Holographic Classical Shadow Tomography

TL;DR

This work introduces holographic shadows, a class of randomized measurements for classical shadow tomography that achieve the optimal $d^k$ scaling for contiguous Pauli observables across length scales without tuning circuit depth or measurement rate. It develops two hierarchical schemes—the tree circuit and holographic random tensor networks (RTN)—and shows, via recursive PLR calculations and a statistical-mechanics mapping, that the sample complexity is governed by minimal boundary/bulk cuts, yielding scale-free performance. The holographic RTN approach connects the scaling to an effective central charge $c_{ ext{eff}}(p,q)$ tied to bulk curvature, with numerical evidence from min-cut analyses supporting the $d^k$-scaling and quantifying tiling-dependent corrections. The results imply practical, scalable quantum state learning across scales and offer a path toward geometry-informed shadow tomography with potential realizations in generalized MERA networks and fusion-based implementations.

Abstract

We introduce "holographic shadows", a new class of randomized measurement schemes for classical shadow tomography that achieves the optimal scaling of sample complexity for learning geometrically local Pauli operators at any length scale, without the need for fine-tuning protocol parameters such as circuit depth or measurement rate. Our approach utilizes hierarchical quantum circuits, such as tree quantum circuits or holographic random tensor networks. Measurements within the holographic bulk correspond to measurements at different scales on the boundary (i.e. the physical system of interests), facilitating efficient quantum state estimation across observable at all scales. Considering the task of estimating string-like Pauli observables supported on contiguous intervals of $k$ sites in a 1D system, our method achieves an optimal sample complexity scaling of $\sim d^k\mathrm{poly}(k)$, with $d$ the local Hilbert space dimension. We present a holographic minimal cut framework to demonstrate the universality of this sample complexity scaling and validate it with numerical simulations, illustrating the efficacy of holographic shadows in enhancing quantum state learning capabilities.

Figures (9)

• Figure S1: Schematic of (a) tree circuit and (b) holographic circuit.
• Figure S2: Comparison of Pauli learning rates in tree circuits and shallow circuits with optimal depth. The support is chosen to be contiguous, with $k = 2^m$. It can be seen that the tree circuit gives a smaller shadow norm when the support size is small, but the growth of the shadow norm with support size is slower in the shallow circuit.
• Figure S3: Schematic of (a) unitary picture, where each gate is unitary; (b) MPO picture, where each MPO layer can be viewed as a layer of unitary gates. Note that this requires measurements represented by red triangles/dots to have different bond dimensions, as indicated by different line thickness; (c) RTN picture, where each MPO tensor is a random tensor.
• Figure S4: Holographic RTN with (a){3,7} tiling and (b) {5,4} tiling. The green triangles and pentagons represent random tensors and the red dots represent measurement on bulk legs.
• Figure S5: Scaling of $\Vert P\Vert_{\mathcal{E}_\sigma}^2$ in holographic circuit with (a) {3,7} tiling and (b) {5,4} tiling.