Jens Eisert

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.

Throughout its history, the theory of quantum error correction has heavily benefited from translating classical concepts into the quantum setting. In particular, classical notions of weight enumerators, which relate to the performance of an error-correcting code, and MacWilliams' identity, which helps to compute enumerators, have been generalized to the quantum case. In this work, we establish a distinct relationship between the theoretical machinery of quantum weight enumerators and a seemingly unrelated physics experiment: we prove that Rains' quantum shadow enumerators - a powerful mathematical tool - arise as probabilities of observing fixed numbers of triplets in a Bell sampling experiment. This insight allows us to develop here a rigorous framework for the direct measurement of quantum weight enumerators, thus enabling experimental and theoretical studies of the entanglement structure of any quantum error-correcting code or state under investigation. On top of that, we derive concrete sample complexity bounds and physically-motivated robustness guarantees against unavoidable experimental imperfections. Finally, we experimentally demonstrate the possibility of directly measuring weight enumerators on a trapped-ion quantum computer. Our experimental findings are in good agreement with theoretical predictions and illuminate how entanglement theory and quantum error correction can cross-fertilize each other once Bell sampling experiments are combined with the theoretical machinery of quantum weight enumerators.

Quantum computers are now on the brink of outperforming their classical counterparts. One way to demonstrate the advantage of quantum computation is through quantum random sampling performed on quantum computing devices. However, existing tools for verifying that a quantum device indeed performed the classically intractable sampling task are either impractical or not scalable to the quantum advantage regime. The verification problem thus remains an outstanding challenge. Here, we experimentally demonstrate efficiently verifiable quantum random sampling in the measurement-based model of quantum computation on a trapped-ion quantum processor. We create and sample from random cluster states, which are at the heart of measurement-based computing, up to a size of 4 x 4 qubits. By exploiting the structure of these states, we are able to recycle qubits during the computation to sample from entangled cluster states that are larger than the qubit register. We then efficiently estimate the fidelity to verify the prepared states -- in single instances and on average -- and compare our results to cross-entropy benchmarking. Finally, we study the effect of experimental noise on the certificates. Our results and techniques provide a feasible path toward a verified demonstration of a quantum advantage.

Extracting tomographic information about quantum states is a crucial task in the quest towards devising high-precision quantum devices. Current schemes typically require measurement devices for tomography that are a priori calibrated to high precision. Ironically, the accuracy of the measurement calibration is fundamentally limited by the accuracy of state preparation, establishing a vicious cycle. Here, we prove that this cycle can be broken and the dependence on the measurement device's calibration significantly relaxed. We show that exploiting the natural low-rank structure of quantum states of interest suffices to arrive at a highly scalable `blind' tomography scheme with a classically efficient post-processing algorithm. We further improve the efficiency of our scheme by making use of the sparse structure of the calibrations. This is achieved by relaxing the blind quantum tomography problem to the de-mixing of a sparse sum of low-rank matrices. We prove that the proposed algorithm recovers a low-rank quantum state and the calibration provided that the measurement model exhibits a restricted isometry property. For generic measurements, we show that it requires a close-to-optimal number of measurement settings. Complementing these conceptual and mathematical insights, we numerically demonstrate that robust blind quantum tomography is possible in a practical setting inspired by an implementation of trapped ions.

The dynamical structure factor is one of the experimental quantities crucial in scrutinizing the validity of the microscopic description of strongly correlated systems. However, despite its long-standing importance, it is exceedingly difficult in generic cases to numerically calculate it, ensuring that the necessary approximations involved yield a correct result. Acknowledging this practical difficulty, we discuss in what way results on the hardness of classically tracking time evolution under local Hamiltonians are precisely inherited by dynamical structure factors; and hence offer in the same way the potential computational capabilities that dynamical quantum simulators do: We argue that practically accessible variants of the dynamical structure factors are BQP-hard for general local Hamiltonians. Complementing these conceptual insights, we improve upon a novel, readily available, measurement setup allowing for the determination of the dynamical structure factor in different architectures, including arrays of ultra-cold atoms, trapped ions, Rydberg atoms, and superconducting qubits. Our results suggest that quantum simulations employing near-term noisy intermediate scale quantum devices should allow for the observation of features of dynamical structure factors of correlated quantum matter in the presence of experimental imperfections, for larger system sizes than what is achievable by classical simulation.