Contractive Unitary and Classical Shadow Tomography

TL;DR

This paper introduces a hybrid measurement protocol using contractive unitaries that reduces the sample complexity of quantum state tomography to approximately 1.8^k, improving efficiency over previous methods for large quantum systems.

Contribution

It proposes a novel hybrid measurement protocol combining local randomness and global deterministic unitaries, notably the contractive unitary, to enhance quantum state tomography efficiency.

Findings

Achieves sample complexity of ~1.8^k for local operator estimation.

Introduces the concept of contractive unitaries for efficient state characterization.

Demonstrates the protocol's compatibility with atom array quantum processors.

Abstract

The rapid development of quantum technology demands efficient characterization of complex quantum many-body states. However, full quantum state tomography requires an exponential number of measurements in system size, preventing its practical use in large-scale quantum devices. A major recent breakthrough in this direction, called classical shadow tomography, significantly reduces the sample complexity, the number of samples needed to estimate properties of a state, by implementing random Clifford rotations before measurements. Despite many recent efforts, reducing the sample complexity below $\mathbf{2^k}$ for extracting any non-successive local operators with a size $\sim \mathbf{k}$ remains a challenge. In this work, we achieve a significantly smaller sample complexity of $\mathbf{\sim 1.8^k}$ using a protocol that hybridizes locally random and globally deterministic unitary operations. The key insight is the discovery of a deterministic global unitary, termed as \textit{contractive unitary}, which is more efficient in reducing the operator size to enhance tomography efficiency. The contractive unitary perfectly matches the advantages of the atom array quantum computation platform and is readily realized in the atom array quantum processor. More importantly, it highlights a new strategy in classical shadow tomography, demonstrating that a random-deterministic hybridized protocol can be more efficient than fully random measurements.