Theory and Experimental Demonstration of Wigner Tomography of Unknown Unitary Quantum Gates

TL;DR

This paper introduces a novel protocol for Wigner tomography of unknown unitary quantum processes, enabling visualization and characterization of quantum gates on IBM devices through combined experimental and classical post-processing methods.

Contribution

The work extends Wigner tomography from known to unknown quantum processes, providing a practical method for experimental reconstruction of arbitrary unitary gates.

Findings

Successful experimental demonstration on IBM quantum devices.

Development of calibration circuits for error quantification.

Implementation of a combined experimental and classical post-processing protocol.

Abstract

We investigate the tomography of unknown unitary quantum processes within the framework of a finite-dimensional Wigner-type representation. This representation provides a rich visualization of quantum operators by depicting them as shapes assembled as a linear combination of spherical harmonics. These shapes can be experimentally tomographed using a scanning-based phase-space tomography approach. However, so far, this approach was limited to $\textit{known}$ target processes and only provided information about the controlled version of the process rather than the process itself. To overcome this limitation, we introduce a general protocol to extend Wigner tomography to $\textit{unknown}$ unitary processes. This new method enables experimental tomography by combining a set of experiments with classical post-processing algorithms introduced herein to reconstruct the unknown process. We also demonstrate the tomography approach experimentally on IBM quantum devices and present the specific calibration circuits required for quantifying undesired errors in the measurement outcomes of these demonstrations.