A Mathematical Structure for Amplitude-Mixing Error-Transparent Gates for Binomial Codes

TL;DR

This paper introduces a mathematical framework for designing continuous, error-transparent amplitude-mixing gates for binomial quantum codes, enhancing active quantum error correction against photon loss.

Contribution

It presents the concept of 'parity nested' operations to achieve error-transparent amplitude-mixing gates for binomial codes, extending error correction beyond phase gates.

Findings

Error-transparency achieved with multiple squeezing orders.

Single-order squeezing can protect against photon jumps.

Framework applicable to experimental implementations.

Abstract

Bosonic encodings of quantum information offer hardware-efficient, noise-biased approaches to quantum error correction relative to qubit register encodings. Implementations have focused in particular on error correction of stored, idle quantum information, whereas quantum algorithms are likely to desire high duty cycles of active control. Error-transparent operations are one way to preserve error rates during operations, but, to the best of our knowledge, only phase gates have so far been given an explicitly error-transparent formulation for binomial encodings. Here, we introduce the concept of 'parity nested' operations, and show how these operations can be designed to achieve continuous amplitude-mixing logical gates for binomial encodings that are fully error-transparent to the photon loss channel. For a binomial encoding that protects against l photon losses, the construction requires $\lfloor$l/2$\rfloor$ + 1 orders of generalized squeezing in the parity nested operation to fully preserve this protection. We further show that error-transparency to all the correctable photon jumps, but not the no-jump errors, can be achieved with just a single order of squeezing. Finally, we comment on possible approaches to experimental realization of this concept.