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arXiv:2507.13237

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

Qingyue Zhang, Dayue Qin, You Zhou, Xu Feng, Jens Eisert, Zhou, You

Published Jul 17, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2507.13237

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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-ZZZ 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.
Reliable and scalable readout strategies are essential for quantum technologies. As quantum processors grow, extracting useful information must remain feasible without measurement circuits becoming a dominant bottleneck. Randomized measurements and classical shadows provide a powerful route, but global estimation is conventionally associated with highly random ensembles that require increasing circuit depth and hence substantial experimental overhead. In this work, we show that substantially less randomness suffices when the readout is meaningfully adapted to the quantities being estimated. We introduce shallow phase shadows, based on a sparse Clifford-IQP ensemble, and prove efficient global estimation of stabilizer-state fidelities despite the ensemble not forming an approximate relative-error design. On all-to-all architectures, the protocol admits a constant-depth implementation using mid-circuit measurements and classical feedforward, or logarithmic depth without auxiliary systems. The protocol requires only controlled-phase entangling gates and offers a tunable trade-off between circuit resources and estimation accuracy, making it particularly amenable to experimentally relevant architectures with long-range connectivity. Our results show that scalable quantum readout need not reproduce generic randomness: task-adapted randomization can enable substantially shallower global characterization protocols.
Abstract We present a robust shadow estimation protocol for wide classes of low-depth measurement circuits that mitigates noise as long as the effective measurement map including noise is locally unitarily invariant. This is in practice an excellent approximation, encompassing for instance the case of ideal single-qubit Clifford gates composing the first circuit layer of an otherwise arbitrary circuit architecture and even non-Markovian, gate-dependent noise in the rest of the circuit. We argue that for weakly-correlated local noise, the measurement channel has an efficient matrix-product representation, and show how to estimate this directly from experimental data using tensor-network tools, eliminating the need for analytical or numeric calculations. We illustrate the relevance of our method with both numerics and proof-of-principle experiments on an IBM Quantum device. Numerically, we show that unmitigated shallow shadows with noisy circuits become more biased as the depth increases. In contrast, using the same number of samples, robust ultra-shallow shadows become more precise with increasing depth for relevant parameter regimes. The gain in sample efficiency is still limited by the noise per gate, resulting in an optimal circuit depth per noise level. Experimentally, we observe improved precision in two simple fidelity estimation tasks using five-qubit circuits with up to two layers of entangling gates, by about an order of magnitude. Under the practical constraints of current and near-term noisy quantum devices, our method maximally realizes the potential of shadow estimation with global rotations and identifies its fundamental limitations in the presence of noise.
Extracting information efficiently from quantum systems is crucial for quantum information processing. Classical shadows enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Introducing a shallow random quantum circuit before measurements improves sample efficiency for high-weight Pauli observables and low-rank properties. However, in practice, these circuits can be noisy and bias the measurement results. Here, we propose the robust shallow shadows, which employs Bayesian inference to learn and mitigate noise in postprocessing. We analyze noise effects on sample complexity and the optimal circuit depth. We provide theoretical guarantees for the success of error mitigation under a wide class of noise processes. Experimental validation on a superconducting quantum processor confirms the advantage of our method, even in the presence of realistic noise, over single-qubit measurements for predicting diverse state properties, such as fidelity and entanglement entropy. Our protocol thus offers a scalable, robust, and sample-efficient method for quantum state characterization on near-term quantum devices. Fast and reliable characterisation of quantum systems is a key part of quantum technologies development. Here, the authors propose and demonstrate a way to embed noise characterisation in classical shadow estimation of nonlocal properties, enabling an efficient way to extract information from noisy quantum systems.
Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.
Quantum state tomography is a fundamental technique for characterizing quantum systems, but its scalability remains a significant challenge. This paper investigates an efficient alternative using classical shadows to reconstruct a Bell state with high fidelity. Classical shadows provide a compressed representation of a quantum state using randomized measurements, reducing the measurement complexity compared to full quantum tomography when full state reconstruction is not required. We employed a quantum circuit to generate a Bell state and utilize 1000 snapshots to construct its classical shadow. The reconstructed density matrix is evaluated using fidelity and norm difference metrics against the ideal Bell state. Our analysis demonstrates that as the number of snapshots increases, the fidelity of reconstruction stabilizes around 0.98-1.0, with the norm difference decreasing accordingly. The results also confirm the convergence of reconstructed states towards the ideal Bell state, highlighting the efficiency and accuracy of classical shadows in quantum state estimation. Shallow shadow tomography uses shallow random circuits to estimate global quantum state properties on noisy hardware. By combining low-depth entangling gates with randomized measurements and noise-aware postprocessing, it reduces sample complexity compared to Pauli shadows while remaining experimentally feasible. Recent experiments on superconducting processors confirmed up to fivefold measurement savings.
We provide practical and powerful schemes for learning properties of a quantum state using a small number of measurements. Specifically, we present a randomized measurement scheme modulated by the depth of a random quantum circuit in one spatial dimension. This scheme interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. We focus on the regime where depth scales logarithmically in the system size and provide evidence that this retains the desirable sample complexity properties of both extremal schemes while also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We achieve our findings by bringing together tools from shadow estimation, random circuits, and tensor networks.