Quantum error correction with higher Gottesman-Kitaev-Preskill codes: minimal measurements and linear optics

TL;DR

This paper introduces efficient linear optical schemes for GKP error syndrome measurement, reducing measurement complexity and analyzing ancilla state preparation, with implications for improved quantum error correction in continuous-variable systems.

Contribution

It presents minimal measurement schemes for GKP code syndromes using linear optics and homodyne detection, and explores ancilla state preparation feasibility for higher GKP codes.

Findings

Only 2n measurements needed for syndrome extraction in concatenated GKP codes.

Feasibility of ancilla state preparation from single-mode states for simple GKP codes.

Proposed methods to prevent crosstalk between ancillae during syndrome measurements.

Abstract

We propose two schemes to obtain Gottesman-Kitaev-Preskill (GKP) error syndromes by means of linear optical operations, homodyne measurements and GKP ancillae. This includes showing that for a concatenation of GKP codes with a $[n,k,d]$ stabilizer code only $2n$ measurements are needed in order to obtain the complete syndrome information, significantly reducing the number of measurements in comparison to the canonical concatenated measurement scheme and at the same time generalizing linear-optics-based syndrome detections to higher GKP codes. Furthermore, we analyze the possibility of building the required ancilla states from single-mode states and linear optics. We find that for simple GKP codes this is possible, whereas for concatenations with qubit Calderbank-Shor-Steane (CSS) codes of distance $d\geq3$ it is not. We also consider the canonical concatenated syndrome measurements and propose methods for avoiding crosstalk between ancillae. In addition, we make use of the observation that the concatenation of a GKP code with a stabilizer code forms a lattice in order to see the analog information decoding of such codes from a different perspective allowing for semi-analytic calculations of the logical error rates.