Fourier-style Quantum State Tomography and Purity Measurement of a Multi-qubit System from Bloch Rotations

TL;DR

This paper introduces a quantum state tomography method using Bloch rotations that reduces measurement complexity for multi-qubit systems, enabling more efficient estimation of global properties like purity.

Contribution

It proposes a simple protocol based on single-pulse X/Y rotations that decreases the exponential measurement growth in quantum state tomography and purity estimation.

Findings

Statistical variance scales as 5^N / N_tot for full tomography.

Variance in purity estimation scales as 7^N / N_tot^2.

The method simplifies global property measurements in many-body quantum systems.

Abstract

Quantum state tomography and other measures of the global properties of a quantum state are indispensable tools in understanding many body physics through quantum simulators. Unfortunately, the number of experimental measurements of the system required to estimate these global quantities scales exponentially with system size. Here, we consider the use of random-axis measurements for quantum state tomography and state purity estimation. We perform a general analysis of the statistical deviation in such methods for any given algorithm. We then propose a simple protocol which relies on single-pulse X/Y rotations only. We find that it reduces the basis of the exponential growth, calculating the statistical variance to scale as $\sum_{a,b}\left\lvert\Delta\rho_{ab}\right\rvert^{2}\sim 5^{N}/N_{\rm tot}$ for full tomography, and $\left(\Delta\mu\right)^{2}\sim 7^{N}/N_{\rm tot}^{2}$ for purity estimation, for $N$ qubits and $N_{\rm tot}$ measurements performed.