Chasing shadows with Gottesman-Kitaev-Preskill codes

TL;DR

The paper develops a GKP-based shadow tomography framework for continuous-variable quantum systems by twirling physical POVMs over the logical GKP Clifford group, effectively projecting measurements onto a depolarizing channel in the encoded logical space. It provides concrete realizations for heterodyne (yielding Gaussian-state decompositions) and photon-parity measurements (leading to randomized Wigner tomography), and introduces a randomized-Wigner scheme that averages over random GKP codes to reconstruct arbitrary bounded observables. The authors establish rigorous bounds via displacement and Gaussian-unitary twirls and Siegel-mean-value techniques for random symplectic lattices, enabling controlled sample complexity and a practical pathway to efficiently describe CV states in terms of logical information. They also discuss applications to classical simulation, decoding procedures, and potential extensions to other bosonic codes, highlighting the deep link between phase-space structure, lattice symmetries, and shadow tomography in quantum information processing.

Abstract

We consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman-Kitaev-Preskill (GKP) error correcting code. Our protocol does not require the input state to be a code state but is implemented by appropriate twirling of the measurement channel, such that the encoded logical tomographic information becomes encoded in the classical shadow. We showcase this protocol for measurements natural in continuous variable (CV) quantum computing. For heterodyne measurement, the protocol yields a probabilistic decomposition of any input state into Gaussian states that simulate the encoded logical information of the input relative to a fixed GKP code where we prove bounds on the Gaussian compressibility of states in this setting. For photon parity measurements, our protocol is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling strategies. Finally, by randomizing over the reference GKP code, we show how Wigner samples of any input state relative to a random GKP codes can be used to estimate any sufficiently bounded observable.

Figures (7)

• Figure 1: Illustration of Gaussian decomposition of arbitrary CV states via twirled heterodyne measurements. Relative to a GKP code described by a lattice $\mathcal{L}\subseteq \mathcal{L}^{\perp}$ is shadow tomography protocol yields a probabilistic decomposition of the input state into Gaussian states that reproduce logical expectation values up to a logical depolarization $\mathcal{M}$.
• Figure 2: The square $\mathbb{Z}^2$ (l.) and hexagonal $A_2$ (r.) GKP codes each encoding a qubit. The logical displacement amplitudes are marked in turqouise and stabilizer displacements are marked in red.
• Figure 3: Different notions of twirling discussed in the main text.
• Figure 4: Figure for $\nu_{M^{\perp}}$ for square and hexagonal GKP codes.
• Figure 5: Phase-space distribution $p_{\sigma}(\cdot, L)$ for the GKP Wigner shadow protocol, which are determined by a randomly chosen lattice $L$.

Theorems & Definitions (18)

• Theorem 1: HKP Huang_2020
• Theorem 2: Full representation of the state
• Lemma 1: Random lattice point sampling
• Lemma 2: Scalar product bound
• Lemma 3: Second moment: random lattice sampling
• Lemma 4: Second moment: random lattice sampling for exact inner means
• Corollary 1: Variance upper bound of protocol \ref{['protocol2']}
• Theorem 3: Random lattice CV shadows
• Theorem 4: Random lattice CV shadows with exact inner means
• Theorem 5: Obstructions against general quantum state tomography CVLearning