Holographic Classical Shadow Tomography

TL;DR

This paper introduces holographic shadow measurement schemes that enable efficient, scale-invariant quantum state estimation of local Pauli operators using hierarchical quantum circuits, achieving optimal sample complexity without fine-tuning.

Contribution

It presents a novel holographic shadow tomography method utilizing hierarchical quantum circuits for optimal, scalable quantum state learning across all length scales.

Findings

Achieves optimal sample complexity scaling for local Pauli observables in 1D systems.

Demonstrates universality of the sample complexity scaling via a holographic minimal cut framework.

Validates the approach with numerical simulations showing improved quantum state estimation.

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.