Efficient Hamiltonian, structure and trace distance learning of Gaussian states

TL;DR

The paper advances quantum Hamiltonian learning for bosonic Gaussian states by introducing a local-inversion framework that recovers both the Hamiltonian and the interaction graph from measurements that are experimentally feasible (heterodyne) with sample complexity that scales polylogarithmically in the number of modes. It delivers three main results: (i) efficient Hamiltonian and graph learning for Gaussian states with bounded temperature, (ii) the first polynomially-scaled, trace-distance learning bound for Gaussian states, and (iii) robust methods to estimate covariance and relate covariance accuracy to state fidelity. Central to the approach are continuity bounds between covariance and Hamiltonian matrices and the local-inversion technique, which enables submatrix-based inference that does not scale with system size. The results place Gaussian-state learning on a firmer footing than spin-based quantum learning, offering scalable, practically implementable algorithms applicable to continuous-variable quantum systems and Gaussian graphical models. Together, these contributions push forward the understanding of structure learning and parameter estimation in quantum CV systems with potential impact on quantum optics, CV quantum computing, and quantum system identification.

Abstract

In this work, we initiate the study of Hamiltonian learning for positive temperature bosonic Gaussian states, the quantum generalization of the widely studied problem of learning Gaussian graphical models. We obtain efficient protocols, both in sample and computational complexity, for the task of inferring the parameters of their underlying quadratic Hamiltonian under the assumption of bounded temperature, squeezing, displacement and maximal degree of the interaction graph. Our protocol only requires heterodyne measurements, which are often experimentally feasible, and has a sample complexity that scales logarithmically with the number of modes. Furthermore, we show that it is possible to learn the underlying interaction graph in a similar setting and sample complexity. Taken together, our results put the status of the quantum Hamiltonian learning problem for continuous variable systems in a more advanced state when compared to spins, where state-of-the-art results are either unavailable or quantitatively inferior to ours. In addition, we use our techniques to obtain the first results on learning Gaussian states in trace distance with a quadratic scaling in precision and polynomial in the number of modes, albeit imposing certain restrictions on the Gaussian states. Our main technical innovations are several continuity bounds for the covariance and Hamiltonian matrix of a Gaussian state, which are of independent interest, combined with what we call the local inversion technique. In essence, the local inversion technique allows us to reliably infer the Hamiltonian of a Gaussian state by only estimating in parallel submatrices of the covariance matrix whose size scales with the desired precision, but not the number of modes. This way we bypass the need to obtain precise global estimates of the covariance matrix, controlling the sample complexity.

Figures (2)

• Figure 1: Schematic workflow for our Hamiltonian learning protocol given we know the interaction graph. We first perform heterodyne measurements on copies of a Gaussian state $\rho$. We then use the outcomes to obtain empirical estimates of the entries of the covariance matrix and the mean. Importantly, for most of our results we will only require that the precision with which we estimate each entry does not scale with the number of modes in the systems. The reason for that is the next step, our local inversion procedure. Here, we are given a neighborhood structure $\{\mathcal{N}_i\}_{i\in V}$. Roughly speaking, for each $i$, $\mathcal{N}_i$ contains all the vertices that are at most a distance $l$ from $i$, where $l$ depends on the target precision. We then only invert the matrix $(2\hat{V}-i\Omega)$ on the submatrix corresponding to each neighborhood $\mathcal{N}_i$ and "stitch together" the global matrix combining all estimates from the various neighborhoods. We then apply the function that computes $H$ from $(2V-i\Omega)^{-1}$ to the resulting matrix to form our guess of the Hamiltonian.
• Figure 2: LearnGraphGaussian

Theorems & Definitions (57)

• Theorem 3.1: Learning Gaussian states in trace distance (informal)
• Theorem 3.2: Gaussian Hamiltonian learning (informal)
• Theorem 3.3: Learning the graph of Gaussian states, informal
• Lemma 4.1
• Lemma 4.2
• Definition 4.1: Problem 1: Gaussian trace distance learning
• Proposition 4.1
• proof
• Theorem 4.1
• proof