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arXiv:2410.19610 · Physical Review X

Complexity of Gottesman-Kitaev-Preskill States

Lukas Brenner, Libor Caha, Xavier Coiteux-Roy, Robert König

Published Aug 19, 2025• 2 citations
arXivHTMLPDF

Abstract

We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations and single- and two-qubit operations as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an approximation to the state in trace distance. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state | G K P κ , Δ ⟩ . Here, κ − 2 is the variance of the envelope, and Δ 2 is the variance of the individual peaks. We show that the circuit accepts with constant probability and—conditioned on acceptance—the output state is polynomially close in ( κ , Δ ) to the state | G K P κ , Δ ⟩ . The size of our circuit is linear in ( log 1 / κ , log 1 / Δ ) . To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states.

DOI
10.1103/4ww5-4yww

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•Apr 16, 2021•3 citations
We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations and single- and two-qubit operations as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an approximation to the state in trace distance. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state | G K P κ , Δ ⟩ . Here, κ − 2 is the variance of the envelope, and Δ 2 is the variance of the individual peaks. We show that the circuit accepts with constant probability and—conditioned on acceptance—the output state is polynomially close in ( κ , Δ ) to the state | G K P κ , Δ ⟩ . The size of our circuit is linear in ( log 1 / κ , log 1 / Δ ) . To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states.
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.
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode is a promising bosonic code for quantum computation due to its tolerance for noise and all-Gaussian gate set. We present a toolkit for phase-space description and manipulation of GKP encodings that includes Wigner functions for ideal and approximate GKP states, for various types of mixed GKP states, and for GKP-encoded operators. One advantage of a phase-space approach is that Gaussian unitaries, required for computation with GKP codes, correspond to simple transformations on the arguments of Wigner functions. We use this fact and our toolkit to describe GKP error correction, including magic-state preparation, entirely in phase space using operations on Wigner functions. While our focus here is on the square-lattice GKP code, we provide a general framework for GKP codes defined on any lattice.
Bosonic modes constitute a central resource in a wide range of quantum technologies, providing long-lived degrees of freedom for the storage, processing, and transduction of quantum information. Such modes naturally arise in platforms including circuit quantum electrodynamics, quantum acoustodynamics, and trapped-ion systems. In these architectures, coherent control and high-fidelity readout of the bosonic degrees of freedom are achieved via coupling to an auxiliary qubit. When operated in the strong dispersive regime, this interaction enables parity measurements of the mode which, in combination with phase-space displacements, constitute a standard experimental tool for full Wigner-function tomography. Here, we propose a protocol based on displaced parity measurements that allows for the preparation of a variety of bosonic quantum states. As a first example, we demonstrate the generation of squeezed states, achieving up to ~9 dB of squeezing after only three parity measurements, and show that the protocol is robust against experimental imperfections. Finally, we generalize our approach to the preparation of other paradigmatic bosonic states, including cat and Gottesman-Kitaev-Preskill states.
Bosonic quantum error correction encodes a logical qubit in an oscillator, avoiding the hardware overhead of large qubit arrays. Among such encodings, Gottesman-Kitaev-Preskill (GKP) states are paticularly powerful because their phase-space grid structure protects against small displacement errors simultaneously in both conjugate quadratures. Here we provide the first protocol for preparing magnonic GKP states, which involves an ellipsoidal magnetic crystal effectively coupled to a superconducting qubit via a microwave cavity. The geometric anisotropy intrinsically squeezes the magnon mode, while the cavity-mediated qubit control realizes an effective conditional-displacement interaction. We show that two rounds of a conditional-displacement interaction and a qubit projective measurement yield three- and four-component magnonic GKP-like states. We also show how to realize single logical qubit gate operations, such as Pauli, Hadamard and phase gates, completing the logical Pauli basis of the approximate GKP code. Our results establish hybrid magnon-qubit systems as a promising platform for preparing bosonic code states, with applications in magnonic fault-tolerant quantum computation and quantum sensing.
Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behavior of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for noisy intermediate-scale quantum devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.
Gottesman-Kitaev-Preskill (GKP) states are a central resource for fault-tolerant optical continuous-variable quantum computing and communication. However, their realization in the optical domain remains to be demonstrated. Here we propose a method for preparing GKP states using a cavity QED system as the only non-Gaussian element. This system can be realized in several platforms such as trapped atoms, quantum dots or diamond color centers. We simulate the protocol with finite-cooperativity systems to evaluate the achievable quality of the produced GKP states in realistic noisy systems. We further combine the protocol with the previously proposed breeding protocol by Vasconcelos et al. to relax the demands on the quality of the QED system. We find that GKP states with more than 10 dB squeezing could be achieved in near-future experiments. Additionally, the approach can be made fully deterministic, thus providing a more scalable and hardware-efficient approach compared to probabilistic approaches based on photon counting.