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

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 $d\ge 3$ 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.

22 pages, 4 figures. Comments are welcome

References in corpus (6)

• Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency
• Demonstration of a quantum nondemolition sum gate
• Analog quantum error correction with encoding a qubit into an oscillator
• Homological Error Correction: Classical and Quantum Codes
• Fault-tolerant quantum computation with static linear optics
• Qudit surface codes and gauge theory with finite cyclic groups

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