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  • Open Access

Complexity of Gottesman-Kitaev-Preskill States

Lukas Brenner1,2,*, Libor Caha1,2,†, Xavier Coiteux-Roy3,4,‡, and Robert Koenig1,2,§

  • *Contact author: lukas.brenner@tum.de
  • †Contact author: cahalibor@me.com
  • ‡Contact author: xavier.coiteuxroy@ucalgary.ca
  • §Contact author: robert.koenig@tum.de
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Phys. Rev. X 15, 031073 – Published 19 September, 2025

DOI: https://doi.org/10.1103/4ww5-4yww

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 |GKPκ,Δ⟩. Here, κ−2 is the variance of the envelope, and Δ2 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 |GKPκ,Δ⟩. The size of our circuit is linear in (log1/κ,log1/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.

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Popular Summary

Quantum information is highly fragile, and protecting it against noise is one of the central challenges in building practical quantum computers. A promising tool for this is the Gottesman-Kitaev-Preskill (GKP) state, a special kind of continuous-variable quantum state that is resilient to small errors in phase space. However, creating high-quality GKP states is notoriously difficult. They cannot be produced with simple optical operations, and the difficulty of preparing them grows rapidly with the degree of error protection one wants. In this work, we introduce a new and efficient protocol for preparing approximate GKP states using experimentally accessible interactions between quantum bits and oscillators.

With just two oscillators and a single qubit, we can prepare GKP states in a time that scales only logarithmically with the level of error protection, quantified by the squeezing parameter of the state. We also prove that this scaling is the best possible, meaning that no more efficient general method exists. Importantly, unlike previous approaches, our protocol comes with rigorous accuracy guarantees in the “trace norm” sense, which provides a strong mathematical measure of their reliability.

The fact that our states come with formal accuracy guarantees means they can safely replace ideal GKP states in any larger quantum information protocol, opening the door to modular designs for quantum error correction and computation. Looking forward, adapting our protocol to realistic experimental noise will be an important step toward creating robust and scalable quantum technologies based on bosonic error correction.

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Phys. Rev. X 15, 031073– Published 19 September, 2025
  • Received 11 April 2025
  • Accepted 19 August 2025
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