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Propagating Gottesman-Kitaev-Preskill states encoded in an optical oscillator
Shunya Konno, Warit Asavanant, Fumiya Hanamura, Hironari Nagayoshi, Kosuke Fukui, Atsushi Sakaguchi, Ryuhoh Ide, Fumihiro China, Masahiro Yabuno, Shigehito Miki, Hirotaka Terai, Kan Takase, Mamoru Endo, Petr Marek, Radim Filip, Peter van Loock, Akira Furusawa
TL;DR
The paper develops criteria for identifying when measured GKP stabilizers exceed classical limits and applies them to reconstructed, Gaussian-corrected states. The analysis obtains a maximum fidelity of 0.551 ± 0.003 to an approximately 2.5 dB-squeezed GKP state and establishes classical-limit benchmarks for the criteria.
Problem
The paper addresses how to determine whether measured stabilizer values surpass the limits attainable by classical mixtures of coherent states.
Method
The authors reconstruct the density matrix from quadrature measurements, optimize a Gaussian correction against an approximated GKP state, and calculate stabilizers from the corrected state's Wigner function.
Results
0.551 ± 0.003 maximum fidelity is obtained between the Gaussian-corrected state and an approximated GKP state with about 2.5 dB squeezing.
Takeaways & Limitations
Stabilizer values above the derived classical maxima indicate that an experimentally generated state surpasses the classical limitation.
Takeaways & Limitations
The analysis ignores an extra phase factor from noncommuting displacements because it does not change the norm.
Abstract
from arXiv · showhide
A quantum computer with low-error, high-speed quantum operations and capability for interconnections is required for useful quantum computations. A logical qubit called Gottesman-Kitaev-Preskill (GKP) qubit in a single Bosonic harmonic oscillator is efficient for mitigating errors in a quantum computer. The particularly intriguing prospect of GKP qubits is that entangling gates as well as syndrome measurements for quantum error correction only require efficient, noise-robust linear operations. To date, however, GKP qubits have been only demonstrated at mechanical and microwave frequency in a highly nonlinear physical system. The physical platform that naturally provides the scalable linear toolbox is optics, including near-ideal loss-free beam splitters and near-unit efficiency homodyne detectors that allow to obtain the complete analog syndrome for optimized quantum error correction. Additional optical linear amplifiers and specifically designed GKP qubit states are then all that is needed for universal quantum computing. In this work, we realize a GKP state in propagating light at the telecommunication wavelength and demonstrate homodyne meausurements on the GKP states for the first time without any loss corrections. Our GKP states do not only show non-classicality and non-Gaussianity at room temperature and atmospheric pressure, but unlike the existing schemes with stationary qubits, they are realizable in a propagating wave system. This property permits large-scale quantum computation and interconnections, with strong compatibility to optical fibers and 5G telecommunication technology.
METHOD
The method uses optical parametric oscillators, homodyne detection, and photon-subtraction-based cat-state generation to produce and measure propagating GKP states. The section also documents the experimental operating parameters and prior work cited alongside the setup.
- METHOD: The optical system uses semi-monolithic OPOs, homodyne detectors, and SNSPDs for state generation and measurement.The OPOs use 10-mm PPKTP crystals; homodyne detectors have about 200 MHz bandwidth, and SNSPD efficiency is about 75%.
- METHOD: Cat-state interference succeeds at about 10 Hz, with homodyne conditioning applied within a ±0.3 quadrature range.The photon-subtraction count rate is about 80–90 kHz per OPO, while the conditioning range corresponds to about 30% for the stated parameters.
- METHOD: Quantum-state generation uses maximum-likelihood reconstruction from measurements at six quadrature phases, with about 20,000 conditioned data points per phase.The phases are 0°, 30°, 60°, 90°, 120°, and 150°.
ADDITIONAL INFORMATION
The paper provides correspondence information for contacting A.F. or W.A.
