Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2203.11182 · Quantum

Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits

Cameron Calcluth, Alessandro Ferraro, Giulia Ferrini

Published Dec 1, 2022• 18 citations
arXivHTMLPDF

Abstract

We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems such as the Gottesman-Knill theorem can be employed to assess the simulatability. We first develop a method to evaluate the probability density function corresponding to measuring a single GKP state in the position basis following arbitrary squeezing and a large set of rotations. This method involves evaluating a transformed Jacobi theta function using techniques from analytic number theory. We then use this result to identify two large classes of multimode circuits which are classically efficiently simulatable and are not contained by the GKP encoded Clifford group. Our results extend the set of circuits previously known to be classically efficiently simulatable.

DOI
10.22331/q-2022-12-01-867

Related papers

Classical Simulation of Circuits with Realistic Odd-Dimensional Gottesman-Kitaev-Preskill States

Cameron Calcluth, Oliver Hahn, Juan Bermejo-Vega +2 more

Classically simulating circuits with bosonic codes is challenging due to the prohibitive cost of simulating quantum systems with many, possibly infinite, energy levels. We propose an algorithm to simulate circuits with encoded Gottesman-Kitaev-Preskill (GKP) states, specifically for odd-dimensional encoded qudits. Our approach is tailored to be especially effective in the most challenging but practically relevant regime, where the codeword states exhibit high (but finite) squeezing. Our algorithm leverages the Zak-Gross Wigner function introduced by Davis, Fabre, and Chabaud, which represents infinitely squeezed encoded stabilizer states positively. The run-time of the algorithm scales with the negativity of the Wigner function, allowing for efficient simulation of certain large-scale circuits-namely, input stabilizer GKP states undergoing generalized GKP-encoded Clifford operations followed by modular measurement-with a high degree of squeezing. For stabilizer GKP states exhibiting 12 dB of squeezing, our algorithm can simulate circuits with up to 1000 modes with less than double the number of samples required for a single input mode, which is in stark contrast to existing simulators. Therefore, this approach holds significant potential for benchmarking early implementations of quantum computing architectures utilizing bosonic codes.

arXiv:2412.13136•Physical Review Letters•May 14, 2025•2 citations

Realistic Gottesman-Kitaev-Preskill stabilizer states enable universal quantum computation

Fariba Hosseinynejad, Pavithran Iyer, Guillaume Dauphinais +1 more

Physical Gottesman-Kitaev-Preskill (GKP) states are inherently noisy as ideal ones would require infinite energy. While this is typically considered as a deficiency to be actively corrected, this work demonstrates that imperfect GKP stabilizer states can be leveraged in order to apply non-Clifford gates using only linear optical elements. In particular, Gaussian operations on normalizable GKP states, combined with homodyne measurements, permit two key primitives: clean projection onto Pauli eigenstates in the normalizable GKP codespace, thereby implementing Clifford gates with high fidelity; and probabilistic projection of unmeasured modes onto non-Pauli eigenstates. These results demonstrate that normalizable GKP stabilizer states combined with Gaussian operations provide a practical framework for computational universality within the measurement-based model of quantum computation in a realistic continuous-variable setting.

quant-pharXiv:2511.03874•Physical Review Letters•Nov 7, 2025•1 citation

Exact simulation of realistic Gottesman-Kitaev-Preskill cluster states

Milica Banić, Valerio Crescimanna, J. Eli Bourassa +3 more

We describe a method for simulating and characterizing realistic Gottesman-Kitaev-Preskill (GKP) cluster states, rooted in the representation of resource states in terms of sums of Gaussian distributions in phase space. We apply our method to study the generation of single-mode GKP states via cat state breeding, and the formation of multimode GKP cluster states via linear optical circuits and homodyne measurements. We characterize resource states by referring to expectation values of their stabilizers and witness operators constructed from them. Our method reproduces the results of standard Fock-basis simulations, while being more efficient and being applicable in a broader parameter space. We also comment on the validity of the heuristic Gaussian random noise model, through comparisons with our exact simulations: We find discrepancies in the stabilizer expectation values when homodyne measurement is involved in cluster state preparation, yet we find close agreement between the two approaches on average.

arXiv:2504.10606•Physical Review A•Nov 13, 2025•3 citations

Efficient characterization of general Gottesman-Kitaev-Preskill qubits

Vojtěch Kuchař, Petr Marek

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.

quant-pharXiv:2604.17303•Apr 21, 2026•0 citations

Vacuum provides quantum advantage to otherwise simulatable architectures

Cameron Calcluth, Alessandro Ferraro, Giulia Ferrini

We consider a computational model composed of ideal Gottesman-Kitaev-Preskill stabilizer states, Gaussian operations including all rational symplectic operations and all real displacements, and homodyne measurement. We prove that such architecture is classically efficiently simulatable by explicitly providing an algorithm to calculate the probability density function of the measurement outcomes of the computation. We also provide a method to sample when the circuits contain conditional operations. This result is based on an extension of the celebrated Gottesman-Knill theorem, via introducing proper stabilizer operators for the code at hand. We conclude that the resource enabling quantum advantage in the universal computational model considered by Baragiola et al. [Phys. Rev. Lett. 123, 200502 (2019)], composed of a subset of the elements given above augmented with a provision of vacuum states, is indeed the vacuum state.

