# Hamiltonian-Driven Shadow Tomography of Quantum States

arXiv: https://arxiv.org/abs/2102.10132 · HTML: https://mlsanity.com/paper/2102.10132/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 48 references that are in the map.
- First posted on arXiv 19 Feb 2021.

## Builds on (most cited first, 5 of 5)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Attention-based Quantum Tomography](https://mlsanity.com/paper/2006.12469/) (2020)
- [Adaptive Quantum State Tomography with Neural Networks](https://mlsanity.com/paper/1812.06693/) (2018)
- [Classical Shadows With Noise](https://mlsanity.com/paper/2011.11580/) (2020)
- [Neural network state estimation for full quantum state tomography](https://mlsanity.com/paper/1811.06654/) (2018)

## Built on by (most cited first, 4 of 4)

- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows](https://mlsanity.com/paper/2402.17911/) (2024)
- [Hardware-efficient learning of quantum many-body states](https://mlsanity.com/paper/2212.06084/) (2022)

## Abstract (arXiv)

Classical shadow tomography provides an efficient method for predicting functions of an unknown quantum state from a few measurements of the state. It relies on a unitary channel that efficiently scrambles the quantum information of the state to the measurement basis. Facing the challenge of realizing deep unitary circuits on near-term quantum devices, we explore the scenario in which the unitary channel can be shallow and is generated by a quantum chaotic Hamiltonian via time evolution. We provide an unbiased estimator of the density matrix for all ranges of the evolution time. We analyze the sample complexity of the Hamiltonian-driven shadow tomography. For Pauli observables, we find that it can be more efficient than the unitary-2-design-based shadow tomography in a sequence of intermediate time windows that range from an order-1 scrambling time to a time scale of D1/6, given the Hilbert space dimension D. In particular, the efficiency of predicting diagonal Pauli observables is improved by a factor of D without sacrificing the efficiency of predicting off-diagonal Pauli observables.