Here we provide experimental details on each major component in this experiment: the laser subsystem, photonic integrated chip, PNR detectors for state heralding and the homodyne detection system used for state tomography. Further details can be found in the Supplementary Information. Laser system The laser subsystem is composed of five lasers: two lasers (P1 and P2) for driving the dual-pump SFWM process that generates squeezing, a local oscillator laser used to perform homodyne detection and also as a frequency reference, a reference laser used to stabilize the optical phase of the quantum state and a probe laser to lock the resonance frequency of the on-chip resonators. The phase and frequency of each of these lasers are stabilized by comparing it to the local oscillator, which thus serves as a frequency and/or phase reference for the entire experiment. A portion of the local oscillator is deeply phase modulated using fibre optic phase modulators to produce an electro-optic frequency comb that spans more than 1.2 THz. Each laser is locked to one tooth of the frequency comb using an optical phase-locked loop. Using amplitude modulators, 0.5-ns pump pulses are prepared from P1 and P2, 25-ns local oscillator pulses are prepared from the local oscillator and interleaved pulses are prepared from the reference laser (20 ns) and the probe laser (100 ns). Pulses are prepared with 200 kHz repetition rate. The P1, P2 and local oscillator pulses are amplified using erbium-doped fibre amplifiers: P1 and P2 pulses are amplified to deliver 300–400 pJ to each of the four on-chip squeezers; the local oscillator is amplified to optimize homodyne detector shot noise clearance. The pulsed light is coherently distributed such that pulsed P1, P2 and a portion of the pulsed reference laser proceed to the chip, while the pulsed local oscillator and the remaining pulsed reference laser proceed to the homodyne detector. The phase of the interferometer formed by these two paths is stabilized using a fibre phase modulator by detecting the reference laser’s interference at the homodyne detector. The phase noise in this interferometer is measured independently by leaking a small portion of the local oscillator through the chip, and is measured to be 1.7° standard deviation. When combined with 1.2° standard deviation phase noise from the initial frequency and/or phase lock between P1, P2 and the local oscillator, the overall phase noise is 2.1° standard deviation. Photonic integrated circuit The chip was fabricated using a custom silicon nitride process, optimized for low linear loss. All samples were fabricated on 300-mm wafers using a manufacturing process that is compatible with state-of-the-art, high-volume semiconductor manufacturing (see Supplementary Information for details). The fabrication process steps were optimized to mitigate the impact of absorption and scattering losses, resulting in ultra-low propagation losses in single-mode waveguides. Reference structures are used to calibrate the loss contributions of each relevant component. Directional couplers in the filters and interferometer have roughly 4 mdB of loss, and the loss from edge coupler to SMF-28-Ultra fibre for the packaged devices is consistently less than 0.5 dB. The squeezers are based on a photonic molecule design29 and are composed of a primary resonator coupled to an auxiliary resonator with a larger free spectral range (FSR). The ratio of primary FSR to auxiliary FSR is chosen to allow the resonances that would contribute to unwanted parametric nonlinear processes to be spectrally split or displaced, thereby suppressing these parasitic effects. The average loaded and intrinsic quality factors of the squeezer resonators on the chip are measured to be 3.33 × 105 and 1.28 × 107, respectively, corresponding to a 580-MHz full-width at half-maximum resonance bandwidth and 97.4% escape efficiency for the signal resonance. The resonances of each resonator can be independently shifted using two integrated thermo-optic phase shifters. The squeezer design is highly engineered to combine high escape efficiency, strong resonant enhancement, suppression of spurious processes and support pulsed single-temporal-mode operation. The input filter, pump filter and post-interferometer filters are all based on an asymmetric MZI design. The programmable interferometer is composed of a cascaded ‘staircase’ of tuneable couplers, with the last coupler being fixed at a 50/50 splitting ratio (see Supplementary Information for a corresponding circuit diagram). Although not universal, this arrangement has been shown to be capable of generating optical GKP qubits using the fewest optical elements (to reduce loss)12. The first two couplers in the interferometer staircase are tunable to allow different state preparation depending on the amount of available squeezing and desired state. The chip itself is fully electro-optically packaged. The input and output waveguides are coupled to fibre array units of medium mode field diameter fibre (6.4 μm MFD, Corning HI 1060 Flex), which are then spliced to Corning SMF-28 Ultra fibre. The combined coupling efficiency from the chip waveguide to SMF-28 Ultra fibre is measured to be 0.45 dB (90%). The chip is wirebonded to a carrier printed circuit board that interfaces with the