Hamiltonian-Driven Shadow Tomography of Quantum States

TL;DR

This paper introduces a Hamiltonian-driven shadow tomography method that uses shallow, chaotic Hamiltonian evolutions to efficiently predict quantum states, outperforming traditional methods in certain regimes.

Contribution

It develops an unbiased estimator for quantum states using shallow Hamiltonian evolutions and analyzes its sample complexity, showing improved efficiency over existing shadow tomography techniques.

Findings

More efficient than unitary 2-designs for certain observables

Improves diagonal Pauli observable prediction by a factor of D

Effective over a range of evolution times from scrambling to D^{1/6}

Abstract

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 $D^{1/6}$, 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.