Abstract We present an experimental scanning-based tomography approach for near-term quantum devices. The underlying method has previously been introduced in an ensemble-based NMR setting. Here we provide a tutorial-style explanation along with suitable software tools to guide experimentalists in its adaptation to near-term pure-state quantum devices. The approach is based on a Wigner-type representation of quantum states and operators. These representations provide a rich visualization of quantum operators using shapes assembled from a linear combination of spherical harmonics. These shapes (called droplets in the following) can be experimentally tomographed by measuring the expectation values of rotated axial tensor operators. We present an experimental framework for implementing the scanning-based tomography technique for circuit-based quantum computers and showcase results from IBM quantum experience. We also present a method for estimating the density and process matrices from experimentally tomographed Wigner functions (droplets). This tomography approach can be directly implemented using the Python-based software package .
Abstract We present an experimental scanning-based tomography approach for near-term quantum devices. The underlying method has previously been introduced in an ensemble-based NMR setting. Here we provide a tutorial-style explanation along with suitable software tools to guide experimentalists in its adaptation to near-term pure-state quantum devices. The approach is based on a Wigner-type representation of quantum states and operators. These representations provide a rich visualization of quantum operators using shapes assembled from a linear combination of spherical harmonics. These shapes (called droplets in the following) can be experimentally tomographed by measuring the expectation values of rotated axial tensor operators. We present an experimental framework for implementing the scanning-based tomography technique for circuit-based quantum computers and showcase results from IBM quantum experience. We also present a method for estimating the density and process matrices from experimentally tomographed Wigner functions (droplets). This tomography approach can be directly implemented using the Python-based software package .
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 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 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.
We study the tomography of propagators for spin systems in the context of finite-dimensional Wigner representations, which completely characterize and visualize operators using shapes assembled from linear combinations of spherical harmonics. The Wigner representation of a propagator can be experimentally recovered by measuring expectation values of rotated axial spherical tensor operators in an augmented system with an additional ancilla qubit. The methodology is experimentally demonstrated for standard one-qubit quantum gates using nuclear magnetic resonance spectroscopy. In particular, this approach provides a direct and compelling visualization of the spinor property of the propagators corresponding to the rotation of a spin-1/2 particle.
We study the tomography of multispin quantum states in the context of finite-dimensional Wigner representations. An arbitrary operator can be completely characterized and visualized using multiple shapes assembled from linear combinations of spherical harmonics [A. Garon, R. Zeier, and S. J. Glaser, Phys. Rev. A 91, 042122 (2015)]. We develop a general methodology to experimentally recover these shapes by measuring expectation values of rotated axial spherical tensor operators and provide an interpretation in terms of fictitious multipole potentials. Our approach is experimentally demonstrated for quantum systems consisting of up to three spins using nuclear magnetic resonance spectroscopy.
In this thesis, we introduce new methodologies for the visualization, evaluation, and optimal control of spin systems based on four publications appended in Appendix A -D. Essential parts are established on the DROPS representation, a phase-space method which exploits generalized Wigner functions, denoted as droplets, to represent and visualize quantum systems.In the first publication "Wigner tomography of multispin quantum states" in Appendix A, a general sampling theorem for spherical functions is derived.This is evolved to a novel Wigner quantum state tomography: A technique to reconstruct the surface of the droplets representing quantum states by measuring expectation values of rotated axial tensor operators.This concept is extended in the second publication "Wigner process tomography: Visualization of spin propagators and their spinor properties" in Appendix B to experimentally recover the phase space representation of propagators by mapping these operators to density matrices of an augmented system with an additional ancillar qubit and by applying a modified form of the Wigner quantum state tomography.Both techniques are experimentally demonstrated for quantum states of systems up to three qubits and for standard 1 qubit gates using nuclear magnetic resonance spectroscopy.In doing so, the droplets are associated to measurable quantities, which were originally the mere result of an abstract mathematical mapping, and an interpretation in form of multipoles is provided.Those tomography schemes can be regarded as universally applicable validation methods for quantum systems.They are particularly well suited for the characterization of spin operators, pulse sequence elements, or entire pulse sequences in spectroscopy and in quantum information processing.In a third publication "Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems" in Appendix C, the concept of the DROPS representation, which is in its original form restricted to systems of up to four spins 1/2 due to computational challenges, is generalized to any number of coupled spins 1/2 and to two coupled spins with arbitrary spin numbers.These obstacles are circumvented by extending and introducing two different approaches based on explicit projection operators and coefficients of fractional parentage.This results in a powerful and versatilely applicable visualization tool for the analysis of highly diverse spin topologies and paves the way for the Wigner tomography schemes in such quantum systems.This thesis is complemented by proposing novel optimal control strategies, which, combined with the previous techniques, provide a rich toolbox for the control, analysis, and verification of spin systems.In a last publication "Linking the rotation of a rigid body to the Schrödinger equation: The quantum tennis racket effect and beyond" in Appendix D, efficient and robust control strategies are introduced by linking the free rotation of a classical rigid body to the control of two-level quantum systems using external electromagnetic fields.In the case of state to state transfer and gate implementation, a new class of optimal controls is derived.This relies on two free parameters and allows for a variable efficiency, time, and robustness of the transfer process.An experimental demonstration is presented using techniques of nuclear magnetic resonance.
We present Tomography-NMR, an open-source Python package that reconstructs quantum density matrices from spectroscopic measurement data. The package implements a complete analysis pipeline for two-qubit quantum state tomography based on the product operator formalism: raw time-domain signals are Fourier-transformed into frequency-domain spectra, spectral peak intensities are mapped to expansion coefficients of the density matrix, and the full quantum state is reconstructed. Three integration methods are provided for different use cases: direct peak height measurement and fixed-parameter numerical integration require no theoretical reference and are suited to unknown states, achieving fidelities of approximately 98\% on a benchmark set of known states, while a systematic parameter optimization against a known target state achieves reconstruction fidelities exceeding 99\% for the same benchmark states. While the detailed theoretical framework for quantum state tomography is well established, the practical procedures for extracting density matrices from experimental spectra are often inadequately documented in the literature and obscured within proprietary software. This package addresses that gap by providing a fully transparent, reproducible implementation of every analysis step, from spectral preprocessing to density matrix visualization. The software has been validated on experimentally prepared two-qubit states measured via nuclear magnetic resonance (NMR) spectroscopy of coupled 31P nuclei. Average reconstruction fidelities range from 0.975 to 0.995 across a benchmark set of 20 two-qubit states, including the computational basis states, Bell states, and the outputs of three fundamental quantum gates (CNOT, H, and T). Although developed for NMR, the modular architecture facilitates adaptation to other spectroscopic platforms and alternative measurement protocols.
The impressive pace of advance of quantum technology calls for robust and scalable techniques for the characterization and validation of quantum hardware. Quantum process tomography, the reconstruction of an unknown quantum channel from measurement data, remains the quintessential primitive to completely characterize quantum devices. However, due to the exponential scaling of the required data and classical post-processing, its range of applicability is typically restricted to one- and two-qubit gates. Here, we present a technique for performing quantum process tomography that addresses these issues by combining a tensor network representation of the channel with a data-driven optimization inspired by unsupervised machine learning. We demonstrate our technique through synthetically generated data for ideal one- and two-dimensional random quantum circuits of up to 10 qubits, and a noisy 5-qubit circuit, reaching process fidelities above 0.99 using several orders of magnitude fewer (single-qubit) measurement shots than traditional tomographic techniques. Our results go far beyond state-of-the-art, providing a practical and timely tool for benchmarking quantum circuits in current and near-term quantum computers.