# Quantum

# Error Correction

# Quantum error-correcting (QEC) codes are essential for realizing large-scale quantum computation. QEC codes protect quantum information by encoding each logical computational-basis state into a higher-dimensional physical subspace in a manner that permits errors to be detected and corrected. The first QEC codes were generic codes, i.e., they made no assumptions about the underlying hardware, hence their error models do not exploit hardware-specific biases towards particular errors. The achievable code rates of generic QEC codes are therefore constrained by fundamental limits, such as the quantum Hamming bound and the quantum singleton bound. Consequently, a generic QEC code demands more encoding overhead than would be necessary when the quantum computation is run on hardware with a small set of dominant errors, rather than the full error set assumed by that code. This overhead excess impedes the implementation of large-scale quantum computation as compared to what could be accomplished with a hardware-efficient QEC code, viz., one that is matched to the chosen physical implementation.

Quantum error-correcting (QEC) codes are essential for realizing large-scale quantum computation. QEC codes protect quantum information by encoding each logical computational-basis state into a higher-dimensional physical subspace in a manner that permits errors to be detected and corrected. The first QEC codes were generic codes, i.e., they made no assumptions about the underlying hardware, hence their error models do not exploit hardware-specific biases towards particular errors. The achievable code rates of generic QEC codes are therefore constrained by fundamental limits, such as the quantum Hamming bound and the quantum singleton bound. Consequently, a generic QEC code demands more encoding overhead than would be necessary when the quantum computation is run on hardware with a small set of dominant errors, rather than the full error set assumed by that code. This overhead excess impedes the implementation of large-scale quantum computation as compared to what could be accomplished with a hardware-efficient QEC code, viz., one that is matched to the chosen physical implementation.

## Hardware-efficient bosonic quantum error-correcting codes based on symmetry operators

Hardware-efficient bosonic quantum error-correcting codes based on symmetry operators

TABLE I. Hardware-efficiency metrics for the GKP code [17], the χ (2) PCC, the χ (2) EECC, and the χ (2) BC when all four encode a single logical qubit. GEE: Gaussian embedded error. PNR: photon number-resolving detection. PNP: photon-number parity measurement. GPNP: generalized photon-number parity measurement. CM: channel monitoring.

# In Phys. Rev. A **97**, 032323, we developed a new framework of quantum error correction that exploits the inherent symmetries in the physical substrate underlying a given quantum device. This was recognized as the first instance of protecting against photon loss errors with a photon number redundancy that scales only linearly (as opposed to quadratically in the prior art) in system size. This work broadened into an overall Thesis on the theoretical design of quantum computation and error correction that harnesses hardware-specific physics to improve system performance, known as “hardware-efficient quantum computation”.

In Phys. Rev. A

**97**, 032323, we developed a new framework of quantum error correction that exploits the inherent symmetries in the physical substrate underlying a given quantum device. This was recognized as the first instance of protecting against photon loss errors with a photon number redundancy that scales only linearly (as opposed to quadratically in the prior art) in system size. This work broadened into an overall Thesis on the theoretical design of quantum computation and error correction that harnesses hardware-specific physics to improve system performance, known as “hardware-efficient quantum computation”.