Abstract
Quantum error correction (QEC) plays an essential role in fault-tolerantly realizing quantum algorithms of practical interest. Among different approaches to QEC, encoding logical quantum information in harmonic oscillator modes has been shown to be promising and hardware efficient. In this work, we study multimode Gottesman-Kitaev-Preskill (GKP) codes, encoding a qubit in many oscillators, through a lattice perspective. In particular, we implement a closest point decoding strategy for correcting random Gaussian shift errors. For decoding a generic multimode GKP code, we first identify its corresponding lattice followed by finding the closest lattice point in its symplectic dual lattice to a candidate shift error compatible with the error syndrome. We use this method to characterize the error-correction capabilities of several known multimode GKP codes, including their code distances and fidelities. We also perform numerical optimization of multimode GKP codes up to ten modes and find three instances (with three, seven, and nine modes) with better code distances and fidelities compared to the known GKP codes with the same number of modes. While exact closest point decoding incurs exponential time cost in the number of modes for general unstructured GKP codes, we give several examples of structured GKP codes (i.e., of the repetition-rectangular GKP code types) where the closest point decoding can be performed exactly in linear time. For the surface-GKP code, we show that the closest point decoding can be performed exactly in polynomial time with the help of a minimum-weight-perfect-matching algorithm (MWPM). We show that this MWPM closest point decoder improves both the fidelity and the noise threshold of the surface-GKP code to 0.602 compared to the previously studied MWPM decoder assisted by log-likelihood analog information, which yields a noise threshold of 0.599.
- Received 19 April 2023
- Accepted 30 October 2023
- Corrected 23 January 2024
DOI:https://doi.org/10.1103/PRXQuantum.4.040334
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Corrections
23 January 2024
Correction: Equation (118) contained an error and has been fixed. Numerous minor typographical errors have been fixed throughout.
Popular Summary
Quantum error correction (QEC) is critical for implementing quantum algorithms of significant practical interest. The idea of QEC is to encode quantum information redundantly onto several physical elements, followed by using a classical decoder to determine likely errors from the syndrome measurement results. Gottesman-Kitaev-Preskill (GKP) code is a type of QEC code that encodes quantum information onto one or multiple oscillators, and its relation to lattices has been well known. Despite that, a lattice-based decoder has not been implemented because decoding a generic GKP code is equivalent to solving the NP-hard closest-point problem. Hence, there is no prior evidence if it is possible to decode large instances of GKP codes from the lattice perspective, or if there is any advantage of using a lattice-based decoder for GKP codes.
In this work, we demonstrate that certain structured GKP codes, such as repetition-GKP codes and surface-GKP codes, do admit efficient lattice-based decoders. For the latter, we show that the lattice-based decoding can be performed exactly in polynomial time, and it achieves higher code fidelity and noise threshold compared to a previously studied decoder. We also use a general-purpose closest-point decoder as a subroutine to search for optimized GKP code and find several instances with better code distances and fidelities compared to the known GKP codes with the same number of modes.
Our work inspires further research of implementing efficient lattice-based decoders for other families of structured GKP codes and finding new GKP codes with better QEC properties.
