Study of the Gottesman-Kitaev-Preskill code and applications in state preparation and measurement
Creators: name: Mensen, Lucas
Publisher: name: RMIT University
Year: 2024
Subjects: classificationCode: 510803
subject: Quantum information, computation and communication

description: In this work, we consider a specific bosonic code named after the original authors: Gottesman, Kitaev and Preskill. The GKP code is a means of encoding a finite-dimensional quantum system within a single bosonic mode, such that in principle, universal quantum computation can be achieved via Gaussian operations and measurements alone. This property makes the GKP code highly compatible with typical operations and measurements found in quantum optics.

Generic GKP states are highly non-Gaussian and currently difficult to experimentally prepare with high fidelity. With respect to bosonic quadrature measurements, the complex amplitudes of a typical GKP state are, quite paradoxically, both highly localized and widely distributed. Although locally sharp features of GKP states appear to “cheat” the canonical uncertainty of a single bosonic mode, they lose sharpness with respect to a set of larger displacement scales in an exact proportion that ultimately respects the uncertainty principle. It is this property that makes GKP states of interest not only in highly-scalable quantum computing applications, but also, as a resource within quantum-enhanced metrological schemes.

This work will be divided into two distinct sections. First, we study the GKP encoding from the perspective of a quantum-mechanical phase space. We chose to study the GKP code from the phase space perspective because the exclusively Gaussian operations and measurements of the GKP code are naturally compatible with a phase space representation. Although the typical Gaussian operations and measurements are simple in phase space, the remaining piece to complete the picture is representing the states themselves. Compared to operations and measurements, representing the intricate features of
GKP encoded states in phase space is far more complicated.

To resolve this, we consider GKP encoded states in the language of a particular class of quasiperiodic functions, known as θ-functions, and construct a basis of completely periodic functions via a well-defined asymptotic limit. As such, we simplify the remaining piece of the phase space picture, such that GKP computation is now described as symplectic transformations on a linear basis of these functions.

Following this, we consider the phase space description of GKP states and computation in a practical context, namely, GKP states that are mixed under decoherence, and the mechanisms of GKP error correction. In supplement to this, we provide a phase space envelope condition which quantitatively relates the precise fine scale structure of the encoded quasiprobability to the broader distribution of this structure over the larger phase space. This condition allows us to explicitly identify cases where a mixture of states is unphysical. As such, one may begin from a impure description of states where the associated pure states satisfy this condition for physicality.

We describe the process in which GKP error correction transforms the phase space of an input bosonic mode to produce GKP states with less embedded error, up to a probabilisitically incurred displacement error which corresponds to a GKP logical Pauli error. When applied to a noisy GKP input state, this sharpens the localized spikes in phase space quasiprobability, and any incurred encoded logical errors may be corrected within a larger, concatenated error correcting code. Additionally, by considering this process acting on the vacuum state, we also demonstrate in phase space how GKP error correction can be used to produce GKP-encoded non-Pauli eigenstates, which are the essential resource required to enable universal quantum computation.

In the second half of this work, we study the application of GKP encoded states as resources for enhanced parameter estimation, namely as single-mode displacement sensors. More specifically, we considered the performance of a range of finite-energy GKP encoded states as a sensing resource within irregular or restricted parametric statistical models.

By tuning the small and large scale features of a finite-energy GKP state, along with proportionate restrictions on the set of single-mode displacements, we can identify sensing schemes that extend beyond estimation of small displacements with a highly localized spike in the quadrature statistics of a single mode. Lower quality GKP states with an intolerable amount of embedded error for the purposes of quantum computing may still be useful in sensing displacements within a single mode. Moreover, we find a symmetric squeezing threshold for which a GKP state has multiquadrature sensing that is superior to a Gaussian state at all length scales. Beyond this threshold, we identify range of squeezing levels for which an infinimum and supremum may be set on the small and large scales respectively, such that precise single-mode sensing remains viable.

Lastly, we consider the performance of asymmetrically squeezed GKP states in unbiased estimation of restricted parameters. Such states are highly phase-sensitive displacement sensors, with single quadrature statistics that are highly localized on both small and large scales.