Martin Kliesch

Energy estimation in quantum many-body Hamiltonians is a paradigmatic task in various research fields. In particular, efficient energy estimation may be crucial in achieving a quantum advantage for a practically relevant problem. For instance, the measurement effort poses a crucial bottleneck in variational quantum algorithms. We aim to find the optimal strategy with single-qubit measurements that yields the highest provable accuracy given a total measurement budget. As a central tool, we establish new tail bounds for empirical estimators of the energy. They are useful for identifying measurement settings that improve the energy estimate the most. This task constitutes an NP-hard problem. However, we are able to circumvent this bottleneck and use the tail bounds to develop a practical efficient estimation strategy which we call ShadowGrouping. As the name suggests, it combines shadow estimation methods with grouping strategies for Pauli strings. In numerical experiments, we demonstrate that ShadowGrouping outperforms state-of-the-art methods in estimating the electronic ground-state energies of various small molecules, both in provable and effective accuracy benchmarks. Hence, this work provides a promising way, e.g., to tackle the measurement bottleneck of variational quantum algorithms.

Properties of quantum systems can be estimated using classical shadows, which implement measurements based on random ensembles of unitaries. Originally derived for global Clifford unitaries and products of single-qubit Clifford gates, practical implementations are limited to the latter scheme for moderate numbers of qubits. Beyond local gates, the accurate implementation of very short random circuits with two-local gates is still experimentally feasible and, therefore, interesting for implementing measurements in near-term applications. In this work, we derive closed-form analytical expressions for shadow estimation using brickwork circuits with two layers of parallel two-local Haar-random (or Clifford) unitaries. Besides the construction of the classical shadow, our results give rise to sample-complexity guarantees for estimating Pauli observables. We then compare the performance of shadow estimation with brickwork circuits to the established approach using local Clifford unitaries and find improved sample complexity in the estimation of observables supported on sufficiently many qubits.

The characterization of quantum devices is crucial for their practical implementation but can be costly in experimental effort and classical post-processing. Therefore, it is desirable to measure only information that is relevant for specific applications and develop protocols that require little additional effort. In this work, we focus on the characterization of quantum computers in the context of stabilizer quantum error correction. We prove that (i) physical and (ii) logical error channels induced by Pauli noise can be estimated from syndrome data under minimal conditions. Essentially, any Pauli channel a code can correct can also be estimated from its syndrome measurements. We also provide a concrete estimation algorithm for this task.