Phase shadow: A noise-tolerant path to global quantum property estimation

TL;DR

The paper introduces a noise-robust phase shadow measurement scheme using controlled-Z circuits, enabling efficient and reliable estimation of global quantum properties in noisy quantum systems, especially suited for current quantum architectures.

Contribution

It develops a new phase shadow framework with controlled-Z circuits, offering noise robustness and computational efficiency improvements over previous methods.

Findings

Achieves performance comparable to Clifford-based shadows.

Supports noise-robust estimation via classical post-processing.

Provides an efficient algorithm for stabilizer state analysis.

Abstract

Measuring global quantum properties-such as the fidelity to complex multipartite states-is both an essential and experimentally challenging task. Classical shadow estimation offers favorable sample complexity, but typically relies on many-qubit circuits that are difficult to realize on current platforms. We propose the robust phase shadow scheme, a measurement framework based on random circuits with controlled-$Z$ as the unique entangling gate type, tailored to architectures such as trapped ions and neutral atoms. Leveraging tensor diagrammatic reasoning, we rigorously analyze the induced circuit ensemble and show that phase shadows match the performance of full Clifford-based ones. Importantly, our approach supports a noise-robust extension via purely classical post-processing, enabling reliable estimation under gate-dependent noise where existing techniques often fail. Additionally, by exploiting structural properties of random stabilizer states, we design an efficient post-processing algorithm that resolves a key computational bottleneck in previous shadow protocols. Our results enhance the practicality of shadow-based techniques, providing a robust and scalable route for estimating global properties in noisy quantum systems.