- ADDITIONAL INFORMATION: Correspondence is directed to A.F. or W.A.
COMPETING FINANCIAL INTERESTS
The supplementary setup information details the optical components, stabilization scheme, and detector timing used in the experiment. The authors also state that they have no competing financial interests.
- COMPETING FINANCIAL INTERESTS: The authors declare no competing financial interests.
- COMPETING FINANCIAL INTERESTS: The setup uses a 1545.32-nm continuous-wave master laser and a second-harmonic generator producing 772.66-nm pump light.The telecom-wavelength light is distributed to local oscillators, OPO alignment, phase references, and cavity locks.
- COMPETING FINANCIAL INTERESTS: The two OPOs contain 10-mm PPKTP crystals in semi-monolithic cavities, with an additional cavity used to match the pump-beam spatial mode.
- COMPETING FINANCIAL INTERESTS: The idler path uses filtering, fiber coupling, and SNSPD detection, resulting in about 50% total idler-path efficiency.
- COMPETING FINANCIAL INTERESTS: Beam-splitter visibility exceeds 96%, while sample-and-hold stabilization blocks reference lights during data acquisition and protects the SNSPDs.
- COMPETING FINANCIAL INTERESTS: The homodyne detectors have about 200 MHz bandwidth, and temporal matching within ±0.6 ns retains mode matching above 93%.The maximum measured mode matching is 94.6%.
II. DATA ANALYSIS
The analysis reconstructs the quantum state from conditioned quadrature measurements and derives Gaussian corrections and stabilizer values from the reconstructed density matrix and Wigner function.
- II. DATA ANALYSIS: HD1 fixes the conditioning basis to x, while HD2 measures six phases for quantum-tomographic state reconstruction.The phases are 0°, 30°, 60°, 90°, 120°, and 150°.
- II. DATA ANALYSIS: The density matrix is reconstructed in the Fock basis using maximum likelihood, and Gaussian corrections maximize fidelity to an approximated GKP state.The maximum fidelity is 0.551 ± 0.003 for an approximated GKP state with about 2.5 dB squeezing.
- II. DATA ANALYSIS: The Gaussian-corrected density matrix yields the Wigner function and stabilizer values, with reported error bars calculated by bootstrapping.
III. GAUSSIAN LIMIT CRITERIA FOR GKP STABILIZER
The paper defines criteria for distinguishing GKP stabilizer behavior from classical and Gaussian states, allowing effective squeezing to align peaks with the GKP grid. The Gaussian bound is obtained by optimizing over squeezing and rotation and coincides with the classical limit.
- Criteria definition: The criteria use phase-space displacement operators related to the GKP stabilizers, with g controlling effective squeezing to align state peaks with the grid.The displacement is by (x0, p0) in the Wigner-function phase space.
- Classical limits: For coherent states and their classical mixtures, the criteria equal the coherent-state values, defining the classical bounds C1,cl(g) and C2,cl(g).The classical limit is obtained by maximizing these functions over g.
- Classical limits: The maximum of C1,cl(g) occurs at 2g^2 = 1, establishing the classical threshold for the first criterion.This value corresponds to the optimum effective squeezing parameter for the classical state.
- Classical limits: The maximum of C2,cl(g) occurs as g → 0 or g → ∞, corresponding to infinitely strong squeezing or antisqueezing.Exceeding this maximum in experiment indicates that the generated state surpasses the classical limitation.
- Gaussian limit: For arbitrary Gaussian states, optimizing over g, rotation θ, and squeezing r gives a maximum at r = 0, equal to the classical limit.Thus squeezing and rotation do not raise the Gaussian bound above the classical criterion.
- Gaussian limit: The plotted criteria compare experimentally calculated values with classical and Gaussian limits, using a Wigner function corrected for Gaussian operations.Blue curves represent experimental criteria, orange curves classical limits, and yellow curves Gaussian limits optimized over θ and r.