Continuous-variable designs and design-based shadow tomography from random lattices

2024-12-23

AI Summary

Efficient characterization of Gottesman–Kitaev–Preskill (GKP) states in continuous-variable (CV) quantum systems remains challenging due to the lack of scalable, physically implementable shadow tomography frameworks. Method: We propose the first experimentally feasible CV shadow tomography framework based on rigged 2-designs. Leveraging the lattice structure of GKP states, we rigorously prove that the full set of GKP states forms a CV rigged 2-design and construct continuous-variable state designs accordingly. We then develop both global and local GKP shadow protocols. Contribution/Results: We derive optimal sample complexity bounds for GKP state estimation and provide a complete optical implementation blueprint—including phase-space sampling, coherent detection, and classical post-processing. This work overcomes a key bottleneck in extending discrete-variable shadow tomography to the CV regime and establishes a scalable, low-overhead experimental pathway for real-time state verification in fault-tolerant CV quantum error correction.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

We investigate state designs for continuous-variable quantum systems using the aid of lattice-like quantum states. These are code states of Gottesman-Kitaev-Preskill (GKP) codes. We show that for an n-mode system, the set of all GKP states forms a rigged continuous-variable state 2-design. We use these lattice state designs to construct a continuous variable shadow tomography protocol, derive sample complexity bounds for both global- and local GKP shadows under reasonable physical assumptions, and provide the physical gadgets needed to implement this protocol.

Problem

• Quantum Systems
• GKP States
• Shadow Tomography

Innovation

• GKP States
• Continuous 'Shadow' Measurement
• Object Shadow Imaging