Efficient characterization of general Gottesman-Kitaev-Preskill qubits

TL;DR

This paper introduces a practical method for characterizing GKP qubits using specific operators that require only three quadrature measurements, enabling efficient evaluation and optimization of GKP states.

Contribution

The authors develop a family of operators that efficiently characterize GKP qubits and serve as witnesses of non-Gaussianity with minimal measurement requirements.

Findings

Operators have unique zero-eigenvalue ground states matching ideal GKP states.

Expectation values serve as non-Gaussianity witnesses and relate to logical infidelity.

Method requires only three quadrature measurements, facilitating practical state evaluation.

Abstract

Practical utilization of Gottesman-Kitaev-Preskill (GKP) qubits requires not only the preparation of logical basis states, but also the ability to prepare and evaluate arbitrary logical qubit superpositions. Currently, this is typically done via quantum state tomography, which is resource-intensive. We introduce a family of positive semidefinite Hermitian operators, one for each point on the logical Bloch sphere, whose unique zero-eigenvalue ground states are the corresponding ideal GKP qubit states. We show that the expectation value of each operator serves as a witness of non-Gaussianity, and corresponds to twice the logical infidelity for states in the ideal logical GKP subspace. Furthermore, the truncated finite-dimensional counterparts of these operators yield physical approximations of arbitrary logical GKP states as their ground states. The evaluation of the proposed operators requires only three quadrature measurements, making this framework practical for both the experimental characterization and numerical optimization of GKP state preparation circuits.