Sample efficient tomography via Pauli Measurements

Abstract

Pauli Measurements are the most important measurements in both theoretical and experimental aspects of quantum information science. In this paper, we explore the power of Pauli measurements in the state tomography related problems. Firstly, we show that the \textit{quantum state tomography} problem of $n$-qubit system can be accomplished with $\mathcal{O}(\frac{10^n}{{\epsilon }^2})$ copies of the unknown state using Pauli measurements. As a direct application, we studied the \textit{quantum overlapping tomography} problem introduced by Cotler and Wilczek in Ref. \cite{Cotler_2020}. We show that the sample complexity is $\mathcal{O}(\frac{10^k\cdot \log ((\genfrac{}{}{0ex}{}{n}{k})/\delta ))}{{\epsilon }^2})$ for quantum overlapping tomography of $k$-qubit reduced density matrices among $n$ is quantum system, where $1-\delta$ is the confidential level, and $\epsilon$ is the trace distance error. This can be achieved using Pauli measurements. Moreover, we prove that $\Omega (\frac{\log (n/\delta )}{{\epsilon }^2})$ copies are needed. In other words, for constant $k$, joint, highly entangled, measurements are not asymptotically more efficient than Pauli measurements.

Comments are welcome. The quantum overlapping tomography part was originally announced in arXiv:1904.03218v3

References in corpus (6)

• Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators
• Quantum Overlapping Tomography
• Collective vs local measurements in qubit mixed state estimation
• Optimal estimation of qubit states with continuous time measurements
• Quantum state estimation and large deviations
• Scalable Bayesian Hamiltonian learning

Cited by in corpus (3)

• Quantum Algorithmic Measurement
• Improved quantum data analysis
• Sample optimal Quantum identity testing via Pauli Measurements