as log log n

Quantum Ising model exhibits a finite threshold for certain noise types

Abstract

The low-energy subspace of a conformal field theory (CFT) can serve as a quantum error correcting code, with important consequences in holography and quantum gravity. We consider generic 1+1D CFT codes under extensive local dephasing channels and analyze their error correctability in the thermodynamic limit. We show that (i) there is a finite decoding threshold if and only if the minimal nonzero scaling dimension in the fusion algebra generated by the jump operator of the channel is larger than $1/2$ and (ii) the number of protected logical qubits $k \geq \Omega( \log \log n)$, where $n$ is the number of physical qubits. As an application, we show that the one-dimensional quantum critical Ising model has a finite threshold for certain types of dephasing noise. Our general results also imply that a CFT code with continuous symmetry saturates a bound on the recovery fidelity for covariant codes.