arXiv:2205.09781•Physical Review A•Jun 15, 2023•13 citations

Bridging Magic and Non-Gaussian Resources via Gottesman-Kitaev-Preskill Encoding

Oliver Hahn, Giulia Ferrini, Ryuji Takagi

Although the similarity between nonstabilizer states—also known as magic states—in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer Rényi entropy—a recent proposal for a magic measure for multiqubit states. We further provide the magic measure with an operational interpretation as a runtime of a classical simulation algorithm. In addition, we employ our results to prove that implementing a multiqubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.

arXiv:2406.06418•PRX Quantum•Feb 18, 2025•15 citations
We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems such as the Gottesman-Knill theorem can be employed to assess the simulatability. We first develop a method to evaluate the probability density function corresponding to measuring a single GKP state in the position basis following arbitrary squeezing and a large set of rotations. This method involves evaluating a transformed Jacobi theta function using techniques from analytic number theory. We then use this result to identify two large classes of multimode circuits which are classically efficiently simulatable and are not contained by the GKP encoded Clifford group. Our results extend the set of circuits previously known to be classically efficiently simulatable.
Classically simulating circuits with bosonic codes is challenging due to the prohibitive cost of simulating quantum systems with many, possibly infinite, energy levels. We propose an algorithm to simulate circuits with encoded Gottesman-Kitaev-Preskill (GKP) states, specifically for odd-dimensional encoded qudits. Our approach is tailored to be especially effective in the most challenging but practically relevant regime, where the codeword states exhibit high (but finite) squeezing. Our algorithm leverages the Zak-Gross Wigner function introduced by Davis, Fabre, and Chabaud, which represents infinitely squeezed encoded stabilizer states positively. The run-time of the algorithm scales with the negativity of the Wigner function, allowing for efficient simulation of certain large-scale circuits-namely, input stabilizer GKP states undergoing generalized GKP-encoded Clifford operations followed by modular measurement-with a high degree of squeezing. For stabilizer GKP states exhibiting 12 dB of squeezing, our algorithm can simulate circuits with up to 1000 modes with less than double the number of samples required for a single input mode, which is in stark contrast to existing simulators. Therefore, this approach holds significant potential for benchmarking early implementations of quantum computing architectures utilizing bosonic codes.
Physical Gottesman-Kitaev-Preskill (GKP) states are inherently noisy as ideal ones would require infinite energy. While this is typically considered as a deficiency to be actively corrected, this work demonstrates that imperfect GKP stabilizer states can be leveraged in order to apply non-Clifford gates using only linear optical elements. In particular, Gaussian operations on normalizable GKP states, combined with homodyne measurements, permit two key primitives: clean projection onto Pauli eigenstates in the normalizable GKP codespace, thereby implementing Clifford gates with high fidelity; and probabilistic projection of unmeasured modes onto non-Pauli eigenstates. These results demonstrate that normalizable GKP stabilizer states combined with Gaussian operations provide a practical framework for computational universality within the measurement-based model of quantum computation in a realistic continuous-variable setting.
We describe a method for simulating and characterizing realistic Gottesman-Kitaev-Preskill (GKP) cluster states, rooted in the representation of resource states in terms of sums of Gaussian distributions in phase space. We apply our method to study the generation of single-mode GKP states via cat state breeding, and the formation of multimode GKP cluster states via linear optical circuits and homodyne measurements. We characterize resource states by referring to expectation values of their stabilizers and witness operators constructed from them. Our method reproduces the results of standard Fock-basis simulations, while being more efficient and being applicable in a broader parameter space. We also comment on the validity of the heuristic Gaussian random noise model, through comparisons with our exact simulations: We find discrepancies in the stabilizer expectation values when homodyne measurement is involved in cluster state preparation, yet we find close agreement between the two approaches on average.
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.
We consider a computational model composed of ideal Gottesman-Kitaev-Preskill stabilizer states, Gaussian operations including all rational symplectic operations and all real displacements, and homodyne measurement. We prove that such architecture is classically efficiently simulatable by explicitly providing an algorithm to calculate the probability density function of the measurement outcomes of the computation. We also provide a method to sample when the circuits contain conditional operations. This result is based on an extension of the celebrated Gottesman-Knill theorem, via introducing proper stabilizer operators for the code at hand. We conclude that the resource enabling quantum advantage in the universal computational model considered by Baragiola et al. [Phys. Rev. Lett. 123, 200502 (2019)], composed of a subset of the elements given above augmented with a provision of vacuum states, is indeed the vacuum state.
Although the similarity between nonstabilizer states—also known as magic states—in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer Rényi entropy—a recent proposal for a magic measure for multiqubit states. We further provide the magic measure with an operational interpretation as a runtime of a classical simulation algorithm. In addition, we employ our results to prove that implementing a multiqubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.