electronics required to drive the thermo-optic phase shifters and program the chip. PNR detectors The PNR detectors are based on a cryogenic transition edge sensor design operated in a dilution fridge at 14 mK, but can be operated at temperatures up to 50 mK (ref. 37). Advancements in device fabrication, simulation, metrology and packaging have resulted in significant improvements in detection efficiency compared to results reported in the literature. The implementation of in situ spectroscopic ellipsometry measurements and new rigorous finite-difference time-domain simulations have enabled precise optimization of stack layer thicknesses during fabrication. A multifaceted metrology approach, incorporating transmission electron microscopy along with complementary techniques, has been implemented to provide comprehensive verification of the fabricated stack. Furthermore, optical packaging of the PNR detectors has been refined with improved concentricity, coupled with either a roughly 70 or 96% larger detection area compared to previously published work12, leading to more misalignment-tolerant light coupling. The three detectors involved in the heralding operation have a measured detection efficiency of \(99.8{9}_{-0.53}^{+0.11}\), 98.40 ± 1.19 and 96.45 ± 1.04%, where uncertainties represent a 95% confidence interval (k = 2). The primary contributor to these uncertainties is the absolute calibration of the optical power meter, which has an uncertainty of 0.42% (k = 2). Many more detectors were yielded with measured detection efficiencies above 99%; the two sensors with slightly lower detection efficiency in this experiment were chosen as they had superior electrical noise performance. This characteristic is related to electronic packaging yield and is not correlated with detection efficiency. Finally, the transmissions of each wavelength division multiplexing filter before each PNR detector are measured to be 93.7, 94.1 and 94.8%. Homodyne detection system The homodyne detector used for state tomography is composed of a pair of high quantum efficiency photodiodes in a custom trans-impedance amplifier circuit. The quadratures of the electro-magnetic field are measured by interfering the quantum state with a strong local oscillator field on a balanced beamsplitter, detecting the two outputs with the two photodiodes and measuring the photocurrent difference. The temporal mode of the quadrature measurement is defined by the local oscillator field. Using an IQ modulator, the local oscillator field is shaped from the initial 25-ns local oscillator pulse to match the temporal mode profile of the squeezing (measured by mode tomography: Supplementary Information). The measured quadrature is defined by the relative phase between the quantum state and local oscillator. For state tomography, the local oscillator phase is varied over 32 different phase settings between 0 and π. At each phase setting, measured quadrature values are recorded for different PNR heralding events. Up to 2 × 106 quadrature measurements are recorded for each heralding event. From the quadrature measurements the density matrix of the quantum state is reconstructed using maximum-likelihood techniques38. The state reconstruction is done without any loss compensation, and the resulting states presented in Fig. 3 include end-to-end loss from squeezing generation to homodyne detection. We estimate the total homodyne detection efficiency to be 97%. This includes photodiode quantum efficiency (less than 99%), 21.3-dB electronic noise clearance (99.2%), mode overlap between shaped local oscillator pulse and quantum state (more than 99%) and polarization visibility (more than 99%). Figures of merit The defining feature of ideal GKP Pauli eigenstates is that they are eigenstates of a pair of displacements that form a parallelogram with a phase-space area 2π. For example, a rectangular lattice GKP qubit \(| 1\rangle \) is stabilized by \({\widehat{S}}_{p}={{\rm{e}}}^{i\alpha \widehat{p}}\) and \({\widehat{S}}_{q}={{\rm{e}}}^{2{\rm{\pi }}i\widehat{q}/\alpha }\), which correspond to α and 2π/α shifts in phase space along the q and p quadratures1. For any approximate GKP state, the absolute value of the stabilizer expectation values will lie between 0 and 1, with 1 only attained by ideal GKP states. The stabilizer expectation values can be related to the effective squeezing of the peaks of the GKP state, with \({\varDelta }_{p(q)}^{2}=-\,\text{ln}(| \langle {\widehat{S}}_{p(q)}\rangle {| }^{2})/{\rm{\pi }}\) (ref. 22). This formula can be understood as the per-peak squeezing of an approximate GKP state were it to be transformed to the lattice defined by \(\alpha =\sqrt{2{\rm{\pi }}}\) by a single-mode Gaussian unitary that implements a symplectic transformation on the phase-space lattice. This allows us to compare the quality of GKP states on different lattices39. Both finite energy effects and sources of decoherence, such as photon loss, contribute to lowering the effective squeezing. We can also consider the symmetric effective squeezing, which is defined as \({\varDelta }_{{\rm{sym}}}^{2}=({\varDelta }_{p}^{2}+{\varDelta }_{q}^{2})/2\), the average variance of the peaks in both quadratures12,39. Effective squeezing can be expressed in dB units through \(-10{\log }_{10}{\varDelta }^{2}\).